Computability Theory and Algebra

Wu Huishan

School of Physical and Mathematical Sciences

A thesis submitted to the Nanyang Technological University in partial fulﬁlment of the requirement for the degree of Doctor of Philosophy

2017

Acknowledgments

I would like to thank my supervisor, Prof. Wu Guohua, for teaching me computability theory and guiding my research in the past few years. His knowledge, suggestions, and discussions inspire me a lot, and I appreciate him for sharing me with his ideas. His style of teaching, writing of papers and presentations influence me a lot. I am grateful to him for all his efforts for guiding me to be a teacher and to be a researcher. I also want to thank him for supporting my PhD study by his research fund from MOE of Singapore. I would like to thank Prof. Ng Keng Meng, for teaching me a course on first- order logic. I also want to thank my peers and friends, Wang Shaoyi, Ru Junren, Yu Hongyuan, Yuan Bowen for many inspiring discussion and various seminars on logic and computability theory. Last, but not the least, I want to thank my parents for their love and support.

i ii Contents

Acknowledgments ...... i Abstract ...... vi

1 Introduction 1 1.1 Basics of computability theory ...... 1 1.2 Part I: Bounded-jump operator and the high/low hierarchy ...... 8 1.3 Part II: Degrees of orders on torsion-free abelian groups ...... 11 1.4 Part III: Reverse mathematics, abelian groups and modules ...... 17

I Bounded-jump operator and the high/low hierarchy 26

2 A bounded-low c.e. set which is low, but not superlow 27 2.1 Requirements and strategies ...... 28 2.2 Construction ...... 32 2.3 Veriﬁcation ...... 33 2.4 A bounded-low set with Turing degree 0′ ...... 35

3 A bounded-high set which is high, but not superhigh 37 3.1 Pseudo-jump inversion via bounded-high sets ...... 38

3.1.1 A Re-strategy ...... 39

3.1.2 A Pe-strategy ...... 42

3.1.3 A Qe-strategy ...... 43

3.1.4 Deﬁne the computable function gr ...... 48 3.1.5 Construction ...... 52 3.1.6 Veriﬁcation ...... 57 3.2 Pseudo-jump inversion via bounded-low c.e. sets ...... 60

iii II Degrees of orders on torsion-free abelian groups 63

4 Group-order-computable degrees and PA degrees 64 4.1 Requirements ...... 65 4.2 Background ...... 66 4.3 The P-strategy ...... 67 4.4 The Q-strategy ...... 70 4.5 The R-strategy ...... 73 4.5.1 Background on PA-complete sets ...... 73 4.5.2 The R′-strategy ...... 75 4.6 Construction and veriﬁcation ...... 82 4.6.1 Construction ...... 82 4.6.2 G is a computable group possessing computable orders . . . . 82 4.6.3 A-computable orders on G which are not computable . . . . . 84 4.6.4 Orders on G which are computable from PA-complete sets . . 90

5 Group orders and c.e. degrees 91 5.1 Requirements and strategies ...... 91

5.1.1 A Pe-strategy with permitting ...... 92

5.1.2 The Q0-strategy ...... 93

5.1.3 The Q1-strategy ...... 97 5.1.4 The Q-strategy ...... 99

5.1.5 A Ne-strategy ...... 100 5.2 Construction ...... 107 5.3 Veriﬁcation ...... 109

III Reverse mathematics, abelian groups and modules 113

6 Reverse mathematics and divisible abelian groups 114 6.1 Divisible subgroups ...... 114 6.2 Two decomposition theorems ...... 118

iv 7 Reverse mathematics and modules 124 7.1 Projective modules ...... 124 7.1.1 Direct sums of projective modules ...... 124 0 7.1.2 Free modules and projective modules over Σ1-PIDs ...... 127 7.2 Injective modules ...... 129 7.2.1 Baer’s criterion for injective modules ...... 129 0 7.2.2 Divisible modules and injective modules over Σ1-PIDs . . . . . 132

References 135

v Abstract

This thesis mainly focuses on classical computability theory and effective aspects of algebra. In particular, we will work on bounded low/high sets, degrees of orders on torsion-free abelian groups, and reverse mathematics of several classic results in modules. In Part I, we will study bounded-low sets and bounded-high sets in terms of the high/low hierarchy. Anderson and Csima proposed in [1] the notion of bounded- jump on wtt-degrees, and proved an analogue of Shoenfield’s jump inversion theorem. In [2], Anderson, Csima and Lange compared the bounded jump with Turing jump and proved the existence of high bounded-low sets and low bounded-high sets. Observing that superlow sets are all bounded-low, Anderson, Csima and Lange asked in [2] whether there exist bounded-low c.e. sets which are low but not superlow, and whether there exist superhigh sets which are not bounded-high. In Part I, we will provide positive answers to these questions. We will also prove that there are bounded-high sets which are high but not superhigh, and that there are bounded-low c.e. sets which are high but not superhigh. In Part II, we study Turing degrees of orders on computable torsion-free abelian groups with infinite rank. Recently, Kach, Lange and Solomon [3] built a noncomputable c.e. set C and a computable torsion-free abelian group G with computable orders such that every C-computable order on the constructed group is computable. Motivated by this result, we call a degree a group-order-computable if there is a computable torsion-free abelian group G with infinite rank which admits computable orders such that every a-computable order on G is computable; in this case, say a is group-order-computable via G . Following this definition, we call a degree a weakly group-order-computable if there is a computable torsion-free abelian group G with

vi infinite rank such that every a-computable group order on G is computable. We will prove that a Turing degree a is group-order-computable iff a is weakly group- order-computable iff a is not a PA degree. In particular, all c.e. degrees except 0′ are group-order-computable. The objective to study group-order-computable degrees is indeed to explore degrees of orders on computable torsion-free abelian groups, because for a nonzero degree a, if it is group-order-computable via G , then G has no orders of degree a. We show that for any nonzero c.e. degree a, there is a computable torsion- free abelian group G and a nonzero c.e. degree c < a such that G has orders of degree ≥ a and c is group-order-computable via G , which means that G has no incomputable orders of degree ≤ c. In Part III, we study the reverse mathematics of classic results of divisible abelian groups and modules. We will prove that over RCA0, the decomposition theorem of divisible abelian groups and the decomposition theorem of torsion abelian groups are both equivalent to ACA0. We then consider projective modules and injective 0 0 modules over Σ1-principal ideal domains (Σ1-PIDs). We will also show that ACA0 proves the following theorems:

• 0 every projective module over a Σ1-PID is free;

• 0 every submodule of a projective module over a Σ1-PID is projective;

• 0 every divisible module over a Σ1-PID is injective;

• 0 every quotient of an injective module over a Σ1-PID is injective.

vii viii Chapter 1

Introduction

1.1 Basics of computability theory

We ﬁrst give a brief introduction of basic concepts and theorems in computability theory. Say that a function f with domain X ⊆ N is partial computable if there is a Turing program P such that for x ∈ X, P running on input x stops after ﬁnitely many steps, with output f(x); and for x ̸∈ X, P running on input x never stops. A subset of natural numbers is computably enumerable (c.e. for short) if it is the domain of a partial computable function. A subset A of natural numbers is computable if both A and its complement are computably enumerable.

Turing programs can be effectively listed, i.e. via their Godel¨ numbers, as {Pe : e ∈ N}. Let φe be the partial computable function computed by the e-th Turing program Pe, and We be the domain of φe. Then {φe : e ∈ N} and {We : e ∈ N} are effective enumerations of partial computable functions and c.e. sets respectively. { k ∈ N} Similarly, we have an effective listing φe : e of all k-ary partial computable functions with k ≥ 2.

Lemma 1.1 (Padding Lemma) For each partial computable function φe, we can ′ eﬀectively ﬁnd an index e > e such that φe′ = φe.

Theorem 1.1 (Parameter Theorem) For a partial computable binary function, say g, there is an injective computable function s such that for all x, y, φs(x)(y) = g(x, y).

Theorem 1.2 (Kleene’s Recursion Theorem) For any total computable function N → N f : , there is an e0 such that We0 = Wf(e0).

1 Chapter 1. Introduction

Theorem 1.3 (Recursion Theorem with parameters, Kleene) For any total computable function f : N2 → N, there is a computable function g : N → N such that for any n, Wg(n) = Wf(n,g(n)).

Similar to Turing programs, all Turing programs with oracles can be eﬀectively listed, and we have a standard enumeration {Φe : e ∈ N} of partial computable X ↓ Turing functionals. For any set X and number x, we write Φe (x) = y if y is the output of the program Φe with oracle X and input x. The use of this computation X X Φe (x), denoted by ϕe (x), is deﬁned as the greatest number z such that the question z ∈ X or not is queried.

Theorem 1.4 (Parameter Theorem for functionals) For any partial computable binary functional, say Θ, there is an injective computable function s such that for any Y ≃ Y oracle Y , and for all x, y, Φs(x)(y) Θ (x, y).

Theorem 1.5 (Recursion Theorem for functionals) For any total computable function f : N2 → N, there is a computable function g : N → N such that for any oracle Y Y Y and number n, Φg(n) = Φf(n,g(n)).

Say that a set A is Turing reducible to B, written as A ≤T B, if there is a Turing program with oracle B that outputs membership of A, and that A is Turing equivalent to B, written as A ≡T B, if A ≤T B and B ≤T A. ≡T is an equivalence relation over subsets of N, and we call the equivalence class of A the Turing degree of A, denoted by deg(A). That is, deg(A) = {B ⊆ N : B ≡T A}. We also use a to denote the Turing degree of A. ≤T gives rise to a natural order ≤ among Turing degrees, i.e., a ≤ b if there are sets A ∈ a and B ∈ b such that A ≤T B. 0 denotes the Turing degree of computable sets. If a degree a contains c.e. sets, then we call this degree a c.e. degree. In computability theory, we also consider strong reducibilities, like one-one reducibility (≤1), many-one reducibility (≤m), truth-table reducibility (≤tt) and weak- truth-table reducibility (≤wtt). Under thse strong reducibilities, we can deﬁne corresponding degrees:

2 Chapter 1. Introduction

• many-one degree (m-degree, for short) of A is deﬁned as {X ⊆ N : X ≡m

A}. Here, X is many-one reducible to Y , denoted by X ≤m Y , if there is a computable function f such that for all x, x ∈ X iﬀ f(x) ∈ Y . Moreover, if f

is one-to-one, X is said to be one-one reducible to Y , denoted by X ≤1 Y .

• truth-table degree (tt-degree, for short) of A is deﬁned as {X ⊆ N : X ≡tt A}.

Let {σn : n ∈ N} be an eﬀective list of all truth tables, i.e., propositional formals obtained from atomic formals n ∈ X with n ∈ N. X is truth-table

reducible to Y , denoted by X ≤tt Y , if there is a computable function f such

that for all x, x ∈ X ⇔ Y σf(x).

• weak-truth-table degree (wtt-degree, for short) of A is deﬁned as {X ⊆ N :

X ≡wtt A}. Here, X is weak-truth-table reducible to Y , denoted by X ≤wtt Y , Y Y if there is an oracle Turing program, say Φe, such that X = Φe with use ϕe recursively bounded (i.e., there is a computable function f such that f(x) ≥ Y ϕe (x) for all x). Weak-truth-table reducibility is also called bounded-Turing reducibility.

Turing’s halting problem is one of the most fundamental results in computability theory and has played a great role in the further development of this area.

Theorem 1.6 (Halting problem) K = {e ∈ N : φe(e) ↓} is computably enumerable, but not computable.

It is known that all c.e. sets are one-one reducible to K and Shoenfield’s limit lemma shows that all sets Turing reducible to K have effective approximations. The definition of Turing jump operator is based on a relativization of the Halting ′ { ∈ N A ↓} ′ problem. For a given set A, define A as the set e :Φe (e) , and we call A the ′ ′ Turing jump of A. The Jump Theorem says that A ≤T B iff A ≤1 B , so A ≡T B ′ ′ implies A ≡T B , and hence the jump operator defined above is well-defined. If a is the degree of A, the degree of A′ is called the jump of a, and is denoted by a′. The degree of Halting problem is just the Turing jump of 0, and denoted by 0′. Obviously, if a is a c.e. degree, then a ≤ 0′. Friedberg’s jump inversion theorem says that the Turing jump, as a mapping on Turing degrees, is surjective over 0′.

3 Chapter 1. Introduction

Let A(0) = A and A(1) = A′, the ﬁrst jump of A. We then deﬁne by induction the n-th jump of A, A(n), as the Turing jump of A(n−1). Let ∅ denote the empty set. Below is the well-known high/low hierarchy in computability theory.

′ (n) (n) (n) Deﬁnition 1.1 A set A ≤T ∅ is called lown if A ≡T ∅ , and highn if A ≡T ∅(n+1).

′ (n) (n) (n) (n+1) Deﬁnition 1.2 A degree c ≤ 0 is lown if c = 0 , and highn if c = 0 .

′ ′ ′ low1, high1 are also called low, high respectively. So A ≤T ∅ is low if A ≤T ∅ , ′ ′′ and high if A ≥T ∅ . Mohrherr [4] named superlow sets and superhigh sets via the ′ ′ ′ ′ ′′ tt-reducibility as: A ≤T ∅ is superlow if A ≤tt ∅ , and superhigh if A ≥tt ∅ . Another hierarchy which is popular in computability theory is the Ershov hierarchy, or the diﬀerence hierarchy.

Deﬁnition 1.3 For n ∈ N, a set A is called n-c.e., if there is a computable function f(x, s) such that for all x,

• f(x, 0) = 0,

• A(x) = lim f(x, s), s→∞ • |{s ∈ N : f(x, s) ≠ f(x, s + 1)}| ≤ n.

A degree is n-c.e. with n ≥ 1 if it contains a n-c.e. set, and properly n-c.e. with n ≥ 2 if it is n-c.e. but not (n − 1)-c.e. 1-c.e. sets (or degrees) are just c.e. sets (or degrees), and 2-c.e. sets (or degrees) are often called d.c.e. sets (or degrees). Cooper constructed a properly d.c.e. degree in 1971 in his PhD thesis [5], and his constructions can be modiﬁed to show the existence of properly n-c.e. degrees for n ≥ 2. The set of all degrees is denoted by D, while the set of all c.e. degrees is denoted by E. The following Figure 1.1 (from Cooper [6]) provides some basic information of these two hierarchies.

Deﬁnition 1.4 A set A is called ω-c.e., if there is a computable function f(x, s) and a computable function g such that

4 Chapter 1. Introduction

Figure 1.1: High/low hierarchy and Ershov hierarchy

• for all x, A(x) = limf(x, s), s • for all x, |{s ∈ N : f(x, s) ≠ f(x, s + 1)}| ≤ g(x).

The following relations between ω-c.e. sets and strong reducibilities are well- known:

′ ′ Proposition 1.1 A set A is ω-c.e. iﬀ A ≤tt ∅ iﬀ A ≤wtt ∅ .

The study of the structure of c.e. degrees stems from a question of Post, now known as the Post’s problem:

Question 1.1 (Post (1944)) Does there exist c.e. degrees diﬀerent from 0 and 0′?

In order to solve his problem, Post himself developed many important concepts like simple sets, hypersimple sets and hyperhypersimple sets, with the attempt of making the complements of the constructed c.e. sets not c.e., by making them sparser and sparser. These concepts are closely related to strong reducibilities, in the sense that even though these sets can be Turing complete, simple sets can never be m-complete, and hypersimple sets can never be tt-complete. Post’s problem was eventually solved by Friedberg in 1957 and Muchnik in 1956, independently, where they introduced the injury method for the ﬁrst time.

5 Chapter 1. Introduction

Theorem 1.7 (Friedberg (1957), Muchnik (1956)) There exist c.e. sets A and B such that A T B and B T A.

A solution for Post’s problem, along Post’s original idea of making the complements sparse, was found in the 1970s, after Ershov’s work on η-maximal sets. Friedberg and Muchnik’s construction has a feature that the injuries from higher priority requirements are recursively bounded. In 1963, Sacks proved that any nonzero c.e. degree splits and this construction has an inherited feature that the injuries from higher priority requirements are ﬁnite, but cannot be bounded by any computable function.

Theorem 1.8 (Sacks Splitting Theorem (1963)) Let B be a noncomputable c.e. set. For any c.e. set A, there are low c.e. sets C and D such that

(1) C ∩ D = ∅, A = C ∪ D;

(2) B T C and B T D.

Theorem 1.8 says that any nonzero c.e. degree splits above 0. Robinson gener- alized Sacks splitting theorem to the splitting of a given c.e. degree above a low c.e. degree, which indicated that low c.e. degrees resemble degree 0.

Theorem 1.9 (Robinson Splitting Theorem (1971) [7]) Let L be a low c.e. set. For any c.e. set A >T L, there are c.e. sets C and D such that

(1) C ∩ D = ∅, A = C ∪ D;

(2) A T C ⊕ L and A T D ⊕ L.

In 1964, Sacks proved that the c.e. degrees are dense, and the method used in the proof is an inﬁnite injury argument. That is, the injuries from higher priority requirements can be inﬁnite.

Theorem 1.10 (Sacks Density Theorem (1964)) For any two c.e. degrees a < b, there is a c.e. degree c such that c ∈ (a, b).

6 Chapter 1. Introduction

The same style of construction was used by Sacks to prove the existence of incomplete high c.e. degrees. In 1966, Lachlan introduced another kind of inﬁnite injury arguments in his proof of the existence of minimal pairs. Here, a pair of nonzero c.e. degrees a and b form a minimal pair, denoted as a ∩ b = 0, if 0 the inﬁmum of a and b, i.e., is the only c.e. degree below both a and b.

Theorem 1.11 (Lachlan (1966), Yates (1966)) There are c.e. degrees forming a minimal pair.

Cooper proved in 1974 that any high c.e. degree bounds a minimal pair, and in this sense, high c.e. degrees resemble 0′.

Theorem 1.12 (Cooper (1974) [8]) Any high c.e. degree bounds a minimal pair of c.e. degrees.

Miller proved in 1981 in his thesis [9] that any high c.e. degree bounds a minimal pair of high c.e. degrees. This work was published by Slaman and Shore in their paper [10], where a new construction was provided.

Deﬁnition 1.5 A c.e. degree c is cappable if it is zero or a half of a minimal pair, that is, there is a c.e. degree a > 0 such that c ∩ a = 0; otherwise, c is noncappable.

It is easy to prove that every nonzero c.e. degree bounds a nonzero cappable degree, which needs a nonuniform proof. The dual of cappable degrees is called cuppable degrees.

Deﬁnition 1.6 (1) A c.e. degree c is cuppable if there is a c.e. degree a < 0′ such that c∪a = 0′; otherwise, c is called noncuppable, i.e., for any c.e. degree a < 0′, c ∪ a < 0′.

(2) A c.e. degree c is called low-cuppable if there is a low c.e. a with c ∪ a = 0′.

Theorem 1.13 (Harrington (1976)) Any high c.e. degree bounds a high noncuppable degree.

7 Chapter 1. Introduction

Ambos-Spies, Jockusch, Shore and Soare proved in [11] that cappable degrees form an ideal and noncappable degrees form a strong ﬁlter, providing a dichotomy of c.e. degrees.

Theorem 1.14 (Ambos-Spies, Jockusch, Shore and Soare [11]) A c.e. degree is noncappable iﬀ it is low-cuppable (i.e., cups to 0′ via a low c.e. degree).

Lachlan proved in 1975 that Sacks splitting theorem and Sacks density theorem cannot be combined, where he developed a new technique, now known as 0′′′-arguments, a quite complicated construction of degrees.

Theorem 1.15 (Lachlan Nonsplitting Theorem (1975)) There are two c.e. degrees a < b such that for any two c.e. degrees c, d, if b = c ∪ d, then b ≤ a ∪ c or b ≤ a ∪ d.

Harrington later proved in his paper Understanding Lachlan’s monster paper that Lachlan’s nonsplitting theorem can be strengthened as follows:

Theorem 1.16 (Harrington Nonsplitting Theorem (1980)) There exists a c.e. degree a < 0′ such that 0′ cannot split above a.

1.2 Part I: Bounded-jump operator and the high/low hierarchy

The notion of bounded-jump operator † was proposed by Anderson and Csima in paper [1], where they tried to ﬁnd an appropriate jump operator on wtt-degrees1. For a set A, the bounded-jump of A is deﬁned as the set

A † φi(e) A = {e ∈ N : ∃i ≤ e[φi(e) ↓ &Φe (e) ↓]},

† ′ where A z:= {x ∈ A : x ≤ z}. Note that A ≤T A ⊕ ∅ . In [1], Anderson and Csima pointed out this bounded-jump operator † for wtt-reducibility behaves like the Turing jump ′ for Turing reducibility, i.e., (1) ∅† and ∅′ are 1-equivalent, (2) † † † for any set A, A

1In paper [1], Anderson and Csima used b for the bounded-jump operator †.

8 Chapter 1. Introduction

† ′ (2) implies that A ≡T A when A ≥T ∅ . In the same paper, Anderson and Csima proved an analogue of Shoenﬁeld’s jump inversion theorem for this bounded-jump operator †. † † † Say that a set A is bounded-low if A ≤wtt ∅ , and bounded-high if A ≤wtt ∅ †† † and ∅ ≤wtt A . There are several immediate consequences: bounded-low sets and bounded-high sets are all ω-c.e.; bounded-low sets are closed downwards under wtt-reducibility; and bounded-high sets are closed upwards under wtt-reducibility on ω-c.e. sets. Furthermore, bounded-low sets cannot be bounded-high because † † † †† †† A ≤wtt ∅ implies A

Proposition 1.2 (Bickford and Mills (1982) [12]) There are two superlow c.e. sets ′ A, B such that A ⊕ B ≡T ∅ .

The proposition above shows that there are two bounded-low sets whose join is bounded-high, thus not bounded-low. One natural question is:

Question 1.2 (1) Whether bounded-low sets are closed downwards under Turing reducibility?

(2) Similarly, whether bounded-high sets are closed upwards under Turing reducibility on ω-c.e. sets?

′ ′ ′ For a superlow A, if B ≤T A, then B ≤1 A ≤tt ∅ . Thus B is also superlow. Superlow sets are closed downwards under Turing reducibility. Similarly, superhigh sets are closed upwards under Turing reducibility. In Chapter 2, we will give negative answer to Question 1.2 by constructing a bounded low c.e. set which Turing computes ∅′. This tells us that the greatest c.e. Turing degree 0′ contains both bounded-low c.e. sets and bounded-high c.e. sets, and one easy consequence is:

Proposition 1.3 (1) There are bounded-low c.e. sets which are superhigh.

9 Chapter 1. Introduction

(2) There are bounded-high c.e. sets which are superhigh.

Recently, in [2], Anderson, Csima and Lange initiated the study of comparing the bounded-jump and the Turing jump, where they constructed high bounded-low sets, and low bounded-high sets, showing that there is no direct relation between these two jumps. Motivated by the fact that superlowness implies bounded-lowness, they asked the following related questions:

Question 1.3 (Anderson, Csima and Lange [2])

(1) Does there exist a bounded-low set which is low but not superlow?

(2) Does there exist a bounded-high set which is high but not superhigh?

(3) Does there exist a superhigh set which is not bounded-high?

Question 1.3.(3) has positive answer because of the existence of bounded-low c.e. sets which are superhigh.

Figure 1.2: Turing degrees of bounded-low sets

We will also provide positive answers to the other two questions. In chapter 2, we will construct a bounded-low c.e. set which is low but not superlow directly. In chapter 3, we show the existence of bounded-high sets which are high but not superhigh by proving a pseudo-jump inversion theorem for bounded-high sets; we will then prove a pseudo-jump inversion theorem for bounded-low c.e. sets, and this implies the existence of bounded-low c.e. sets which are high but not superhigh.

10 Chapter 1. Introduction

Figure 1.3: Turing degrees of bounded-high sets

Finally, we point out here that all bounded-high sets we constructed are ω-c.e., and that ∅′ is the only bounded-high c.e. set we know. It seems not easy to construct bounded-high c.e. sets, the questions related to constructions of bounded-high c.e. sets are still open as far as we know. For example, the following two questions still open.

Question 1.4 (Anderson and Csima [1]) Whether the analogue of Sacks’s jump † †† inversion is true or not? That is, for any set A with ∅ ≤wtt A ≤wtt ∅ , does there † † exist a c.e. set C ≤wtt ∅ such that C ≡wtt A?

Question 1.5 (Anderson, Csima and Lange [2]) Does there exist a low c.e. set which is bounded-high?

1.3 Part II: Degrees of orders on torsion-free abelian groups

An abelian group G = (G, +G , 0G ) is orderable if there is a linear order ≤G on G such that for all x, y, z in G, x ≤G y implies x +G z ≤G y +G z; say ≤G is a group order (or simply order) on G . It is known that an abelian group is orderable iﬀ it is torsion-free (Levi’s theorem). So we will restrict ourselves to torsion-free abelian groups. ≤ G { ∈ ≥ } From an order G on , one can form the set P≤G := g G : g G 0G of all ≤ ≤ positive elements under G . P≤G is a positive cone of G . On the other hand, if

11 Chapter 1. Introduction

X ⊆ G satisﬁes the following conditions: (1) X is a semigroup, (2) X ∩−X = {0G },

(3) X ∪ −X = G, where −X = {−g : g ∈ X}, then we can deﬁne an order ≤X by

x ≤X y ⇔ y − x ∈ X

and X is the positive cone of the order ≤X . Indeed, this entails a one-to-one correspondence between orders and positive cones on G .

An abelian group G = (G, +G , 0G ) is computable if its domain G and addition

+G are both computable. Note that under this deﬁnition, the inverse function on G is computable. If G is a computable torsion-free abelian group, we will use X(G ) to denote the set of positive cones of G , and we will study Turing degrees of members X G G ≤ of ( ). Note that for such a group , P≤G and G are Turing equivalent.

C 0 Deﬁnition 1.7 A class of sets is called a Π1 class if it can be represented as the set of all inﬁnite paths of some computable binary tree.

For a computable torsion-free abelian group G , the positive cones of G are deﬁned 0 X G via Π1 conditions, and one can show that ( ) can be represented as the set of all X G 0 inﬁnite paths of a computable binary tree, which means that ( ) is a Π1 class. We are mainly interested in the degrees of members of X(G ), i.e., the degrees G 0 of orders on . Classical results on degrees of members of Π1 classes are due to Jockusch and Soare [13] in 1972, where the low basis theorem and the hyperimmune- free basis theorem are provided. These two basis theorems imply that every computable torsion-free abelian group has orders of low degree, and of hyperimmune-free degree, respectively.

Deﬁnition 1.8 For a torsion-free abelian group (G, +, 0G), g0, ··· gn ∈ G are linearly independent if for any a0, ··· an ∈ Z,

a0g0 + ··· + angn = 0G → a0 = ··· = an = 0.

An inﬁnite subset B of G is linearly independent if any ﬁnite subset of B is linearly independent. A maximal linearly independent set is called a basis of G.

12 Chapter 1. Introduction

For 1 ≤ k ≤ ∞, let Qk denote the direct sum of k many copies of additive group Q. In particular, Q∞ = ⊕ Q, its elements are just ﬁnite Q-linear sums over a i∈N computable basis {ei : i ∈ N}, where ei = (··· , 1, 0, ··· ) is the sequence whose i-th column is 1 and all other columns are 0. The rank of G, rank(G), is deﬁned as the size of a basis which is the least number k ≤ ∞ such that G embeds into Qk. Here are some easy examples:

• rank(Z) = rank(Q) = 1.

• rank(Z ⊕ Z) = rank(Q ⊕ Q) = 2.

• Q⊕Q { 1 1 } Let G be the subgroup of generated by (1, 0), (0, 1), ( 2 , 2 ) . Elements of 1 1 ∈ Z G are of the form m(1, 0)+n(0, 1)+k( 2 , 2 ) with m, n, k . Then rank(G) = 2.

We use O(G ) to denote the set of all degrees of orders on a computable torsion- free abelian group G . Solomon [14] proved the following results about O(G ):

(1) if rank(G ) = 1, then O(G ) contains exactly one degree 0;

(2) if 2 ≤ rank(G ) < ∞, then O(G ) contains all degrees;

(3) if rank(G ) = ∞, then O(G ) contains all degrees ≥ 0′.

G 0 C X G For a computable torsion-free abelian group and a Π1 class , we say ( ) degree represents C if O(G ) = {deg(C): C ∈ C}, where deg(C) is the Turing degree of C. Jockusch and Soare in [13] constructed an 0 S inﬁnite Π1 class, say , in which any two distinct elements are Turing incomparable. Then for any computable torsion-free abelian group G , O(G ) ≠ {deg(C): C ∈ S}, because either O(G ) = {0} or O(G ) ⊇ {d : d ≥ 0′}. 0 It turns out that the Π1 class of positive cones on computable torsion-free abelian 0 groups cannot degree represent all Π1 classes. However, an earlier result of Metakides 0 and Nerode [15] in 1979 showed that any Π1 class can be degree represented by the 0 Π1 class of positive cones on some computably formally real ﬁeld in a much stronger manner.

13 Chapter 1. Introduction

For a computable torsion-free abelian group G with ﬁnite rank, we know that O(G ) is either {0} or D, the set of all degrees. However, for a computable torsion- free abelian group G with inﬁnite rank, although O(G ) ⊇ {d : d ≥ 0′}, we don’t know what the set O(G ) exactly is. Here are two related results:

• (Solomon [14]) If B is a basis of G , then {d : d ≥ deg(B)} ⊆ O(G ).

• (Kach, Lange, Solomon and Turetsky [3]) O(G ) must contain inﬁnitely many low degrees as well as inﬁnitely many hyperimmune-free degrees.

In [3], Kach, Lange and Solomon considered the question whether O(G ) is upwards closed for a computable torsion-free abelian group G with inﬁnite rank, and provided a negative answer to it.

Theorem 1.17 (Kach, Lange and Solomon [3]) There is a computable torsion-free abelian group G with inﬁnite rank, and a noncomputable c.e. set C such that G has exactly two computable orders and every C-computable order on G is computable.

G in Theorem 1.17 has no orders in deg(C), but has orders in 0. By deg(C) > 0, O(G ) is not upwards closed.

Deﬁnition 1.9 A set C is called group-order-computable via G if G is a computable torsion-free abelian group with inﬁnite rank which admits computable orders, and every C-computable order on G is computable. C is group-order-computable if it is group-order-computable via some G .

Deﬁnition 1.10 A Turing degree c is called group-order-computable if it contains a set C which is group-order-computable.

0 is obviously group-order-computable, and group-order-computable degrees are downwards closed. On the one hand, Kach, Lange and Solomon’s theorem above implies the existence of nonzero c.e. group-order-computable degrees, and hence can be low. In his thesis [16], C. J. Martin showed the existence of high c.e. degrees which are group-order-computable. On the other hand, any a ≥ 0′ cannot be group- order-computable because every computable torsion-free abelian group with inﬁnite rank has an order of degree a. Our main objective in chapter 4 is to investigate the characterization of group- order-computable degrees.

14 Chapter 1. Introduction

Deﬁnition 1.11 A set C is said to be weakly group-order-computable via a computable torsion-free abelian group G with inﬁnite rank if every C-computable order on G is computable. C is weakly group-order-computable if C is weakly group-order- computable via some G .

A Turing degree is weakly group-order-computable if it contains a weakly group- order-computable set. Clearly, group-order-computable degrees are weakly group- order-computable. We will show that these two families of degrees, and also non- PA-degrees, coincide. A set is called PA-complete if it computes a complete extension of Peano Arith- metic. An equivalent deﬁnition of PA-complete sets is that:

• a set A is PA-complete iff it computes a {0, 1}-valued diagonally nonrecursive function, where a function f is diagonally nonrecursive if ∀x[φx(x) ↓→ f(x) ≠ φx(x)]. A degree is PA if it contains a PA-complete set. The notion of PA-degrees plays a vital role in the study of randomness. For example, Stephan [17] proved that a PA degree a is Martin-Löfrandom iff a ≥ 0′. The following facts for PA-degrees are well-known:

Proposition 1.4 (Scott and Solovay [18]) Let a be a Turing degree. The following are equivalent:

(1) a computes a complete extension of Peano Arithmetic.

(2) a computes a consistent extension of Peano Arithmetic.

0 (3) Each Π1 class contains an a-computable element.

We ﬁrst characterize weakly group-order-computable degrees in terms of PA- degrees.

Proposition 1.5 Let a be a Turing degree. The following are equivalent.

(1) a is a PA degree.

15 Chapter 1. Introduction

(2) a is not weakly group-order-computable.

Proof: (1) ⇒ (2). Let a be a PA degree, and G be a computable torsion-free abelian group with infinite rank (not necessarily possessing computable orders). We only need to show that G has an a-computable order which is not computable. X G 0 C G Martin proved in [16] that ( ) has a Π1 subclass, say ( ), containing no computable orders. As a computes a positive cone in C(G ), G has an a-computable order which is not computable. (2) ⇒ (1). We first mention a result of Hatzikiriakou and Simpson [19] in reverse mathematics: the statement “every countable torsion-free abelian group is orderable” is equivalent to WKL0 over RCA0. This result has a consequence that there is a computable torsion-free abelian group, say G0, with a property that all orders of it are of PA degree. So G0 has no computable orders, hence the rank of G0 is infinite. By definition, a degree a is not weakly group-order-computable, if any computable torsion-free abelian group with infinite rank has an a-computable order which is not computable. In particular, G0 has an a-computable order, and hence a computes an order on G0, which is of PA degree. This implies that a itself is a PA degree. By Proposition 1.5, all PA-degrees are not group-order-computable. We will construct in chapter 4 a computable torsion-free abelian group G with infinite rank such that for any set A, G admits exactly two A-computable orders iff A is not PA-complete. As computable sets are not PA-complete, G above admits exactly two computable orders. Then for any set A which is not PA-complete, every A-computable order on G is computable; for any set A which is PA-complete, A computes a noncomputable order on G . This shows that for any degree a, if it is now not group-order-computable, then it is a PA degree. Hence, a degree is PA iff it is not weakly group-order-computable iff it is not group-order-computable. 0 At the end of [3], by applying the low basis theorem of Π1 classes infinitely many times, Kach, Lange, Solomon and Turetsky showed that every computable torsion- free abelian group of infinite rank has orders of infinitely many low degrees. We would like to know whether G in Theorem 1.17 has orders of other degrees:

16 Chapter 1. Introduction

Question 1.6 Given a nonzero c.e. degree a, can we build a group G such that: (1) it has exactly two computable orders, (2) G has an order of degree a, and (3) there is a noncomputable c.e. C such that every C-computable group order on G is computable?

We will give a positive answer to Question 1.6 in chapter 5. Our results show that for any nonzero c.e. degree a, there is a nonzero c.e. degree c < a, and a computable torsion-free abelian group G such that G has orders of degree ≥ a but has no incomputable orders of degree ≤ c.

1.4 Part III: Reverse mathematics, abelian groups and modules

In this part, we study reverse mathematics of some classic theorems in module theory. We ﬁrst brieﬂy introduce common subsystems of second order arithmetic studied in reverse mathematics. L2, the language of second order arithmetic, contains two-sorted variables, namely, number variables and set variables. The full system Z2 of second order arithmetic contains basic axioms of ordered semiring of natural numbers, induction axioms of sets, i.e., for any set X,

(0 ∈ X ∧ (n ∈ X → n + 1 ∈ X)) → ∀n(n ∈ X), and comprehension axioms of formulas φ(n) in L2, i.e.,

∃X∀n(n ∈ X ↔ φ(n)), where X does not occur freely in φ. Many theorems in non-set-theoretic mathematics are classified by five big sub- 1 systems of Z2, i.e., RCA0, WKL0, ACA0, ATR0 and Π1-CA0. Simpson’s book [20] provides a comprehensive introduction of these five subsystems. We introduce RCA0 and ACA0 in detail here, as we will mainly work with these two subsystems. 0 RCA0 contains basic axioms of ordered semiring of natural numbers, Σ1-induction 0 0 and ∆1-comprehension. Σ1-induction is the axiom of the following form with φ re- 0 stricted to Σ1 formulas:

(φ(0) ∧ (φ(n) → φ(n + 1))) → ∀nφ(n).

17 Chapter 1. Introduction

0 0 ∆1-comprehension is the axiom of the following form with φ restricted to ∆1 formulas: ∃X∀n(n ∈ X ↔ φ(n)), where X does not occur freely in φ.

ACA0 contains basic axioms of ordered semiring of natural numbers, induction 0 0 axioms of sets, and Σk-comprehension for all k.Σk-comprehension axioms say

∃X∀n(n ∈ X ↔ φ(n)),

0 where φ is a Σk-formula and X does not occur freely in φ(n). The following theorem is widely used in reverse mathematics when proving a theorem in ordinary mathematics implies ACA0.

Theorem 1.18 The following are equivalent over RCA0.

(1) ACA0;

(2) Let f : N → N be a one-to-one function, then the range of f exists.

We now deﬁne basic concepts of module theory using the language L2 of second order arithmetic.

Deﬁnition 1.12 (RCA0) Let R be a commutative ring with identity 1R, a module M over R is an abelian group together with a scalar operation ◦ from R × M to M, satisfying for all m, m1, m2 ∈ M and r, r1, r2 ∈ R,

M1. (r1 + r2) ◦ m = r1 ◦ m + r2 ◦ m;

M2. (r1r2) ◦ m = r1 ◦ (r2 ◦ m);

M3. 1R ◦ m = m;

M4. r ◦ (m1 + m2) = r ◦ m1 + r ◦ m2.

For convenience, we write rm for r ◦ m.

Deﬁnition 1.13 (RCA0) Let M1 and M2 be two R-modules. A map φ : M1 → M2 is a R-module homomorphism if for any x, y ∈ M1 and r ∈ R, φ(x+y) = φ(x)+φ(y), φ(rx) = rφ(x).

18 Chapter 1. Introduction

Deﬁnition 1.14 (RCA0) Let φ : A → B and ψ : B → C be two R-module homo- φ ψ morphisms. The sequence A → B → C is exact at B if ker(ψ) = im(φ). Moreover, if φ is a monomorphism and ψ is a surjective homomorphism, then the sequence

φ ψ 0 → A → B → C → 0 is also exact at A, C, and often called a short exact sequence.

We only consider countable modules over a commutative ring with identity and mainly deal with inﬁnite modules whose domain are subsets of N. Within RCA0, the set N

Deﬁnition 1.15 (RCA ) For 0 ≤ i ≤ n, let (M , + , ◦ , 0 ) be a R-module. ⊕ 0 i Mi Mi Mi The direct sum Mi of modules Mi(0 ≤ i ≤ n) is a R-module containing ﬁnite 0≤i≤n sequences of length n + 1 of the form ⟨m0, m1, ··· , mn⟩ with each mi ∈ Mi, together with componentwise addition

⟨ ··· ⟩ ⟨ ′ ··· ′ ⟩ ⟨ ′ ··· ′ ⟩ m0, , mn + m0, , mn = m0 +M0 m0, , mn +Mn mn for abelian groups and componentwise scalar multiplication

⟨ ··· ⟩ ⟨ ◦ ··· ◦ ⟩ r m0, , mn = r M0 m0, , r Mn mn ⊕ ⟨ ··· ⟩ ≤ ≤ for modules. The zero of Mi is 0M0 , 0M1 , , 0Mn . If Mi = M for all 0 i 0≤i≤n ⊕ n, we often use M n+1 to denote M. 0≤i≤n

In general, as in Solomon [21], we can deﬁne the direct sum of countably many R- modules as a module whose nonzero elements are ﬁnite sequences with last column nonzero, and zero element is just the empty sequence.

19 Chapter 1. Introduction

⊕ Deﬁnition 1.16 (RCA0) Let Mi(i ∈ N) be R-modules. The direct sum Mi is i∈N a R-module containing ﬁnite sequences of the form ⟨m0, m1, ··· , mn⟩ with n ∈ N, ∈ ≤ ≤ ̸ mi Mi for 0 i n, mn = 0Mn , and the empty sequence, together with componentwise addition and componentwise scalar multiplication. Empty sequence is the ⊕ ⊕ ⊕ ∞ zero element of Mi. We simply write M (or M ) for Mi when Mi = M i∈N i∈N i∈N for all i ∈ N.

µ ε Deﬁnition 1.17 (RCA0) A short exact sequence 0 → A → B → C → 0 splits if there exists a homomorphism σ : C → B such that εσ is the identity homomorphism on C. σ is called a splitting for this short exact sequence.

µ ε Lemma 1.2 (RCA0) If the short exact sequence 0 → A → B → C → 0 splits with ∼ a splitting σ : C → B, then B = A ⊕ C. Moreover, B = µ(A) ⊕ σ(C).

Proof: Deﬁne a map ψ : A ⊕ C → B; ⟨a, c⟩ 7→ µ(a) + σ(c). We now show that ψ is an isomorphism between A ⊕ C and B. For any a, a′ ∈ A, c, c′ ∈ C, and r ∈ R,

ψ(⟨a, c⟩ + ⟨a′, c′⟩) = ψ(⟨a + a′, c + c′⟩) = µ(a + a′) + σ(c + c′) = µ(a) + µ(a′) + σ(c) + σ(c′) = ψ(⟨a, c⟩) + ψ(⟨a′, c′⟩) and

ψ(r⟨a, c⟩) = ψ(⟨ra, rc⟩) = µ(ra) + σ(rc) = rµ(a) + rσ(c) = rψ(⟨a, c⟩).

So ψ is a R-module homomorphism. ψ is injective. First, ψ(⟨a, c⟩) = µ(a) + σ(c) = 0 implies 0 = ε(µ(a) + σ(c)) = 0 + εσ(c) = c. Then µ(a) = 0, and this implies a = 0. So ⟨a, c⟩ = 0. ψ is surjective. For any b ∈ B, ε(b) ∈ C and ψ(⟨0, ε(b)⟩) = σε(b), εσε(b) = ε(b). So σε(b)−b ∈ ker(ε) = im(µ). Then there exists an a ∈ A such that µ(a) = σε(b)−b, and then b = µ(a) + σε(b) = ψ(⟨a, ε(b)⟩). This also implies that B = µ(A) + σ(C). Now we show B = µ(A) ⊕ σ(C). It is enough to prove µ(A) ∩ σ(C) = {0}. Let b ∈ B. Assume b = µ(a) = σ(c) for some a ∈ A and c ∈ C. Then ψ(⟨a, −c⟩) =

20 Chapter 1. Introduction

µ(a) − σ(c) = 0. Similarly, 0 = ε(µ(a) − σ(c)) = 0 − εσ(c) = −c. Then c = 0, and thus b = 0. In Chapter 6, we ﬁrst consider reverse mathematics of divisible abelian groups.

Deﬁnition 1.18 (RCA0) Let G be an abelian group, and g ∈ G.

(1) g is torsion if there is a natural number n ≥ 1 such that ng = 0G.

(2) The order o(g) of an element g ∈ G is the least natural number n ≥ 1 such

that ng = 0G if g is torsion, and ∞ otherwise.

An abelian group G is torsion if any element of G has a finite order, and torsion- free if any nonzero element of G has an infinite order. Notions like torsion groups, torsion-free groups are all definable within RCA0.

Deﬁnition 1.19 (RCA0) An abelian group G is divisible if for every g ∈ G and for every nonzero n ∈ N, there is a g′ ∈ G such that g = ng′.

Z ∞ { n ∈ Z} Z For a prime number p, (p ) = pm : m, n / exists in RCA0.

We will prove that over RCA0, the following are equivalent:

(1) ACA0.

(2) Every divisible subgroup of an abelian group is a direct summand.

(3) Every torsion-free abelian group has the largest divisible subgroup.

(4) Every divisible abelian group is isomorphic to a direct sum of the form ⊕ ⊕ ∞ Z ln ⊕ Q ⊆ N ̸ ∅ ∈ ( (pn )) for some I,J , and if I = , for each n I, n∈I n∈J 1 ≤ ln ≤ ∞.

For an R-module M and a subset A of M, the set

⟨A⟩ = {r0a0 + ··· + rnan : n ∈ N, 0 ≤ i ≤ n, ri ∈ R, ai ∈ A} of R-linear combinations of elements of A forms a submodule of M. If M = ⟨A⟩, we say A generates M or A is a set of generators of M.

A finite set {m0, m1, ··· , mn} of M is linearly independent if for any ri ∈ R, 0 ≤ i ≤ n, r0m0 + ··· + rnmn = 0 implies ri = 0 for all 0 ≤ i ≤ n. An infinite subset A of M is linearly independent if any finite subset of A is linearly independent. A R-module M has a basis B if B is linearly independent and B generates M.

21 Chapter 1. Introduction

Deﬁnition 1.20 (RCA0) A R-module is free if it has a basis.

Proposition 1.6 (RCA ) Let F be a free R-module with an inﬁnite basis, say B = 0 ⊕ {bi : i ∈ N}. Then F is isomorphic to R. i∈N ⊕ Proof: Let ci = ⟨0, ··· , 0, 1R⟩ ∈ R of length i + 1. That is, the j-th column is i∈N the zero element of R for j ≤ i − 1 and the i-th column is the identity of R. Then ⊕ ⊕ ∑ ∑ {ci : i ∈ N} is a basis of R. Deﬁne φ : F → R by φ( ribi) = rici for i∈N i∈N 0≤i≤n 0≤i≤n any n ∈ N and ri ∈ R with 0 ≤ i ≤ n.

Proposition 1.7 (RCA ) Let M be a R-module. Then M is isomorphic to a quo- 0 ⊕ tient of the free module F = R. i∈N

Proof: List M as m0, m1, ··· . Deﬁne a map φ : F → M as follows: ﬁrst, for all i,

φ(ci) = mi, where ci = ⟨0, ··· , 0, 1R⟩ of length i + 1; second, extend φ linearly to all ∑ ∑ ∼ elements of F . That is, φ( rici) = rimi. As φ is surjective, M = F/ker(φ), 0≤i≤n 0≤i≤n where ker(φ) = {x ∈ F : φ(x) = 0}. ⊕ Proposition 1.8 (RCA0) Let F = R. For every surjective homomorphism i∈N ε : B → C of R-modules and every homomorphism φ : F → C, there exists a homomorphism ψ : F → B such that φ = εψ.

Proof: Let c0, c1, ··· be a basis of F . For each φ(ci) ∈ C, ﬁnd an element di of B such that ε(di) = φ(ci). Deﬁne for all i ∈ N, ψ(ci) = di.

Deﬁnition 1.21 (RCA0) Let P be a R-module. P is called projective if for every surjective homomorphism ε : B → C of R-modules and every homomorphism φ : P → C, there exists a homomorphism ψ : P → B such that φ = εψ.

The following proposition is an immediate consequence of Proposition 1.8.

Proposition 1.9 (Yamazaki [22])(RCA0) A free R-module is projective.

Theorem 1.19 (Yamazaki [22]) Over RCA0, a R-module is projective iﬀ it is isomorphic to a direct summand of a free R-module.

22 Chapter 1. Introduction

A free R-module is projective, but the converse fails in general. However, classically, for a principal ideal domain R, projective R-modules are always free.

Deﬁnition 1.22 (RCA0) A commutative ring R with identity 1R is an integral domain if it has no zero divisors, that is, for any a, b ∈ R − {0}, ab ≠ 0.

Downey, Lempp and Mileti in [23] showed that there is a computable integral domain, say S, which is not a field such that S has no computable non-trivial proper ideals which are finitely generated. In particular, S has no computable non-trivial proper ideals which are principal. Hence, within RCA0, one cannot show that any integral domain which is not a field has a principal non-trivial proper ideal. Indeed, Downey, Lempp, and Mileti [23] proved that the statement that “every commutative ring which is not a field has a finitely generated nontrivial proper ideal” is equivalent to ACA0 over RCA0. This result of Downey, Lempp, and Mileti also implies that “every commutative ring which is not a field has a principal non-trivial proper ideal” is equivalent to ACA0 over RCA0, refer to Sato’s thesis [24] (Theorem 6.11). In reverse mathematics, to prove basic properties on principal ideal domains

(PIDs) even within weak systems like RCA0, we had better to deﬁne a PID as an 0 integral ideal domain whose Σ1-ideals are generated by one element. Such PIDs are 0 known as Σ1-PIDs, refer to Simpson [25] and Sato [24] for more details.

Deﬁnition 1.23 (RCA0)

(1) Let R be a commutative ring with 1R. A sequence ⟨rn : n ∈ N⟩ ⊆ R is a 0 ∈ N ∈ N Σ1-ideal if for any i, j , there is a k such that rk = ri + rj and for any

i ∈ N and s ∈ R, there is a j ∈ N such that rj = ris.

0 0 (2) An integral domain R is a Σ1-PID, if for any Σ1-ideal I of R, there is an element r ∈ I such that I = (r), the ideal generated by r.

We will prove in Section 7.1 some results related to projective modules within

ACA0. In particular, we will show that ACA0 proves:

0 (1) every projective module over a Σ1-PID is free;

0 (2) every submodule of a projective module over a Σ1-PID is projective.

23 Chapter 1. Introduction

As a dual to projective modules, we have the following deﬁnition for injective modules.

Deﬁnition 1.24 (RCA0) Let I be a R-module. I is injective if for any monomorphism µ : A → B and any homomorphism φ : A → I, there exists a homomorphism ψ : B → I such that φ = ψµ.

The following classical lemma for injective modules is known as Baer’s criterion.

Lemma 1.3 (Baer’s Criterion) A module I over a commutative ring R with 1R is injective iﬀ for every ideal J of R and every R-module homomorphism f : J → I, there is a homomorphism g : R → I such that g J = f.

Proposition 1.10 (RCA0) A module I over a commutative ring R with 1R is injective implies for every ideal J of R and every R-module homomorphism f : J → I, there is a homomorphism g : R → I such that g J = f.

Theorem 1.20 (Yamazaki [22]) The following are equivalent over RCA0.

(1) ACA0.

(2) If for every ideal J of R and every R-module homomorphism f : J → I, there

is a homomorphism g : R → I such that g J = f, then I is injective.

Deﬁnition 1.25 (RCA0) Let R be an integral domain. A R-module D is divisible if for every d ∈ D and nonzero r ∈ R, there is a c ∈ D such that d = rc.

Classically, a module over a principal ideal domain is injective iﬀ it is divisible.

Proposition 1.11 (RCA0) An injective module I over an integral domain R is divisible.

Proof: For any nonzero r ∈ R and d ∈ I, let µ : R → R; 1R 7→ r and φ : R → ′ ′ ′ I; 1R 7→ d. As µ(r ) = r r = 0 implies r = 0, µ is a monomorphism. As I is an injective R-module, there is a homomorphism ψ : R → I such that φ = ψµ. Now d = φ(1R) = ψµ(1R) = ψ(r) = rψ(1R) with ψ(1R) ∈ I. The following theorem about reverse mathematics of divisible abelian groups is due to Friedman, Simpson and Smith [26] and can be found in Simpson’s book [20].

24 Chapter 1. Introduction

Theorem 1.21 [26] The following are equivalent over RCA0.

(1) ACA0.

(2) Every divisible abelian group is injective.

In Section 7.2, we will extend Theorem 1.21 to modules, and show that the 0 statement that “every divisible module over a Σ1-PID is injective” is equivalent to

ACA0 over RCA0. We will also show that ACA0 proves the statement that “every 0 quotient of an injective module over a Σ1-PID is injective”.

25 Part I

Bounded-jump operator and the high/low hierarchy

26 Chapter 2

A bounded-low c.e. set which is low, but not superlow

In this chapter, we will answer two questions raised by Anderson, Csima and Lange in [2] by proving: (1) there is a bounded-low c.e. set which is low, but not superlow; (2) 0′ contains a bounded-low c.e. set. Anderson, Csima and Lange considered in [2] the interaction between the bounded- jump operator and the Turing jump, and constructed a high bounded-low set. They asked whether bounded-lowness and superlowness agree with each other on low c.e. sets. We will give a negative answer to this question by showing the existence of bounded-low c.e. sets which are low, but not superlow.

Theorem 2.1 There are low c.e. sets which are bounded-low but not superlow.

After seeing that superlow sets are always bounded-low, Anderson, Csima and Lange [2] asked whether any superhigh set is also bounded-high. Our second result in this chapter will provide a negative answer to this question. That is, we show the existence of bounded-low c.e. sets which are of degree 0′. Thus, such bounded-low c.e. sets are superhigh.

Theorem 2.2 0′ contains a bounded-low c.e. set.

We will give a proof of Theorem 2.1 ﬁrst, and then prove Theorem 2.2 in Section 2.4. To prove Theorem 2.1, we will construct a low c.e. set A such that A† is ω-c.e. (so A is bounded-low), and A′ is not ω-c.e. (so A is not superlow). One approach

27 Chapter 2. A bounded-low c.e. set which is low, but not superlow of constructing a set low, but not superlow, is to apply Sacks splitting theorem, to split ∅′ into two low c.e. sets A and B. A and B cannot both be superlow, as if A is superlow, then ∅′ is wtt-reducible to B [by a theorem of Bickford and Mills in [12]]. In same paper [12], Bickford and Mills also pointed out that there are superlow c.e. sets A and B such that A ⊕ B computes ∅′. Most of the results in [12] can be found in Nies’ book [27]. One feature of Sacks splitting is that there is no eﬀective bound on the number of injuries from a strategy with higher priority, and hence we cannot have a recursive bound on the number of changes of A′(e) in advance. It also turns out to be an obstacle for us to obtain a bounded-low c.e. set which is low, but not superlow, by using Sacks splitting. Instead, we will construct a low, but not superlow, c.e. set directly.

2.1 Requirements and strategies

We will construct a c.e. set A meeting the following requirements:

L A ↓ A ↓ e : If there are inﬁnitely many stages s such that Φe (e)[s] , then Φe (e) .

† 2 Re : The number of changes of A at e is bounded by 2(e + 1) .

P 2 ⟨i,j⟩ : If φi and φj are both total, then there is some x such that either

′ ̸ 2 A (x) = lim φj (x, t) t→∞

or |{ ∈ N 2 ̸ 2 }| ≥ t : φj (x, t) = φj (x, t + 1) φi(x).

{ ∈ N} { 2 ∈ N} Here, φi : i and φj : j are standard enumerations of all partial computable functions in one variable and two variables respectively, {Φe : e ∈ N} is a standard enumeration of all partial computable functionals, and ⟨·, ·⟩ is an eﬀective bijection between N2 and N. We assign the priority of requirements in our ﬁnite injury construction as

L0 ≺ R0 ≺ P0 ≺ L1 ≺ R1 ≺ P1 ≺ · · · ≺ Le ≺ Re ≺ Pe ≺ · · · .

28 Chapter 2. A bounded-low c.e. set which is low, but not superlow

If all the L-requirements are satisfied, then A is low. If all the R-requirements are satisfied, then the whole construction will ensure that A† is ω-c.e., which ensures that A is bounded-low. If all the P-requirements are satisfied, then A′ is not ω-c.e., and hence, A is not superlow.

The strategy of satisfying one Le is standard, i.e., we will set restraints to preserve the desired computations we have seen and this strategy can only be injured by P- strategies with higher priority, which will happen at most ﬁnitely many times. Thus, after a stage large enough, after which no P-strategies with higher priority act, if the strategy sets restraints to preserve computations, these computations will be preserved forever, and Le is satisﬁed.

The strategy for satisfying one Re is also to set restraints to preserve computations. At each stage s, when we see that φi(e) converges, where i ≤ e, we set a restraint to protect A φi(e). Note that P-strategy with higher priority can injure

Re by enumerating small numbers x into A. But as φi(e) will be ﬁxed, once P ′ enumerates x into A, P selects a new number, x say, bigger than φi(e), and the ′ further enumeration of x will not injure Re [as this enumeration will not aﬀect A A φi(e) φi(e) the computation Φe (e) if Φe (e) converges]. Note that there are at most e + 1 many such restraints, and each P-strategy with higher priority can enumerate numbers less than φi(e) at most once, the total number of such enumerations is at most (e + 1)2. As one such an enumeration entails at most two changes of A†(e), the number of changes of A†(e) is bounded by 2(e + 1)2.

We now describe how to satisfy a P⟨i,j⟩-requirement. As in [27] for the construction of nonsuperlow sets, we will apply the recursion theorem for the construction. That is, our construction will be uniform in parameters, r say, and in the r-th construction, we will build a partial computable functional Γr, a c.e. set of axioms ⟨ ⟩ ⟨ ⟩ A σ, n, m , where if σ, n, m is enumerated into Γr at stage s,Γr (n)[s] is deﬁned as m with use A A = σ. From Γ , if Γ = W , we can have a computable function s γr (n)[s] r r r pr such that ∀ ∀ X ≃ X X x[Γr (x) Φpr(x)(pr(x)) ].

A ↓⇐⇒ ∈ ′ A This will ensure that Γr (x) pr(x) A . Hence, by controlling Γr , we can 2 ′ defeat φj being an approximation of A with number of changes bounded by φi.

29 Chapter 2. A bounded-low c.e. set which is low, but not superlow

Since the construction is uniform in r, there is a computable function g such that Γr = Wg(r). By recursion theorem, there is a r0 such that Wg(r0) = Wr0 , i.e.,

Γr0 = Wr0 . In the following, assume that we are in the r0-th construction, just ﬁx the computable function pr0 in advance.

A P⟨i,j⟩-strategy proceeds as follows:

(i) Choose a fresh witness x, let z = pr0 (x).

(ii) Wait for φi(z) ↓ at some stage s.

(If we wait at (2) forever, then φi(z) ↑, and φi is partial and P⟨i,j⟩ is satisﬁed.)

(iii) Set N = φi(z) + 1, and run the following modules C(n) and D(n) for n ≤ 2 ↓ N. We start with C(0) in which case φj (z, 0) = 1 or D(0) in which case 2 ↓ − φj (z, 0) = 0, and let ℓ( 1) = 0. We will end when reaching C(N) or D(N), 2 because this shows that φi(z) cannot bound the number of changes of φj (z, q),

showing that P⟨i,j⟩ is satisﬁed.

− 2 C(n): There is a q1 > ℓ(n 1) such that φj converges (at stage s) to 1 on all − ≤ ≤ 2 (z, q) with ℓ(n 1) < q q1, and for those q2 with q1 < q2 s, if φj (z, q2) 2 converges at stage s, then φj (z, q2) = 1.

If such a q1 does not exist, then do nothing.

2 − – If this is because of the fact that φj (z, ℓ(n 1) + 1) never converges, 2 P then φj is partial and ⟨i,j⟩ is satisﬁed.

2 – If it is because of the existence of q2 with φj (z, q2) = 0, then wait for 2 ≤ a bigger stage at which φj (z, q) converges for all q q2, leave C(n)

with no actions and switch to D(n + 1) with ℓ(n) = q2. 2 2 (Again, if φj (z, q) never converges for some q < q2 , then φj is partial

and P⟨i,j⟩ is satisﬁed.)

2 That is, at stage s, we guess that φj (z, q) has limit 1, and we should not 2 − keep such a guess if we see φj (z, q) converges to 0 after ℓ(n 1).

A − Action for C(n): If Γr0 (x) has deﬁnition at stage s 1, then enumerate A γr0 (x) into A, to undeﬁne Γr0 (x). Otherwise, do nothing.

30 Chapter 2. A bounded-low c.e. set which is low, but not superlow

A – Wait for a stage t > s such that Φz (z)[t] diverges.

(For r0-th construction, such a stage t exists, because otherwise, we A A would have Γr0 (x) diverges, but Φz (z) converges.)

2 – Wait for a bigger stage at which we see that φj (z, q3) converges to ≥ 2 ≤ 0, where q3 q1, and φj (z, q) converges for all q q3. Let ℓ(n) = q3 and switch to D(n + 1).

2 2 (Again, if φj (z, q) never converges for some q < q3 , then φj is partial P 2 ≥ and ⟨i,j⟩ is satisﬁed. Or, if φj (z, q) converges to 1 for all q q1, 2 ̸∈ ′ 2 then lim φj (z, q) = 1, while z A , meaning that lim φj (z, q) is not x→∞ x→∞ ′ the characteristic function of A , and P⟨i,j⟩ is satisﬁed.)

− 2 D(n): There is a q1 > ℓ(n 1) such that φj converges (at stage s) to 0 on all − ≤ ≤ 2 (z, q) with ℓ(n 1) < q q1, and for those q2 with q1 < q2 s, if φj (z, q2) 2 converges at stage s, then φj (z, q2) = 0.

If there is no such a q1, then do nothing.

2 − – If this is because of the fact that φj (z, ℓ(n 1) + 1) never converges, 2 P then φj is partial and ⟨i,j⟩ is satisﬁed.

2 – If it is because of the existence of q2 with φj (z, q2) = 1, then wait 2 ≤ for a bigger stage at which φj (z, q) converges for all q q2, leave

D(n) with no actions and switch to C(n + 1) with ℓ(n) = q2.(Again, 2 2 if φj (z, q) never converges for some q < q2 , then φj is partial and

P⟨i,j⟩ is satisﬁed.)

2 That is, at stage s, we guess that φj (z, q) has limit 0, and we should not 2 − keep such a guess if we see φj (z, q) converges to 1 after ℓ(n 1).

A − A Action for D(n): If Γr0 (x) has no deﬁnition at stage s 1, deﬁne Γr0 (x)

with use γr0 (x) big.

A – Wait for a stage t > s such that Φz (z)[t] converges.

(For r0-th construction, such a stage t exists, because otherwise, we A A would have Γr0 (x) converges, but Φz (z) diverges.)

31 Chapter 2. A bounded-low c.e. set which is low, but not superlow

2 – Wait for a bigger stage at which we see that φj (z, q3) converges to ≥ 2 ≤ 1, where q3 q1, and φj (z, q) converges for all q q3. Let ℓ(n) = q3 and switch to C(n + 1).

2 2 (Again, if φj (z, q) never converges for some q < q3 , then φj is partial P 2 ≥ and ⟨i,j⟩ is satisﬁed. Or, if φj (z, q) converges to 0 for all q q1, 2 ∈ ′ 2 then lim φj (z, q) = 0, while z A , meaning that lim φj (z, q) is not x→∞ x→∞ ′ the characteristic function of A , and P⟨i,j⟩ is satisﬁed.)

Thus, a P⟨i,j⟩-strategy either waits at some C(n) or D(n) for some n < N, or reaches C(N) or D(N), both of which will show that P⟨i,j⟩ is satisﬁed, as explained above. Thus, in the construction, a P⟨i,j⟩-strategy can injure those strategies with lower priority at most ﬁnitely often.

2.2 Construction

Construction of A.

Stage 0: Let A0 = ∅, and initialize all strategies.

Stage s > 0: Find a requirement with the highest priority, Q say, among

L0 ≺ R0 ≺ P0 ≺ · · · Ls ≺ Rs ≺ Ps that requires attention at stage s, and act accordingly. L A − ↑ A ↓ Say that e requires attention at stage s if Φe (e)[s 1] and Φe (e)[s] . Say

Re requires attention at stage s if there is a n ≤ e ≤ s such that φn(e) ↓ at stage s and Re has not set A-restraint φn(e) yet. For both cases, we act as follows:

Action: Set A-restraint as s, and initialize all strategies with lower priority.

Say that P⟨i,j⟩ requires attention at stage s if one of the following applies:

(i) P⟨i,j⟩ has no witness currently.

Action: Pick a fresh number x as a witness for P⟨i,j⟩.

32 Chapter 2. A bounded-low c.e. set which is low, but not superlow

↓ (ii) x is selected, pr0 (x) converges to z, φi(z)[s] and N is not deﬁned yet.

Action: Deﬁnite N as φi(z) + 1.

P ≤ 2 (iii) N is deﬁned, ⟨i,j⟩ is not in C(n) or D(n) for any n N, and φj (z, 0)[s] converges.

P 2 ↓ Action: Let ⟨i,j⟩ start by running C(0), if φj (z, 0)[s] = 1, or D(0), if 2 ↓ − φj (z, 0)[s] = 0. Here we let ℓ( 1) = 0.

(iv) P⟨i,j⟩ switches from C(n) to D(n + 1), where n + 1 < N.

Action: If ΓA(x) has no deﬁnition at stage s−1, then deﬁne ΓA(x)[s] = 0 with use γ(x)[s] larger than all numbers used before.

(v) P⟨i,j⟩ switches from D(n) to C(n + 1), where n + 1 < N.

Action: If ΓA(x) has deﬁnition at stage s − 1, then enumerate γ(x) into A, undeﬁning ΓA(x).

(vi) P⟨i,j⟩ switches from C(N − 1) to D(N) or from D(N − 1) to C(N).

2 Action: Declare that φi(z) cannot bound the number of changes of φj (z, q),

and that P⟨i,j⟩ is satisﬁed.

If P⟨i,j⟩ acts as above, then all strategies with lower priority are initialized. This ends the construction of A at stage s.

2.3 Veriﬁcation

Lemma 2.1 For each requirement, Q say,

(1) Q can be initialized at most ﬁnitely often;

(2) Q acts at most ﬁnitely often and is satisﬁed;

(3) The restraint set by Q is ﬁnite, and Q can initialize all strategies with lower priority ﬁnitely often.

33 Chapter 2. A bounded-low c.e. set which is low, but not superlow

Proof: We will show that for each e, (1)-(3) are true for all Le, Re, P⟨i,j⟩, where ⟨i, j⟩ = e. We prove it by induction.

When e = 0, L0 can never be initialized, due to its highest priority. (1) holds. L A 0 acts and sets a restraint only when Φ0 (0) converges, and once we set such a A restraint, Φ0 (0) converges at any further stage, and (2) holds. In the construction, L A 0 sets a restraint only when Φ0 (0) converges, and after it acts, it will never act again. (3) holds.

Note that R0 can be initialized by L0 at most once. (1) holds for R0. Note that R A φ0(0) 0 acts only when φ0(0) converges, and after this, if Φ0 (0) converges, then it will converge forever, as no P can enumerate numbers less than φ0(0) into A. So A φ0(0) R if Φ0 (0) converges again, then it converges forever, which means that 0 is satisﬁed. (2) holds. The restraint set by R0 is φ0(0), if it converges, and after it acts, it will never act again. (3) holds.

For P⟨0,0⟩, it can be initialized by L0 and R0, so P⟨0,0⟩ can be initialized by L0 and R0 at most three times. (1) holds. Let s0 be the least stage after which P⟨0,0⟩ can not be initialized again, and suppose that P⟨0,0⟩ selects a witness x as a big number at stage s1 > s0. Then x cannot be canceled later. Then after stage s1,

P⟨0,0⟩ acts only when

(i) it selects N; or

(ii) it starts module C(0) or D(0); or

(iii) it switches from C(n − 1) to D(n), or from D(n − 1) to C(n); or

(iv) it reaches C(N) or D(N),

which can happen at most N +2 many times. If it reaches C(N) or D(N), then φ0(z) 2 P cannot be a bound of the number of changes of φ0(z, q), and ⟨0,0⟩ is satisﬁed. If it never selects N, then φ0(z) never converges (recall that pr0 is assumed to be total, m by the relativized sn -theorem). Here z = pr0 (x). If it never starts module C(0) or 2 D(0), then φ0(z, 0) diverges. If it stays at some module C(n) or D(n) forever, then 2 2 ̸ ′ P either φ0(z, q) diverges for some q, or we will have lim φ0(z, q) = A (z). ⟨0,0⟩ is q→∞ satisﬁed for all cases. (2) holds for P⟨0,0⟩. Thus, after a stage s3 large enough, P⟨0,0⟩

34 Chapter 2. A bounded-low c.e. set which is low, but not superlow

will have no actions, and hence the restraint set by P⟨0,0⟩ is ﬁnite, and it will not initialize strategies with lower priority. (3) holds. For e > 0, we assume that the lemma is true for e′ < e. The proof that the lemma is true for e is very similar to the proof for the basic case, i.e. e = 0, and we will assume that after a stage s big enough, no strategies Le′ , Re′ , and Pe′ can act. Then we can show that after stage s, Le, Re, and Pe will behave exactly like

L0, R0, and P0 above, and hence the lemma is true for e. This completes the proof of Lemma 2.1. By Lemma 2.1, A is low but not superlow. Lemma 2.2 below shows that A is bounded-low.

Lemma 2.2 A† is ω-c.e., and hence A is bounded-low.

Proof: To show that A† is ω-c.e., we need to show that there is a computable function bounding the changes of e in A† for all e.

Fix e, and n ≤ e. Then in the construction, whenever we see φn(e) converges, at stage s say, Re acts and initializes all strategies with lower priority. Thus, no strategy with lower priority can put a number less that φn(e) into A later. So if

Aφn(e) † Φe converges later (e enters A ), this computation can only be changed by Pe′ - ′ A strategies (where e < e, which have higher priority) by enumerating γr0 (y)[t] into A A. After this, if γr0 (y) is deﬁned later, then it will be deﬁned as a number bigger Aφn(e) than φn(e). Thus, because of this, Φe is injured at most e + 1 many times, then A†(e) can change at most 2(e + 1) many times, for this particular n. Thus, in total, A†(e) can change at most 2(e + 1)2 many times, and A† is ω-c.e. This completes the proof of Theorem 2.1.

2.4 A bounded-low set with Turing degree 0′

In this section, we give a sketchy proof of Theorem 2.2. That is, we will construct a bounded-low c.e. set A Turing computing ∅′. We have seen how to construct a bounded-low c.e. set in the proof of Theorem 2.1, so to prove Theorem 2.2, we only need to show how to make A Turing complete. We will construct a partial computable functional Γ such that ∅′ = ΓA. Of course, the construction of Γ will be consistent with the R-strategies of making A bounded-low.

35 Chapter 2. A bounded-low c.e. set which is low, but not superlow

For the construction of Γ, we will make sure that for each e,ΓA(e) is defined and computes ∅′(e) correctly. The idea of constructing Γ is quite standard. That is, in the construction, at stage s, we find the least x with ΓA(x) not defined and define ΓA(x)[s] = ∅′(x)[s] with use γ(x) big, like s. If later, at stage t > s, x enters ∅′, then at this stage, we enumerate γ(x) into A, undefining ΓA(x). In general, the construction of Γ satisfies the following rules:

A A (i) For any x1, x2 with x1 < x2, if Γ (x2) has deﬁnition at stage s, then Γ (x1)

also has deﬁnition at this stage, with uses γ(x1) < γ(x2).

A A (ii) For any x1, x2 with x1 < x2, if Γ (x1) is undeﬁned at stage s, then Γ (x2) is

also undeﬁned at this stage, due to a number less than γ(x1) enters A.

A (iii) For any x, if Γ (x) has deﬁnition at stage s1 and s2, then γ(x)[s1] ≤ γ(x)[s2], A and if γ(x)[s1] < γ(x)[s2], Γ (x) must have been undeﬁned between these two stages.

(iv) For any x,ΓA(x) can be undeﬁned at most ﬁnitely often during the construction.

(v) For any x,ΓA(x) = ∅′(x).

We now show how to modify a Re-strategy in the proof of Theorem 2.1 so that it is consistent with the rules above. Recall that the Re-strategy can act finitely many times, and each time when it acts, because of φi(e) ↓ for some i ≤ e, it initializes all strategies with lower priority. To make it consistent with the construction of Γ, it needs to have an additional action: enumerate γ(e)[s] into A. This enumeration undefines ΓA(e), so when it is defined again, the use will be defined as a big number, which is bigger than the value φi(e). Thus, for a fixed e, the membership of e in † ′ A can be changed by Re′ -strategies with e ≤ e, i.e., by further enumerations of the γ-uses, or by rectifying ΓA(x) = 1 when some x < e enumerating into ∅′. Then A†(e) changes at most 2(e + 1)2 many times, which guarantees that A† is ω-c.e. The construction and verification parts are standard finite injury arguments.

36 Chapter 3

A bounded-high set which is high, but not superhigh

Note that ∅′ is a bounded-high set which is also superhigh. In [2], Anderson, Csima and Lange further asked whether there are bounded-high sets which are high but not superhigh. Similarly, based on the existence of superhigh bounded-low sets, we can also ask whether there are high but not superhigh bounded-low sets. In sections 3.1 and 3.2, we will provide two pseudo-jump inversion theorems for bounded-high sets and bounded-low sets, which will provide positive answers to these questions.

Let We be a c.e. operator, the corresponding pseudo-jump operator Ve is deﬁned A ⊕ A as Ve = A We for any set A. One way to obtain high, but not superhigh, c.e. sets is through the pseudo-jump inversion theorem of Jockusch and Shore [28] which C ≡ ∅′ says: for any c.e. operator We, there is a c.e. set C such that Ve T . This idea was ﬁrst used by Mohrherr in her paper [4]. The procedure to obtain high but not superhigh c.e. sets are:

(1) Through the construction of low but not superlow c.e. sets, one can obtain a c.e. operator W such that for any set A, A ⊕ W A is low over A, but not A ′ ′ A ′ ′ superlow over A, i.e., (A ⊕ W ) ≤T A , but (A ⊕ W ) tt A .

(2) By applying the pseudo-jump inversion theorem to this operator W , there C ′ exists a c.e. set C such that C ⊕ W ≡T ∅ .

′′ C ′ ′ ′′ C ′ ′ Then ∅ ≡1 (C ⊕ W ) ≤T C (C is high), and ∅ ≡1 (C ⊕ W ) tt C (C is not superhigh).

37 Chapter 3. A bounded-high set which is high, but not superhigh

In this chapter, we will provide two strengthened versions of Jockusch and Shore’s pseudo-jump inversion theorem. That is, C in Jockusch and Shore’s theorem can be bounded-high (section 3.1) or bounded-low (section 3.2).

Theorem 3.1 (1) For any c.e. operator W , there is a bounded-high set C such C ′ that C ⊕ W ≡T ∅ .

(2) For any c.e. operator W , there is a bounded-low c.e. set C such that C ⊕ C ′ W ≡T ∅ .

The following two results follow Theorem 3.1 immediately:

Corollary 3.1 (1) There are bounded-high sets which are high but not superhigh.

(2) There are bounded-low c.e. sets which are high but not superhigh.

Proof: Let W be the c.e. operator such that for any set A, A⊕W A is low but not C ′ superlow over A, then any C satisfying C ⊕ W ≡T ∅ is high but not superhigh.

By Theorem 3.1, there are a bounded-high set C1 and a bounded-low c.e. set C2

C1 ′ C2 ′ such that C1 ⊕ W ≡T ∅ and C2 ⊕ W ≡T ∅ . Thus C1 is a bounded-high set which is high but not superhigh, while C2 is a bounded-low c.e. set which is high but not superhigh. We will prove Theorem 3.1.(1) in Section 3.1, and then prove Theorem 3.1.(2) in Section 3.2. For a c.e. operator W , let Ψ be the corresponding partial computable functional. That is, for any oracle A, e ∈ W A[s] ⇔ ΨA(e)[s] ↓. We view the use ψA(e)[s] of A A Ψ (e)[s] as the use for e ∈ W [s]. That is, if A ψA(e)[s] does not change later, e will stay in W A. Ψ-uses satisfy the following usual use rules: if ΨA(e)[s] ↓ and ΨA(e′)[s] ↓ with e < e′, then ψA(e)[s] < ψA(e′)[s]; if ΨA(e)[s] ↓, then ψA(e)[s] ≤ s; and if ΨA(e)[s] ↑, then ψA(e)[s] = 0.

3.1 Pseudo-jump inversion via bounded-high sets

† C ′ We will construct a ω-c.e. set C (so C ≤wtt ∅ ) such that C ⊕ W ≡T ∅ and †† † ∅ ≤wtt C . C will meet the following requirements:

38 Chapter 3. A bounded-high set which is high, but not superhigh

′ C⊕W C Pe : ∅ (e) = Γ (e), where Γ is a computable functional built by us.

†† C† Qe : ∅ (e) = ∆ g(e) (e), where ∆ is a computable functional built by us and g is † a computable function such that the use δC (e) ≤ g(e) for all e.

C C Re : If there are inﬁnitely many stages s with e ∈ W [s], then e ∈ W .

R C 0 If all the -requirements are satisfied, then W is ∆2. By construction, C will C ′ be ω-c.e., and hence C ⊕ W ≤T ∅ . If all the P-requirements are satisfied, then ′ C ′ C ∅ ≤T C ⊕ W . Thus, ∅ ≡T C ⊕ W . If all the Q-requirements are satisfied, then †† † ∅ ≤wtt C , and C will be bounded-high. We will apply the bounded-high strategy developed by Anderson, Csima and Lange in [2], and combine it with Jockusch and Shore’s pseudo-jump inversion theorem. Our construction is a finite injury argument, where the priority of requirements is

R0 ≺ P0 ≺ Q0 ≺ R1 ≺ P1 ≺ Q1 ≺ · · · Re ≺ Pe ≺ Qe ≺ · · · .

3.1.1 A Re-strategy

C C The original Re-strategy is just setting a C-restraint ψ (e)[s] when e ∈ W [s] − W C [s − 1] (i.e., ΨC (e)[s] ↓ and ΨC (e)[s − 1] ↑). We ﬁrst deﬁne C-restraint

r(e, s) = max{ψC (e)[s′]: s′ ≤ s ∧ ΨC (e)[s′] ↓}.

Then r(e, s) ≤ r(e, s + 1). Suppose that e ∈ W C [s] − W C [s − 1]. e will be removed out of W C [t] at some least stage t > s only if

Ct ψC (e)[s]≠ Cs ψC (e)[s] .

C C So if e ∈ W [s] and Cs r(e,s)= Ct r(e,s) for all t ≥ s, then e ∈ W [t] for all stages t ≥ s, and thus e ∈ W C .

We say that Re is injured at stage s if either some Pi enumerates the Γ-use C⊕W C ′ ′ γ (i)[s] ≤ r(e, s) into C when i ∈ ∅ [s] − ∅ [s − 1]; or some Qi enumerates its coding indicator c ≤ r(e, s) into C or extracts its coding indicator c ≤ r(e, s) out of † †† C when redefining ∆C g(i) (i)[s] = ∅ [s](i) (the definition of coding indicators will be clarified in the following basic Q-strategy).

39 Chapter 3. A bounded-high set which is high, but not superhigh

During the construction, we do not allow positive requirements with lower priority, namely Pj, Qj with j ≥ e, to enumerate elements ≤ r(e, s) into C or to extract elements ≤ r(e, s) out of C, at any stage s (this will be clear in Case 1 for Pj and

Case 2 for Qj), and hence these requirements cannot injure Re later. Assume that e ∈ W C [s] − W C [s − 1] with C-restraint r(e, s). Since we have appointed a C-restraint r(e, s − 1), we only need to consider the case r(e, s) > r(e, s − 1). So we assume that at stage s, r(e, s) > r(e, s − 1).

Case 1. At stage s with r(e, s) > r(e, s − 1), for Pj with j ≥ e, we will show that ΓC⊕W C (e)[s] ↑. That is, for all stages s′ < s at which ΓC⊕W C (e)[s′] ↓, either

C ̸ C ′ (a) : W [s] γC⊕W C (e)[s′]= W [s ] γC⊕W C (e)[s′] or

̸ ′ (b) : Cs γC⊕W C (e)[s′]= Cs γC⊕W C (e)[s′] holds. Indeed, for stages s′ < s, if ΓC⊕W C (e)[s′] ↓ with ΨC (e)[s′] ↑, then e∈ / W C [s′]. As e ∈ W C [s], C C ′ W [s] e≠ W [s ] e

(recall that the symbol X z denotes the set {x ∈ X : x ≤ z}). As we will choose γC⊕W C (e)[s′] > e,(a) holds.

If ΓC⊕W C (e)[s′] ↓ with ΨC (e)[s′] ↓. By the assumption that e ∈ W C [s]−W C [s−1] with new computation ΨC (e)[s] ↓, we have

Cs ψC (e)[s′]≠ Cs′ ψC (e)[s′] .

Claim γC⊕W C (e)[s′] > ψC (e)[s′], then (b) holds. We now prove γC⊕W C (e)[s′] > ψC (e)[s′] for all s′ < s with ΓC⊕W C (e)[s′] ↓ and ΨC (e)[s′] ↓. Suppose that the computation ΓC⊕W C (e)[s′] is ﬁrst deﬁned at stage t′ ≤ s′ and returns back at stage s′. This means that

C ′ C ′ ′ ′ ∧ Cs γC⊕W C (e)[t′]= Ct γC⊕W C (e)[t′] W [s ] γC⊕W C (e)[t′]= W [t ] γC⊕W C (e)[t′] .

There are two cases:

40 Chapter 3. A bounded-high set which is high, but not superhigh

(1) ΨC (e)[t′] ↓. As γC⊕W C (e)[t′] is deﬁned to be bigger than all used numbers,

γC⊕W C (e)[t′] > ψC (e)[t′].

Then ΨC (e)[t′] also comes back at stage s′, we have

γC⊕W C (e)[s′] = γC⊕W C (e)[t′] > ψC (e)[t′] = ψC (e)[s′].

(2) ΨC (e)[t′] ↑. Then e∈ / W C [t′], but by assumption, e ∈ W C [s′]. As γC⊕W C (e)[t′] > e, C ′ ̸ C ′ W [t ] γC⊕W C (e)[t′]= W [s ] γC⊕W C (e)[t′] . This contradicts with the assumption that the old computation ΓC⊕W C (e)[t′] ↓ returns back at stage s′, so this case is indeed impossible.

In construction, at each stage t, if ΓC⊕W C (e)[t] ↓ and ΓC⊕W C (j)[t] ↓ with e < j, we will make the corresponding uses satisfying γC⊕W C (e)[t] < γC⊕W C (j)[t]. So when ΓC⊕W C (e)[s] is undefined, then all ΓC⊕W C (j) with j > e are also undefined at stage s. Hence, when the C-restraint r(e, s) moves to a higher level, all the previous ΓC⊕W C (j)-computations with j ≥ e are undefined, we will define new computations ΓC⊕W C (j) with use > r(e, s). Then the enumerations of such uses into C can not injure Re as long as r(e, s) does not increase later.

Case 2. At stage s with r(e, s) > r(e, s − 1), for Qj with j ≥ e, we ﬁrst consider

Qe. Say that Qe is injured at stage s if there is an i ≤ e such that the C-restraint r(i, s) on Ri is larger than or equal to the coding indicator of Qe.

The strategy of Qe to deal with this injury from Ri is still to undefine previ- † † ously defined ∆C g(e) (e)-computations and then define ∆C g(e) (e) via new coding indicators > r(i, s). Then Qe will not injure Ri as long as r(i, s) does not increase later. † † To undefine ∆C g(e) (e)-computations, we need to change C below g(e). In order to enumerate elements into or extract elements out of C†, we will build a partial computable function q and a partial computable functional Θ, and assume that the recursion theorem provides us the graph of q and the c.e. relation of Θ.

41 Chapter 3. A bounded-high set which is high, but not superhigh

Then by controlling q and Θ, we can move elements (i.e., indices of Θ obtained m † by sn -theorem) into or out of C (this will be much clear in the following basic

Qe-strategy).

Hence, during the construction, Re can only be injured by positive requirements with higher priority. Each Pi(i < e) can injure Re at most once, and each Qi(i < e) can injure Re ﬁnitely often by enumerating coding indicators. So Re can be injured ﬁnitely often, and thus limr(e, s) < ∞. s

3.1.2 A Pe-strategy

We will build a functional Γ such that ∅′ = ΓC⊕W C . At each stage s, when e ∈ ∅′[s + 1] − ∅′[s] and ΓC⊕W C (e)[s] ↓, we will enumerate the use γC⊕W C (e)[s] into C to undeﬁne ΓC⊕W C (e), and then redeﬁne ΓC⊕W C (e) = 1 with big use.

When the C-restraints r(i, s + 1) on Ri with i ≤ e move up, i.e. r(i, s + 1) > C⊕W C r(i, s), as in a basic Ri-strategy, we see Γ (i) is undefined at the beginning of stage s+1. Thus, ΓC⊕W C (j) ↑ for all j ≥ i, and such ΓC⊕W C (j) will be defined later with new use > r(i, s + 1). During the construction, we will define at most one ΓC⊕W C (y) at each stage. At the last step of stage s + 1, we find the least y with ΓC⊕W C (y) ↑ currently. If ΓC⊕W C (y) is never defined before, or there is a largest stage w < s+1 at which ΓC⊕W C (y)[w] ↓ and also at least one of the following three conditions holds:

(1) y ∈ ∅′[w + 1] − ∅′[w];

(2) y ∈ W C [t + 1] − W C [t] with r(y, t + 1) > r(y, t) for some t ∈ [w, s];

(3) γC⊕W C (y)[w] ≤ γC⊕W C (y − 1)[s].

Then define ΓC⊕W C (y)[s + 1] = ∅′[s + 1](y) with use γC⊕W C (y)[s + 1] bigger than all numbers used before. Otherwise, there is a largest stage w < s + 1 at which ΓC⊕W C (y)[w] ↓ and none of three conditions above holds, just define ΓC⊕W C (y)[s + 1] = ∅′[s + 1](y) with use γC⊕W C (y)[s + 1] = γC⊕W C (y)[w]. We now show that limγC⊕W C (e)[s] < ∞. Assume that limγC⊕W C (e − 1)[s] < ∞, s s and that limr(i, s) < ∞ for all i ≤ e. We can find a stage s0 such that for all i ≤ e, s

42 Chapter 3. A bounded-high set which is high, but not superhigh

• limr(i, s) = r(i, s0), s

′ ′ • i ∈ ∅ ⇔ i ∈ ∅ [s0].

C⊕W C Let s1 ≥ s0 + 1 be the least stage such that Γ (e)[s1] ↓. Then after stage s1, C⊕W C C⊕W C C⊕W C γ (e)[s] will not move up, so limγ (e)[s] = γ (e)[s1]. s

3.1.3 A Qe-strategy

Fix the canonical notation for ω2. That is, for each β = ω · i + j, we view β coded as ⟨i, j⟩, and the corresponding order on the set of all ordered pairs of natural numbers is just the lexicographic order, i.e., ⟨i, j⟩ < ⟨k, l⟩ iﬀ i < k or i = k with j < l. Anderson and Csima proved in [1] that ∅†† is ω2-c.e. under this notation, i.e., there is a partial computable function χ : ω × ω2 → {0, 1} such that

2 †† • for each e, there is a least ordinal βe < ω such that χ(e, βe) ↓= ∅ (e).

Then

• · for each e, there is a ﬁrst stage se and a least ordinal βe,se = ω ise + jse such ↓ that χ(e, βe,se )[se] ;

• for each e and stage s ≥ se, there is a least ordinal βe,s = ω · ie,s + je,s such

that χ(e, βe,s)[s] ↓.

†† Now βe = limβe,s, and ∅ (e) = χ(e, βe) = limχ(e, βe,s). As χ is partial computable, s s ise and jse are computable functions on e; ie,s and je,s are computable functions on e, s. For convenience, we can also choose χ such that

• for each e and stage s ≥ se, if βe,s ≠ βe,s+1, then χ(e, βe,s+1) ≠ χ(e, βe,s).

In this case, βe,s+1 < βe,s, that is, either ie,s+1 < ie,s, or ie,s+1 = ie,s with

je,s+1 < je,s.

43 Chapter 3. A bounded-high set which is high, but not superhigh

Fix a partial computable function χ : ω × ω2 → {0, 1} which satisﬁes conditions †† above. At stage s ≥ se, we view ∅ [s](e) = χ(e, βe,s). †† C† ′ Consider the requirement Qe : ∅ (e) = ∆ g(e) (e). At stage s ≥ se, we deﬁne C† ′ ′ ∆ g(e) (e)[s ] = χ(e, βe,s′ ). If there is a least stage s > s at which βe,s < βe,s−1 or Qe is injured by R-strategies with higher priority (i.e., the current coding indicator of

Qe is less than or equal to the C-restraint on R), then we need to rectify functional † † ∆C g(e) (e) at stage s by enumerating some x ≤ g(e) into C or extracting some y ≤ g(e) out of C†. In order to enumerate elements into or extract elements out of C†, we will build additional partial computable functions and partial computable functionals. By using the recursion theorem, we can change C† on indices of such partial computable functionals. The construction indeed relies on parameters, say r. In the r-th construction, we will build an auxiliary partial computable function qr, an auxiliary partial computable functional Θr, and also an auxiliary c.e. set Qr, where Qr en- codes the graph of qr as well as the c.e. relation of Θr. † Assume that we are in the r-th construction. To encode ∅††(e) into ∆C (e), we will reserve three types of markers of Qe, namely injury markers, location markers and coding markers [such markers will be deﬁned in Case 1 and Case 2 below]. † We build the functional ∆ such that ∆C (e) only depends on the membership of † such markers in C . Let Ue,r be the set of all injury markers, location markers and coding markers deﬁned during the r-th construction. Then the use function C† δ (e) = maxUe,r. We will determine a strictly increasing computable function gr †† C† (in section 3.1.4) such that gr(e) ≥ maxUe,r. Then ∅ (e) = ∆ gr(e) (e).

Case 1. We ﬁrst consider the situation when Qe is injured from negative requirements with higher priority.

Injury markers. When the C-restraint r(i, s) on Ri(i ≤ e) increases to a new level at stage s with r(i, s) ≥ c, the coding indicator of Qe [the deﬁnition of coding indicators will be given in the following Case 2.1], we will enumerate an injury † marker < gr(e) into C and never remove it out. This enumeration injures all old C† C† convergence ∆ gr(e) (e)-computations, and ∆ gr(e) (e) will arrive at new computations. In this case, just cancel existed coding indicators and coding markers of Qj with j ≥ e.

44 Chapter 3. A bounded-high set which is high, but not superhigh

In order to define injury markers, we first choose a fresh witness, x say. Now Y qr(x) and Θr (x) (for any oracle Y ) are undefined currently. Define qr(x) = 0 and Y ⟨⟨ ⟩ ⟨ ⟩⟩ Θr (x) = 0 for all oracles Y at stage s, and then enumerate x, 0 , λ, x, 0 into

Qr,s (note we also enumerate other elements of the form ⟨⟨y, c⟩, ⟨σ, y, 0⟩⟩ with σ a binary string into Qr when defining location markers and coding markers), where ⟨·, ·⟩ : N2 → N and ⟨·, ·, ·⟩ : N3 → N are fixed effective one-to-one functions; λ stands for the empty string [we indeed fix an effective coding of 2<ω (the set of binary strings) into N and view a binary string σ in ⟨σ, y, 0⟩ as its code].

We now provide more details for the use of recursion theorem.

The application of recursion theorem.

m First, by sn -theorem, there are computable functions f1 and f2 such that for { ∃ ⟨ ⟩ ∈ } { ∃ ⟨ ⟩ ∈ } each x, Wf1(x) = z : y[ z, y Wx] , and Wf2(x) = y : z[ z, y Wx] , where

Wx is the x-th c.e. set of natural numbers.

In the r-th construction, if Qr = Wr, then

{ ∃ ⟨ ⟩ ∈ } Wf1(r) = z : y[ z, y Qr] and { ∃ ⟨ ⟩ ∈ } Wf2(r) = y : z[ z, y Qr] .

So the graph of qr is Wf1(r), and the c.e. relation of Θr is just Wf2(r). Then again, m m we can use sn -theorem as well as the relativized sn -theorem to obtain the injective computable functions kr and hr such that for n, m and Y ,

Y ≃ Y φkr(n)(m) = qr(n), Φhr(n)(m) Θr (n).

Moreover, by Padding Lemma, we can further choose kr and hr such that kr(n) < hr(n) for all n. Such kr and hr will be used to deﬁne injury markers, location markers as well as coding markers.

However, the assumption that Qr = Wr may be incorrect. Then our guess on qr and Θr are wrong, and hence kr and hr are also incorrect. Thus, the r-th construction does not work in general.

Observe that the c.e. sets Qr(r ∈ N) are uniform in r, then Qr = Wf(r) for some computable function f. By recursion theorem, there is at least one r0 (indeed

45 Chapter 3. A bounded-high set which is high, but not superhigh

inﬁnitely many) such that Qr0 = Wf(r0) = Wr0 . Then in the r0-th construction, our guess on qr0 and Θr0 are both correct, and thus, the injective functions kr0 and hr0 really satisfy for all n, m and Y ,

Y Y φk (n)(m) = qr (n), Φ (m) ≃ Θ (n). r0 0 hr0 (n) r0

Then Qe can be satisﬁed via the r0-th construction.

This ends the description of applying recursion theorem.

Now return back to the fresh witness x above which devotes to deﬁne injury Y markers. At stage s, after deﬁning qr(x) = 0 and Θr (x) = 0 for all oracles Y , we view Φ∅ (h (x)) ↓= Θ∅(h (x)) ↓= 0, and for all m, φ (m) ↓= q (x) ↓= 0. hr(x) r r r kr(x) r Then Cφ (h (x)) Φ kr(x) r (h (x)) ↓ ∧ k (x) < h (x). hr(x) r r r † This means that hr(x) enumerates into C forever. Call hr(x) an injury marker of

Qe.

As we will determine an increasing computable function gr with gr(e) bigger than C† all injury markers of Qe, ∆ gr(j) (j) ↑ for all j ≥ e. We can omit such Qj’s coding indicators and coding markers, and Qj will obtain new coding indicators larger than the C-restraint r(i, s).

Case 2. We now consider the situation when βe,s < βe,s−1. There are two subcases.

†† Case 2.1. If the approximation of ∅ (e) changes at stage s via ie,s < ie,s−1, then C† we will undefine all old definitions and then define ∆ gr(e) (e) via new computations.

Location markers. To undefine old computations, we will enumerate a new num- † ber < gr(e) into C and never remove it out. Such a number is called a location †† marker at level ie,s, which indicates that approximations of ∅ (e) move to new level C† ie,s and that ∆ gr(e) (e) will be defined at that new level. To define location markers, we choose a big number, say y, and then define Y ⟨⟨ ⟩ ⟨ ⟩⟩ qr(y) = 0, Θr (y) = 0 for all oracles Y , and enumerate y, 0 , λ, y, 0 into Qr,s. Then we view Cφ (h (y)) Φ kr(y) r (h (y)) ↓ ∧ k (y) < h (y); hr(y) r r r

46 Chapter 3. A bounded-high set which is high, but not superhigh

∈ † ≥ by φkr(y)(hr(y)) = 0, hr(y) C [t] for all t s. Say that hr(y) is a location marker C† at level ie,s. Now ∆ gr(e) (e) is undeﬁned, because we will deﬁne gr(e) bigger than all location markers of Qe.

Coding markers and coding indicators. After obtaining location markers above, we immediately define coding markers by picking a new number, say c, and then define a coding marker via c as follows. Again, choose a number z larger than all used numbers. Define

qr(z) = c and { 0, if c∈ / Y ΘY (z) := r ↑, otherwise.

We enumerate all elements of the form ⟨⟨z, c⟩, ⟨σ, z, 0⟩⟩ with σ ∈ 2<ω, |σ| = c + 1 and σ(c) = 0 into Qr,s, where |σ| is the length of σ. We think for all m,

↓ ↓ Y ≃ Y φkr(z)(m) = qr(z) = c, Φhr(z)(m) Θr (z).

So for all stages t ≥ s,

ΦC c (h (z))[t] ↓⇔ ΘC c (z)[t] ↓⇔ c∈ / C . hr(z) r r t

† Then we view hr(z) ∈ C [t] ⇔ c∈ / Ct. Such a hr(z) is called a coding marker at level ie,s, and such a witness c is called the coding indicator of hr(z). C† We will deﬁne ∆ gr(e) (e) via the coding marker hr(z) and the coding indicator c at level ie,s. If χ(e, βe,s) = 1, then keep c∈ / Cs and deﬁne

† C g (e) † ∆ r (e)[s] = C [s](hr(z)) = 1.

Cs Otherwise, χ(e, βe,s) = 0, to undeﬁne Θr (z), just enumerate the coding indicator c into C . Then ΦCs (m) ↑ for all m; in particular, ΦCs (h (z)) ↑, and thus s hr(z) hr(z) r † hr(z) ∈/ C [s]. Deﬁne

† C g (e) † ∆ r (e)[s] = C [s](hr(z)) = 0.

47 Chapter 3. A bounded-high set which is high, but not superhigh

†† Case 2.2. If ie,s = ie,s−1, but the approximation χ(e, βe,s) of ∅ (e) changes at stage s via je,s < je,s−1. As the approximation stays in the previous level, we have deﬁned the coding indicator c and the coding marker hr(z) at level ie,s, and deﬁned C† ∆ gr(e) (e) via c and hr(z). At stage s − 1, we have

† C g (e) † ∆ r (e)[s − 1] = χ(e, βe,s−1) = C [s − 1](hr(z)) = 1 − Cs−1(c).

If χ(e, βe,s) = 0, then χ(e, βe,s−1) = 1, so c∈ / Cs−1. Just enumerate c into Cs.

If our construction is the r0-th construction provided by recursion theorem, this Cs ∈ † enumeration undeﬁnes Φ (hr0 (z)), so hr0 (z) / C [s]. We can deﬁne hr0 (z)

C† † gr0 (e) − ∆ (e)[s] = 0 = χ(e, βe,s) = C [s](hr0 (z)) = 1 Cs(c).

If χ(e, βe,s) = 1, then χ(e, βe,s−1) = 0, so c ∈ Cs−1. Now extract c out of C c ↓ C. Similarly, if we are in the r0-th construction, then Φ (hr0 (z))[s] with hr0 (z) φ (h (z)) = c and k (z) < h (z), and hence h (z) ∈ C†[s]. We can deﬁne kr0 (z) r0 r0 r0 r0

C† † gr0 (e) − ∆ (e)[s] = 1 = χ(e, βe,s) = C [s](hr0 (z)) = 1 Cs(c).

3.1.4 Deﬁne the computable function gr

We now specify a computable function gr in the r-th construction such that for all e, gr(e) is bigger than maxUe,r, where Ue,r is the set of injury markers, location markers and coding markers of Qe deﬁned in the r-th construction.

We first determine a recursive upper bound for the size |Ue,r| of Ue,r. For injury markers, we define them only if some Ri(i ≤ e) has its C-restraint r(i, s) increase. So we also determine a number µ(i) which exceeds the number of times r(i, s) increases. C As R0 has the highest priority, when 0 is enumerated into W [s] for first time, we set C-restraint r(0, s) = ψC (0)[s] and preserve it forever. Thus, r(0, s) increases at most once, and let µ(0) = 1.

Now consider |U0,r|. Q0 is injured at most once when r(0, s) increases, so it has at most one injury marker. For location markers, we deﬁne them when the †† approximation of ∅ (0) decreases to a new level. Recall that s0 is the ﬁrst stage at · ↓ which there is a least ordinal β0,s0 = ω is0 + js0 such that χ(0, β0,s0 )[s0] , so there Q are at most is0 + 1 many levels for 0. Thus, the number of location markers of

48 Chapter 3. A bounded-high set which is high, but not superhigh

Q Q 0 is at most is0 . For coding markers, we change them when 0 is injured or the approximation of ∅††(0) arrives at a new level, then the number of coding markers Q Q of 0 (= the number of coding indicators of 0) is at most µ(0) + is0 + 1. Hence,

| | ≤ U0,r µ(0) + is0 + (µ(0) + is0 + 1).

In order to ﬁnd an upper bound for general |Ue,r|, we also determine a number Q ν(e) which bounds the number of coding markers of e. Set ν(0) = µ(0) + is0 + 1. Assume that we have already obtained µ(e′) and ν(e′) for all e′ < e. Now determine µ(e) which bounds the number of times C-restraints r(e, s) on

Re increases. Re is injured at stage s if some Pi(i < e) enumerates Γ-uses ≤ r(e, s) into C, or some Qi(i < e) enumerates coding indicators ≤ r(e, s) into C or removes coding indicators ≤ r(e, s) out of C. Note that each Pi(i < e) will enumerate at e most one use into C. Re will be injured by P-requirements at most 2 many times.

For a sequence of ﬁxed coding indicators, say ci, of Qi with i < e, the number of conﬁgurations of C at this sequence is 2e, and there are at most ν(i) many coding ∏ e indicators of Qi(i < e). Re will be injured by Q-requirements for at most 2 ν(i) 0≤i

49 Chapter 3. A bounded-high set which is high, but not superhigh

This ends the description of bounding the size of Ue,r.

C† Our initial target is to bound maxUe,r, the use of ∆ (e). In order to recursively bound maxUe,r, during the r-th construction, we will choose Γ-uses for P- requirements and the witnesses at which we deﬁne qr and Θr for Q-requirements in a neat way. Fix an eﬀective injection ⟨, ⟩ : N × N → N such that for all i, j ∈ N, ⟨i, j⟩ is strictly bigger than both i and j, and if i < k and j < l, then ⟨i, j⟩ is strictly less than ⟨k, l⟩. Let ⟨i, N⟩ := {⟨i, j⟩ : j ∈ N}.

All Γ(e)-uses for Pe are chosen from ⟨3e, N⟩. Γ(e)-uses will be moved up only if ′ some i ≤ e enumerates into ∅ , or the C-restraints on some Ri with i ≤ e increases. The witnesses for deﬁning injury markers, location markers, and coding markers of Qe are chosen from ⟨3e + 1, N⟩, and there are at most 2ν(e) many such witnesses.

Then during the construction, we choose such witnesses of Q0 from

{⟨1, x⟩ : x ≤ 2ν(0)}

one by one in increasing order, and choose such witnesses of Qe from ∑ ∑ {⟨3e + 1, x⟩ : e + 2 ν(i) ≤ x ≤ e + 2 ν(i)} 0≤i≤e−1 0≤i≤e one by one in increasing order.

The coding indicators for coding markers of Qe are chosen from ⟨3e + 2, N⟩, and there are at most ν(e) many such coding indicators. During the construction, when the C-restraint on some Ri(i ≤ e) exceeds the coding indicators of Qe or the approximations of ∅††(e) decrease into a new level, we omit old coding indicators of Qe, and appoint new coding indicators form ⟨3e + 2, N⟩. During the r-th construction:

(1) When deﬁning injury markers or location markers for Qe, we ﬁrst pick a least unused witness, say ⟨3e + 1, y⟩, with y satisfying:

∑ ∑ – e + 2 ν(i) ≤ y ≤ e + 2 ν(i), 0≤i≤e−1 0≤i≤e

– qr(⟨3e + 1, y⟩) ↑,

50 Chapter 3. A bounded-high set which is high, but not superhigh

Y ⟨ ⟩ ↑ – Θr ( 3e + 1, y ) .

| | ≤ Y By Ue,r 2ν(e), during the r-th construction, we will deﬁne qr and Θr on no more than 2ν(e) many arguments inside ⟨3e + 1, N⟩, so such a y exists. We then deﬁne

qr(⟨3e + 1, y⟩) = 0

and Y ⟨ ⟩ Θr ( 3e + 1, y ) = 0

for all oracles Y , and view

∅ Φ (h (⟨3e + 1, y⟩)) ↓= 0 ∧ φ ⟨ ⟩ ↓= 0 hr(⟨3e+1,y⟩) r qr( 3e+1,y )

with qr(⟨3e+1, y⟩) < hr(⟨3e+1, y⟩). That is, the injury marker or the location † marker hr(⟨3e + 1, y⟩) is enumerated into C forever.

(2) When deﬁning coding markers of Qe, we ﬁrst choose a new coding indicator, say ⟨3e + 2, x⟩, with x bigger than all numbers used before, and then pick a least unused witness, say ⟨3e + 1, z⟩, with z satisfying:

∑ ∑ – e + 2 ν(i) ≤ z ≤ e + 2 ν(i), 0≤i≤e−1 0≤i≤e

– qr(⟨3e + 1, z⟩) ↑,

Y ⟨ ⟩ ↑ – Θr ( 3e + 1, z ) .

Similarly, such a z exists. Then we deﬁne

qr(⟨3e + 1, z⟩) = ⟨3e + 2, x⟩,

and for all oracles Y , deﬁne

Y ⟨ ⟩ ↓⇔ ⟨ ⟩ ∈ Θr ( 3e + 1, z ) 3e + 2, x / Y.

† We view the coding marker hr(⟨3e + 1, z⟩) is enumerated into C iﬀ its coding indicator ⟨3e + 2, x⟩ is outside C.

51 Chapter 3. A bounded-high set which is high, but not superhigh

Hence, all markers of Qe are of the form hr(⟨3e + 1, y⟩) for some ∑ ∑ e + 2 ν(i) ≤ y ≤ e + 2 ν(i). 0≤i≤e−1 0≤i≤e

As hr is a computable function, we can deﬁne ∑ gr(0) = hr(⟨1, y⟩). 0≤y≤2ν(0)

For e > 0, deﬁne gr(e) by induction as ∑ gr(e) = 1 + gr(e − 1) + hr(⟨3e + 1, y⟩). ∑ ∑ e+2 ν(i)≤ y ≤ e+2 ν(i) 0≤i≤e−1 0≤i≤e

Then gr is a strictly increasing computable function such that for all e, maxUe,r < gr(e).

In the r-th construction, fix such a gr first, and for each e, we try to ensure †† C† C† ∅ (e) = ∆ gr(e) (e) with the actual use δ (e) = maxUe,r. In the following formal construction and verification, assume that we are in the r0-th construction, which is provided by recursion theorem. Then the injective computable functions kr0 , hr0 , gr0 are all correct, we denote them respectively as k, h, g for simplicity.

3.1.5 Construction

2 Recall that se is the ﬁrst stage at which there is an ordinal α < ω such that ↓ ≥ · χ(e, α)[se] . Assume se e, and let βe,se = ω ise + jse be the least such an α. At stage s ≥ se, let βe,s = ω · ie,s + je,s be the least ordinal such that χ(e, βe,s)[s] ↓, and we assume ie,s, je,s ≤ s. Assume that there is at most one element ≤ s enumerating into ∅′ at each stage s.

Construction of C.

Let C0 := ∅, and all functionals are totally undeﬁned at stage 0.

52 Chapter 3. A bounded-high set which is high, but not superhigh

Stage s + 1:

(1) Coding ∅′. If there is an i ≤ s + 1 such that i ∈ ∅′[s + 1] − ∅′[s] with ΓC⊕W C (i)[s] ↓, then put γC⊕W C (i)[s] into C. Thus, ΓC⊕W C (i) is undeﬁned.

†† (2) Coding ∅ . Say Qe requires attention at stage s + 1 if s + 1 ≥ se and one of the following conditions holds.

(Q0) Qe has no coding indicators at the end of stage s.

(Q1) (Q0) fails and Qe has an uncanceled coding indicator c, and some Ri with i ≤ e has its C-restraint r(i, s) ≥ c.

(Q2) (Q0) and (Q1) both fail, and βe,s+1 < βe,s with s + 1 > se, in this case

χ(e, βe,s+1) ≠ χ(e, βe,s).

Find all Qe(e ≤ s + 1) which require attention at stage s + 1, and consider them in order of priority.

Now assume that we arrive at Qe which requires attention.

(2.1) When (Q0) holds.

– If there are no location markers at level ie,s+1, ﬁrst deﬁne a location

marker at level ie,s+1 as in (⋆) of (2.3.1).

– Deﬁne coding indicator and coding marker as in (⋆⋆) of (2.3.1).

† – Deﬁne computation ∆C g(e) (e) via the new coding indicator and coding marker as in (⋆ ⋆ ⋆) of (2.3.1).

Check whether there is a requirement Qj(e < j ≤ s + 1) which requires

attention at stage s + 1. If there is such a Qj, ﬁnd the highest priority one, act correspondingly. Otherwise, go to step (3).

53 Chapter 3. A bounded-high set which is high, but not superhigh

(2.2) When (Q1) holds.

First deﬁne an injury marker to undeﬁne all old convergence computations of † ∆C g(e) (e) as follows:

– pick the least unused number z in ∑ ∑ {⟨3e + 1, y⟩ : e + 2 ν(i) ≤ y ≤ e + 2 ν(i)}; 0≤i≤e−1 0≤i≤e

Y – deﬁne qr0 (z) = 0 and Θr0 (z) = 0 for all oracles Y .

↓ ∅ ↓ By recursion theorem, for all m, φk(z)(m) = 0, Φh(z)(m) = 0, we also choose k(z) < h(z). So h(z) ∈ C†[t] for all t ≥ s + 1. h(z) is the desired injury marker.

† † As we choose g such that h(z) < g(e), ∆C g(e) (e) ↑, thus all ∆C g(j) (j) ↑ with

j ≥ e. Say Qj(j ≥ e) is injured at stage s + 1. Cancel all existed coding

indicators and coding markers of Qj(j ≥ e), go to step (3).

(2.3) When (Q2) holds, βe,s+1 < βe,s. There are two subcases (2.3.1) and (2.3.2).

†† (2.3.1) ie,s+1 < ie,s, the approximation of ∅ (e) now moves to a new level. Cancel

the coding indicator and coding marker at level ie,s. Define a location marker C† at level ie,s+1 to undefine all old definitions of ∆ g(e) (e), and then define a †† coding marker to encode ∅ [s + 1](e), which is χ(e, βe,s+1).

(⋆) Deﬁne the location marker. Pick the least unused number z in ∑ ∑ {⟨3e + 1, y⟩ : e + 2 ν(i) ≤ y ≤ e + 2 ν(i)}, 0≤i≤e−1 0≤i≤e

Y deﬁne qr0 (z) = 0 and Θr0 (z) = 0 for all oracles Y . By recursion theorem, h(z) ∈ C†[t] for all t ≥ s + 1, and h(z) is the desired location marker.

54 Chapter 3. A bounded-high set which is high, but not superhigh

(⋆⋆) Deﬁne the coding indicator and the coding marker. First choose a coding indicator c ∈ ⟨3e+2, N⟩ which is larger than all numbers used so far, and then pick the least unused number x in ∑ ∑ {⟨3e + 1, y⟩ : e + 2 ν(i) ≤ y ≤ e + 2 ν(i)}. 0≤i≤e−1 0≤i≤e

Y ↓⇔ ∈ Deﬁne qr0 (x) = c and Θr0 (x) c / Y for all oracles Y . By recursion † theorem, h(x) ∈ C [t] iﬀ c∈ / Ct for all t ≥ s + 1. h(x) is the desired coding marker with the coding indicator c.

C† (⋆ ⋆ ⋆) Deﬁne ∆ g(e) (e). If χ(e, βe,s+1) = 1, deﬁne

† C g(e) † ∆ (e)[s + 1] = C [s + 1](h(x)) = 1 − Cs+1(c) = 1.

Otherwise, χ(e, βe,s+1) = 0. First enumerate the coding indicator c into † Cs+1, so C [s + 1](h(x)) = 0, and then deﬁne

† C g(e) † ∆ (e)[s + 1] = C [s + 1](h(x)) = 1 − Cs+1(c) = 0.

†† (2.3.2) ie,s+1 = ie,s and je,s+1 < je,s, the approximation of ∅ (e) changes at same

level. Let c be the uncanceled coding indicator at level ie,s, and h(x) be the corresponding coding marker.

If χ(e, βe,s+1) = 1, then χ(e, βe,s) = 0. As

† C g(e) † ∆ (e)[s] = C [s](h(x)) = 1 − Cs(c) = 0,

c ∈ Cs. Now remove c out of C, then

† C g(e) † ∆ (e)[s + 1] = C [s + 1](h(x)) = 1 − Cs+1(c) = 1.

If χ(e, βe,s+1) = 0, then χ(e, βe,s) = 1, and

† C g(e) † ∆ (e)[s] = C [s](h(x)) = 1 − Cs(c) = 1,

So c∈ / Cs. Now enumerate c into C. Then

† C g(e) † ∆ (e)[s + 1] = C [s + 1](h(x)) = 1 − Cs+1(c) = 0.

55 Chapter 3. A bounded-high set which is high, but not superhigh

Check whether there is a requirement Qj(e < j ≤ s + 1) which requires

attention at stage s + 1. If there is such a Qj, ﬁnd the highest priority one, act correspondingly. Otherwise, go to step (3).

C⊕W C (3) Deﬁning Γ . Now we obtain Cs+1. Calculate C-restraints

r(i, s + 1) = max{ψC (i)[t]: t ≤ s + 1 ∧ ΨC (i)[t] ↓}

on each Ri with i ≤ s + 1.

– If there is a r(i, s + 1) such that r(i, s + 1) > r(i, s), then

i ∈ W C [s + 1] − W C [s]

and ΨC (i)[s + 1] is deﬁned with a new computation. If ΓC⊕W C (i)[s] ↓, C⊕W C then according to a basic Pi-strategy, Γ (i) is undeﬁned currently.

When we deﬁne ΓC⊕W C (i) at stage t ≥ s + 1, its use γC⊕W C (i)[t] will be of the form ⟨3e, x⟩ with x larger than all numbers used so far, especially, γC⊕W C (i)[t] > r(i, s + 1).

Find the least y with ΓC⊕W C (y) ↑ currently.

If ΓC⊕W C (y) is never deﬁned before,

– deﬁne ΓC⊕W C (y)[s + 1] = ∅′[s + 1](y) with use ⟨3e, x + 1⟩, where x is the least number larger than all numbers used so far.

Otherwise, let w < s + 1 be the largest stage at which ΓC⊕W C (y)[w] ↓.

– If y ∈ ∅′[w + 1] − ∅′[w], or

∃t ∈ [w, s] such that y ∈ W C [t + 1] − W C [t] with r(y, t + 1) > r(y, t), or

γC⊕W C (y)[w] ≤ γC⊕W C (y − 1)[s],

then deﬁne ΓC⊕W C (y)[s + 1] = ∅′[s + 1](y) with use ⟨3e, x + 1⟩, where x is the least number larger than all numbers used so far.

56 Chapter 3. A bounded-high set which is high, but not superhigh

– Otherwise, γC⊕W C (y)[w] > γC⊕W C (y−1)[s], just deﬁne ΓC⊕W C (y)[s+1] = ∅′[s + 1](y) with use γC⊕W C (y)[w].

This ends the construction of C at stage s + 1.

3.1.6 Veriﬁcation

Here, we verify the r0-th construction which is provided by the recursion theorem.

Recall that we have already deﬁned computable function g(e), which is gr0 (e), in Q ≥ C† a basic e-strategy with g(e) δ (e) = maxUe,r0 , where Ue,r0 is the set of injury markers, location markers and coding markers of Qe deﬁned in the r0-th construction.

Lemma 3.1 For each requirement, Q say,

(1) Q is injured at most ﬁnitely often;

(2) Q acts at most ﬁnitely often and is satisﬁed;

(3) Q injures all lower priority strategies ﬁnitely often.

Proof: We will prove the lemma by induction on e that all Re, Pe, and Qe are true.

For e = 0. R0 has the highest priority, it is not injured. (1) holds. If 0 ∈ W C [s]−W C [s−1] at some first stage s, then r(0, s) = ψC (0)[s] and the computation C Ψ (0)[s] is never destroyed later [indeed, after stage s, all Pj and Qj with j ≥ 0 can C only deal with witnesses larger than r(0, s)]. Then 0 ∈ W [t] for all t ≥ s. So R0 is satisfied, (2) holds. R0 acts at most once by setting C-restraints, thus initializes lower priority strategies at most once. (3) holds. C C For P0. P0 is injured by R0 at most once when 0 ∈ W [s] − W [s − 1] at some first stage s, (1) holds. In this case, ΓC⊕W C (0) ↑, we will define it with a big use later. Let w0 be the least stage such that P0 is not injured after stage w0. ′ ′ After stage w0, P0 acts at most once if 0 ∈ ∅ [t] − ∅ [t − 1] at some stage t with C⊕W C C⊕W C Γ (0)[t − 1] ↓. In this case, γ (0)[t − 1] is enumerated into Ct, and we will define ΓC⊕W C (0) = 1 with a large use later. So γC⊕W C (0) increases at most twice

57 Chapter 3. A bounded-high set which is high, but not superhigh

C⊕W C after its first definition, there is a stage w1 ≥ w0 such that Γ (0)[w1] is defined and is never undefined later. (2) holds. P0 never acts after stage w1. (3) holds.

For Q0. Recall that we use s0 to denote the first stage at which there is an 2 ↓ · ordinal α < ω such that χ(0, α)[s0] , and let β0,s0 = ω is0 + js0 be the least such ordinals. We will appoint a coding indicator c at level is0 , and then define corresponding coding marker h(z). Define

† † C g(0) − ∆ (0)[s0] = C [s0](h(z)) = 1 Cs0 (c) = χ(0, β0,s0 ) for first time. C C When 0 ∈ W [s] − W [s − 1] at some first stage s ≥ s0 with r(0, s) bigger than coding indicators of Q0, Q0 is injured. We will enumerate an injury marker into † C† C to undefine all old definitions of ∆ g(0) (0). Note P0 may act at most once by enumerating γC⊕W C (0) into C, but this enumeration has no essential influences to C† the definition of ∆ g(0) (0) by the choice of Γ-uses and coding indicators. So Q0 is injured at most once by R0. (1) holds.

Let t0 ≥ s0 be the least stage such that Q0 is not initialized after stage t0, and let β0,t = ω · i0,t + j0,t be the least ordinal such that χ(0, β0,t)[t] ↓ at stage t ≥ s0.

After stage t0, if β0,s < β0,s−1 via i0,s < i0,s−1, then all the previous deﬁnitions of † ∆C g(0) (0) are injured at stage s, because we enumerate a location marker at level † ≥ i0,s into C . Now assume that after stage t1 t0, i0,t never decreases, i.e., i0,t = i0,t1 for all t ≥ t1 (such a t1 exists since limβ0,s exists). s ′ ′ Let c be the coding indicator at level i0,t1 and h(z ) be the corresponding coding ′ ′ † ′ marker. We deﬁne h(z ) such that h(z ) ∈ C [s] ⇔ c ∈/ Cs for all stage s ≥ t1. At stage t1, we maintain

† † ′ ′ C g(0) − ∆ (0)[t1] = C [t1](h(z )) = 1 Ct1 (c ) = χ(0, β0,t1 ).

′ When j0,t < j0,t−1 at some stage t > t1, we will enumerate c into Ct if χ(0, β0,t−1) = ′ 1, and extract c out of Ct−1 if χ(0, β0,t−1) = 0, keeping

† C g(0) † ′ ′ ∆ (0)[t] = C [t](h(z )) = 1 − Ct(c ) = χ(0, β0,t).

≥ ≥ Let t2 t1 be the least stage such that j0,t = j0,t2 for all t t2 (again, such a t2 exists since limβ0,s = β0; in this case, β0 = ω · i0,t + j0,t ). Then for all t ≥ t2, s 1 2

58 Chapter 3. A bounded-high set which is high, but not superhigh

∅†† Q β0,t = β0,t2 and (0) = χ(0, β0,t). So 0 never acts after stage t2. Hence,

† † ′ ′ †† C g(0) − ∅ ∆ (0) = C [t2](h(z )) = 1 Ct2 (c ) = χ(0, β0,t2 ) = (0).

We have showed that Q0 is satisfied after finitely many actions, and thus injures lower priority strategies finitely often. (2) and (3) hold. 2 Recall that sl is the first stage at which there is an ordinal α < ω such that

χ(l, α)[sl] ↓. Let βl,s = ω · il,s + jl,s be the least ordinal such that χ(l, βl,s)[s] ↓ at stage s ≥ sl.

Assume that the lemma is true for Ri, Pi and Qi with i < e. Let ve > max{si : i ≤ e} be the least stage such that for i < e,

• limr(i, s) = r(i, ve), s

C⊕W C • Γ (i)[ve] is deﬁned and is never undeﬁned after stage ve,

† • C g(i) ∆ (i)[ve] is deﬁned with value χ(i, βi,ve ) and is never changed later.

C⊕W C Then no Ri with i < e has C-restraints r(i, s) increase after stage ve, limγ (i)[s] = s C⊕W C γ (i)[ve] for all i < e, and limβi,s = βi,v for all i < e. s e For Re. As no Γ-uses or coding indicators of positive requirements are enumerated into or removed out of C after stage ve, Re is not injured after stage ve. (1) holds. After stage ve, there are two cases.

(i) r(e, s) never increases after stage ve. Then limr(e, s) = r(e, ve), Re is satisﬁed s and never acts after stage ve.

C C (ii) There is a least stage t > ve at which e ∈ W [t] − W [t − 1] with new computation. Then r(e, t) > r(e, s′) for all s′ ≤ t−1. During the construction, no positive requirements with lower priority will enumerate numbers ≤ r(e, t) into C or extract numbers ≤ r(e, t) from C after stage t, so ΨC (0)[t] is never C ′ ′ injured after stage t. Hence, e ∈ W [s ] for all s ≥ t. Re acts at most once

after stage ve and is satisﬁed.

59 Chapter 3. A bounded-high set which is high, but not superhigh

Hence, (2) and (3) hold.

Similar to P0 and Q0 below R0. After stage ve, for Pe and Qe below Re, Pe is injured at most once, and acts at most once by enumerating γC⊕W C (e) into C when ′ e goes into ∅ ; Qe is injured at most once, and will be satisﬁed after ﬁnitely many actions. So the lemma holds for Pe and Qe.

Lemma 3.2 For all x, |{s : Cs(x) ≠ Cs+1(x)}| ≤ x + 1. Thus C is ω-c.e.

Proof: We can eﬀectively check whether x is a Γ-use or a coding indicator or not. If no, Cs(x) = 0 for all stages s. Assume that x is a Γ(i)-use for some i. It enumerates into C at most once when ′ i goes into ∅ , so |{s : Cs(x) ≠ Cs+1(x)}| ≤ 1.

Assume that x is a coding indicator of some Qe appointed at stage s. Consider the least ordinal βe,s = ω·ie,s +je,s such that χ(e, βe,s)[s] ↓. Then x > je,s, and x may be enumerated into Cs. At stage t > s, Ct(x) ≠ Ct−1(x) only if je,t < je,t−1 ≤ je,s.

So |{s : Cs(x) ≠ Cs+1(x)}| ≤ 1 + je,s ≤ x. †† † ′ C C ′ By Lemma 3.1, ∅ ≤wtt C , ∅ ≤T C ⊕ W , and W ≤T ∅ . By Lemma 3.2, C ′ C is ω-c.e. So C is a bounded-high set with C ⊕ W ≡T ∅ . This completes the proof of Theorem 3.1.(1).

3.2 Pseudo-jump inversion via bounded-low c.e. sets

To prove Theorem 3.1.(2), for the c.e. operator W , we need to build a bounded-low † C ′ † c.e. set C (i.e., C is ω-c.e.) such that C ⊕ W ≡T ∅ . As C will be c.e., C has the following natural computable approximations:

{ C φn(x) 1, if ∃n ≤ x[φn(x)[s] ↓ ∧ Φx (x)[s] ↓]; f † (x, s) := C 0, otherwise.

We only need to add negative requirements of the form

2 Ne : |{s : fC† (e, s) ≠ fC† (e, s + 1)}| ≤ 2(e + 1) + 1

60 Chapter 3. A bounded-high set which is high, but not superhigh to the standard pseudo-jump inversion theorem. Then C will be bounded-low. We will build a functional Γ such that ΓC⊕W C = ∅′. If e enumerates into ∅′, C⊕W C C⊕W C C ′ then rectify Γ (e) = 1 by enumerating γ (e) into C. To make W ≤T ∅ C 0 ∈ C C (i.e., W is ∆2), as usual, we will preserve e W by setting a C-restraint ψ (e) if ΨC (e) ↓. When deﬁning ΓC⊕W C , the Γ-uses satisfy the following rules.

(i) For each e, and stage s, if ΓC⊕W C (e) is newly deﬁned at stage s, its use γC⊕W C (e)[s] is deﬁned to be larger than all numbers used before, especially, γC⊕W C (e)[s] > e.

(ii) For each e, and stage s, if ΓC⊕W C (e)[s] ↓, then ΓC⊕W C (e)[t] is undeﬁned at some stage t > s only if

̸ ∨ C ̸ C Ct γC⊕W C (e)[s]= Cs γC⊕W C (e)[s] W [t] γC⊕W C (e)[s]= W [s] γC⊕W C (e)[s] .

(iii) For e < e′ and stage s, if ΓC⊕W C (e)[s] and ΓC⊕W C (e′)[s] are both deﬁned, then

γC⊕W C (e)[s] < γC⊕W C (e′)[s].

C⊕W C ′ (iv) For each e, there is a stage se such that for all s ≥ se,Γ (e)[s] ↓= ∅ [s](e) C⊕W C C⊕W C with use γ (e)[s] = γ (e)[se].

′ C⊕W C ′ C This ensures that ∅ = Γ , and thus ∅ ≤T C ⊕ W .

As in the standard pseudo-jump inversion theorem, the deﬁnition of ΓC⊕W C (e) is compatible with preserving W C (e). † Now consider the bounded-low requirement Ne. In order to preserve C (e), at each stage s, we need to preserve a C-restraint

N r (e, s) = max{φn(e)[s]: n ≤ e ∧ φn(e)[s] ↓}.

C⊕W C C⊕W C N C⊕W C If Γ (e)[s] ↓ with γ (e)[s] ≤ r (e, s), Ne will act to enumerate γ (e)[s] into C, thus making ΓC⊕W C (e) undeﬁned; in this case, ΓC⊕W C (e) will be redeﬁned

61 Chapter 3. A bounded-high set which is high, but not superhigh

N N with a new use > r (e, s). Since r (e, s) can move up at most e + 1 times, Ne will enumerate at most e + 1 many such uses. C φn(e) Now computations of the form Φe (e)[s] ↓ (of course, φn(e)[s] ↓) for some n ≤ C⊕W C e are injured at stage s+1 only if there is some i ≤ e such that γ (i)[s] ≤ φn(e) C⊕W C is enumerated into Cs+1. Now Γ (i) ↑ currently, it will be deﬁned with new use C⊕W C > φn(e). So the enumeration of γ (i) later will not injure a new computation C φn(e) C⊕W C C φn(e) of the form Φe (e), and the enumeration of γ (i)-uses can injure Φe (e) at most once for ﬁxed i and n. Thus, approximations for C†(e) are injured at most (e + 1)2 many times. Note that each such an injury entails at most two changes of fC† (e, s). So 2 |{s : fC† (e, s) ≠ fC† (e, s + 1)}| ≤ 2(e + 1) + 1.

Hence, C† is ω-c.e., i.e., C is bounded-low. This completes the description of basic strategies and possible interactions between them. We omit the remaining construction and veriﬁcation, which are ﬁnite injury arguments. This completes the proof of Theorem 3.1.(2).

62 Part II

Degrees of orders on torsion-free abelian groups

63 Chapter 4

Group-order-computable degrees and PA degrees

In Proposition 1.5, we have proved that PA degrees and degrees which are not weakly group-order-computable coincide. Then all PA degrees are not group-order- computable. In this chapter, we will show that the converse is also true, that is, degrees which are not PA are group-order-computable.

Theorem 4.1 There is a computable torsion-free abelian group G with inﬁnite rank such that for any set A, G admits exactly two A-computable orders iﬀ A is not PA- complete.

For G above, as any computable set is not PA-complete, G admits exactly two computable orders. Then for a set A which is not PA-complete, every A-computable order on G is computable; for a set A which is PA-complete, A computes a noncomputable order on G . Let a be a degree which is not group-order-computable. Then for the group G in Theorem 4.1, a computes a noncomputable order on G . Thus, a is a PA degree. Now we obtain the following characterizations of PA degrees.

Proposition 4.1 Let a be a Turing degree. The following are equivalent.

(1) a is a PA degree.

(2) a is not weakly group-order-computable.

(3) a is not group-order-computable.

64 Chapter 4. Group-order-computable degrees and PA degrees

Note that 0′ is the only c.e. degree which is PA. An immediate corollary of Theorem 4.1 is:

Corollary 4.1 There is a computable torsion-free abelian group G with inﬁnite rank such that

(1) G has exactly two computable orders;

(2) for any c.e. degree a < 0′, every a-computable order on G is computable.

The following sections are devoted to prove Theorem 4.1.

4.1 Requirements

To prove Theorem 4.1, we will build a computable group G and a computable order

≤G on G to meet the following requirements.

∼ ∞ P : G = Q .

∗ Q : ≤G and ≤G are exactly two computable orders on G .

R : For any set A,

A A A ∗ (1) if ≤ is an A-computable order on G , then either ≤ =≤G , or ≤ =≤G , or A is PA-complete;

(2) if A is PA-complete, then there is an A-computable order on G which is not computable.

∞ ∗ Here, Q is the direct sum of inﬁnitely many copies of Q. ≤G denotes the reversal ∗ order of ≤G , that is, for x, y ∈ G , x ≤G y ⇔ y ≤G x. If three requirements above are satisﬁed, then G is a computable group with exactly two computable orders. Moreover, for any set A, if A is not PA-complete, then every A-computable order on G is computable, so there are exactly two A- computable orders on G ; if A is PA-complete, then there are A-computable orders which are not computable, so there are more than two A-computable orders on G .

65 Chapter 4. Group-order-computable degrees and PA degrees

4.2 Background

We now review a few concepts on binary strings which are needed for constructing G . Let 2<ω be the set of binary strings. For each binary string σ, the symbol |σ| denotes the length of σ. For the empty string λ, |λ| = 0. For a nonempty string σ, |σ| ≥ 1; in this case, write σ := σ(0) ··· σ(|σ| − 1) and for 1 ≤ i ≤ |σ|, let

σ i:= σ(0) ··· σ(i − 1). Set σ 0:= λ. <ω 2 has a natural lexicographical order ≤lex deﬁned as follows: 0

σ(i)

σ ≤len−lex τ ⇔ |σ| < |τ| ∨ (|σ| = |τ| ∧ σ ≤lex τ).

Figure 4.1: 2<ω ∪ 2<ω@

Let 2<ω@ be the set of symbols σ@ with σ a binary string. Deﬁne a lexicograph- <ω <ω ical order ≤lex on 2 ∪ 2 @ based on 0

σ@ ≤len−lex τ@ ⇔ |σ| < |τ| ∨ (|σ| = |τ| ∧ σ@ ≤lex τ@).

66 Chapter 4. Group-order-computable degrees and PA degrees

4.3 The P-strategy

We will build a computable group G stage by stage such that G is isomorphic to Q∞ 0 via a ∆2-isomorphism. During the construction, we maintain a finite set Bs at each stage s which serves as an approximation of an infinite basis B of G . Then ∼ ∼ ∞ G = ⊕ Qb = Q . b∈B Now we start to define G . For each σ@ ∈ 2<ω@, we introduce a base-element qσ@ for G . qσ@ will be enumerated into the constructed group at stage |σ|. qσ@ has two totally different fates.

(1) If the R-strategy adds a dependence relation of the form qσ@ = rqτ@ for some

positive rational r and some qτ@ with τ@ |σ|,

then qσ@ is said to be non-active at stage t. In this case, qσ@ ∈/ Bt′ for all t′ ≥ t.

(2) If no dependence relations of the form qσ@ = rqτ@ are added during stages in

[|σ|, t], qσ@ is said to be active at stage t. In this case, qσ@ ∈ Bt.

Hence, either qσ@ is active forever in which case qσ@ ∈ B, or qσ@ becomes non-active at some stage > |σ| in which case qσ@ ∈/ B.

Let H be the computable additive group ⊕ Qqσ@. Elements of H are just σ∈2<ω <ω ﬁnite Q-linear sums of elements in {qσ@ : σ ∈ 2 }. The constructed group G is indeed a quotient group of H .

Fix a computable enumeration Hs of H such that at most one element is enu- H ··· H merated into s at stage s, and for each nonzero x0qσ0@ + + xmqσm@ in s, it satisﬁes xm ≠ 0, and for each i ≤ m, |σi| ≤ s. Assume that H0 = {0H }.

Construction of G = (G, +, 0G ).

At stage 0.

G0 = {0G , q@}, where 0G is the zero element of G and q@ is the active base- element at stage 0. B0 = {q@}.

At the end of stage s, Gs is the set containing all elements of G enumerated by stage s. Bs is the set of active base-elements at stage s. Each nonzero g ∈ Gs can

67 Chapter 4. Group-order-computable degrees and PA degrees be expressed as a unique Q-linear sum of the form

··· r0qσ0@ + + rnqσn@ of active base-elements at stage s with rn ≠ 0 and σ0@

At stage s + 1.

Deﬁne domain Gs+1. There are two cases depending on whether the R-strategy adds dependence relations.

Case 1. If no dependence relations are added among active base-elements in Bs by the R-strategy at stage s + 1, then each base-element in Bs is still active at stage s + 1. For each σ with length s + 1, add a new base-element qσ@ into Bs+1. That is,

Bs+1 = Bs ∪ {qσ@ : |σ| = s + 1}.

H H ∪ ··· If s+1 = s, let Gs+1 = Gs Bs+1. If there is an element, x0qσ0@ + +xmqσm@ say, in Hs+1 − Hs. Note that xm ≠ 0, and for each i ≤ m, |σi| ≤ s + 1.

(1) If there are qσi@s such that qσi@ is non-active at stage s + 1, just update the Q -linear sum by replacing such qσi@s by active base-elements at stage s + 1 with corresponding scaling.

Q (2) Otherwise, each qσi@ in the -linear sum is active at stage s + 1.

Q ··· In both cases, the -linear sum has a new expression, say a0qτ0@ + + anqτn@, as a Q-linear sum of elements in Bs+1. If all ai with i ≤ n are 0, then the Q-linear sum is indeed the zero element of G , just let Gs+1 = Gs ∪ Bs+1. Otherwise, ﬁnd the largest m ≤ n with am ≠ 0, there are two cases:

68 Chapter 4. Group-order-computable degrees and PA degrees

··· G (1) If a0qτ0@ + + amqτm@ is not an expression of elements enumerated into ∪ ∪ { ··· } before, then let Gs+1 = Gs Bs+1 a0qτ0@ + + amqτm@ .

(2) Otherwise, let Gs+1 = Gs ∪ Bs+1.

Case 2. If there are two active base-elements, say qσ@, qτ@ in Bs, such that the pair (σ@, τ@) is enumerated into link at stage s + 1, i.e., the relation link(σ@, τ@) holds at stage s + 1 [here, link is a c.e. equivalence relation on 2<ω@ which will be deﬁned in the basic R-strategy].

We will add dependence relations of the form qη@ = rqρ@ with r > 0, ρ@

η@, i.e., |ρ@| < |η@| or |ρ@| = |η@| with ρ@

Now add new active base-elements qσ@ with |σ| = s + 1 at stage s + 1. Let Bs+1 be the set of active base-elements at stage s + 1 (the concert deﬁnition of Bs+1 will be clear in the following R-strategy).

If there is a new x enumerating into Hs+1, ﬁrst update x as a Q-linear sum of active base-elements at stage s + 1, and then add this Q-linear sum into Gs+1 as in case 1.

This ends the construction of Gs+1.

Deﬁne the partial addition +s+1 on Gs+1. Note that each element of Gs+1 has a unique Q-linear sum of active base-elements in Bs+1. For x, y, z ∈ Gs+1, if the addition of the unique Q-linear sum of x and y is the same as that of z, then let x +s+1 y = z.

If we already deﬁned x +s y = z at stage s, then x +s+1 y = z. Indeed, after substituting one dependence relation of the form qη@ = rqρ@ into the old expression of x, y, z, we obtain the new expression of them as a unique Q-linear sum of current active base-elements respectively. It is a direct checking that the new expression

69 Chapter 4. Group-order-computable degrees and PA degrees of them also satisﬁes the condition that the addition of the new expression of x, y equals that of z.

This ends the construction of +s+1.

4.4 The Q-strategy

Due to the one-to-one correspondence between orders and positive cones on computable groups, when building an order on a group, it is enough to specify the set of positive elements or alternatively to specify the sign of elements in the group. We build ≤G by specifying the sign of elements in G under ≤G . That is, for each g ∈ G , the sign of g is   +1, if g >G 0G sign≤ (g) := G  0, if g = 0G −1, if g

At each stage s, we approximate ≤G through a type of lexicographical order on

Gs in the following sense. The sign of 0G is always 0. Now assign the sign of nonzero elements in Gs. First, each nonzero element g ∈ Gs has a unique Q-linear sum of active base-elements of the form

··· r0qσ0@ + r1qσ1@ + + rnqσn@ with rn ≠ 0 and σ0@

ﬁnd the least m with rm ≠ 0 in the expression, and then deﬁne the sign of g to be the usual sign of rm as rational numbers (i.e., for r ∈ Q, if r > 0, it has sign +1; if r = 0, it has sign 0; if r < 0, it has sign -1). During the construction, we maintain the sign of elements unchanged.

Construction of sign≤G .

At stage 0.

G0 = {0G , q@}. Let the sign of 0G be 0 and the sign of q@ be +1.

At the end of stage s, we have already appointed the sign of elements in Gs.

Now deﬁne the sign of elements in Gs+1.

70 Chapter 4. Group-order-computable degrees and PA degrees

At stage s + 1.

Case 1. If no dependence relations are added via the R-strategy, then the set of active base-elements at stage s + 1 is Bs ∪ {qσ@ : |σ| = s + 1}. The sign of elements in Gs is unchanged at stage s + 1. | | For each new added active base-element qσ@ with σ = s + 1, let sign≤G (qσ@) =

+1. If there is one more added element, x say, in Gs+1. Then x is of the form ··· ̸ ··· a0qσ0@ + a1qσ1@ + + amqσm@ with am = 0 and σ0@

Case 2. If there are two active base-elements, say qσ@, qτ@ in Bs, such that the pair (σ@, τ@) is enumerated into link at stage s + 1, i.e., the relation link(σ@, τ@) holds at stage s + 1, we will add dependence relations to satisfy the R-strategy. To make sign≤G computable, we will ensure that such dependence relations added at stage s + 1 will not change the sign of elements in Gs.

Without loss of generality, let σ@

(1) σ@ ≤lex η@ ≤lex τ@;

(2) there is an element g ∈ Gs and a nonzero coeﬃcient r such that rqη@ appears

in the unique expression of g as a Q-linear sum of elements in Bs.

Now list all such η@s under the lexicographical order as

σ@ = ξ0@

By the deﬁnition of link, link(σ@, τ@) holds implies for all i, j ≤ n, link(ξi@, ξj@) holds. In the following R-strategy, we will add dependence relations of the form

qη@ = rqρ@ with proper r > 0 and ρ@

··· ··· g = + rξ0 qξ0@ + rξ1 qξ1@ + .

71 Chapter 4. Group-order-computable degrees and PA degrees

Let { ξ0@, if ξ0@

qγ0@ = c0qδ0@ with proper c0 > 0 such that the sign of elements in Gs does not change after substituting this relation.

When ξ0@

··· ··· + (rξ0 + rξ1 c0)qξ0@ + 0qξ1@ + .

(1) If rξ0 = 0, then the sign of rξ1 equals the sign of rξ1 c0.

̸ (2) If rξ0 = 0, then we will choose c0 such that the sign of rξ0 equals the sign of

rξ0 + rξ1 c0.

Then the sign of g computed according to the new expression is the same as before.

When ξ1@

··· ··· + 0qξ0@ + (rξ0 c0 + rξ1 )qξ1@ + .

(1) If rξ0 = 0, then the expression of g does not change.

̸ (2) If rξ0 = 0, then we will choose c0 such that the sign of rξ0 equals the sign of

rξ0 c0 + rξ1 .

So the sign of g does not change after substituting the relation.

Now each element in Gs has a unique expression as a Q-linear sum of elements − { } in Bs qγ0@ and there are no active base-elements qζ@ with δ0@

We then deal with qδ0@ and qξ2@ in a similar manner as dealing with qξ0@ and ≤ qξ1@. Continue this process until all qξi@ with i n are considered. Once adding a dependence relation, we preserve the sign of elements in Gs. So the whole process does not change the sign of elements in Gs.

72 Chapter 4. Group-order-computable degrees and PA degrees

In general, for 1 ≤ i ≤ n − 1, deﬁne by induction that δi@ is the length- lexicographically smaller element in {δi−1@, ξi+1@} and that γi@ is the other element in {δi−1@, ξi+1@}. Now the set of active base-elements is

− { ≤ ≤ − } ∪ { | | } Bs+1 = (Bs qγi@ : 0 i n 1 ) qσ@ : σ = s + 1 .

The sign of each new active base-element qσ@ is +1, and the sign of the new added element (if such an element exists) is determined as in case 1.

4.5 The R-strategy

The R-requirement says that for any set A,

A A (1) if the constructed group G has an A-computable order, say ≤ , with ≤ ≠ ≤G A ∗ and ≤ ≠ ≤G , then A is PA-complete;

(2) if A is PA-complete, then G has an A-computable order which is neither ≤G ∗ nor ≤G .

4.5.1 Background on PA-complete sets

We will use an equivalent deﬁnition for a set being PA-complete that A is PA- complete iﬀ it computes a {0, 1}-valued diagonally nonrecursive function, i.e., a

{0, 1}-valued function f such that for all x, if φx(x) ↓, then f(x) ≠ φx(x). For a binary tree T , an infinite binary string f is an infinite path on T if for all n, its prefix f(0)f(1) ··· f(n), denoted by f n+1, is in T ; set f 0:= λ, the empty string. We view f as a {0, 1}-valued function (i.e., the characteristic function of {n ∈ N : f(n) = 1}). We use [T ] to denote the set of infinite paths of T . One can build a co-c.e. binary tree T (i.e., the complement of T is c.e.) such that [T ] is just the set of {0, 1}-valued diagonally nonrecursive functions. Then for a set A, A is PA-complete iff it computes an infinite path of T . Now based on this T , to satisfy R, we only need to meet the following R′:

′ A R : For any set A, if ≤ is an A-computable order on G which is neither ≤G nor ∗ ≤G , then A computes an inﬁnite path on T . If f ∈ [T ], then f computes an order on G which is not computable.

73 Chapter 4. Group-order-computable degrees and PA degrees

We now construct the co-c.e. tree T by stages.

The construction of T . <ω At stage 0. T0 = 2 , the full binary tree. F0 = ∅, the complement of T0.

At the end of stage s, we already obtained approximations Ts and Fs such that <ω Ts ∪ Fs = 2 .

At stage s+1. For each σ ∈ Ts of length ≤ s + 1, if there is a x < |σ| such that φx,s+1(x) is deﬁned with value σ(x), then enumerate σ into Fs+1. Let Ts+1 = <ω 2 − Fs+1. This ends the construction of T at stage s + 1.

Clearly, T = ∩ Ts is a co-c.e. set. For any binary string σ, σ ∈ T iff it satisfies s∈N ∀x < |σ| [ φx(x) ↓→ φx(x) ≠ σ(x)], then any initial segment of σ is also in T . So T is a tree. Furthermore, f ∈ [T ] iff f is a {0, 1}-valued diagonally nonrecursive function. By using infinite paths on T , we will define an equivalence relation on 2<ω@ <ω <ω <ω <ω (recall that 2 @ = {σ@: σ ∈ 2 }). The lexicographical order ≤lex on 2 ∪ 2 @ is already defined in section 4.2, where for any binary string σ, σ0

• say that f is lexicographically less than σ@, denoted by f

is an i < |σ| + 1 such that f i= σ@ i and f(i)

lexicographically less than f, denoted by σ@

Hence, if f

Deﬁnition 4.1 Let T be the co-c.e. tree, where [T ] is the set of {0, 1}-valued diagonally nonrecursive functions. For σ@, τ@ ∈ 2<ω@, link(σ@, τ@) holds if there is no f ∈ [T ] such that f is lexicographically between σ@ and τ@.

We use link to denote the set of pairs (σ@, τ@) such that link(σ@, τ@) holds. One can directly check that link is an equivalence relation on 2<ω@.

Proposition 4.2 link is a c.e. equivalence relation.

74 Chapter 4. Group-order-computable degrees and PA degrees

Proof: Fix a computable approximation Ts of T (as in the construction of T above) such that Ts+1 ⊆ Ts and T = ∩ Ts. s∈N <ω For σ@, τ@ ∈ 2 @, assume that σ@

<ω ∃ n ∃ s ∀ ρ ∈ 2 [(|ρ| = n ∧ σ@

0 So link is Σ1.

4.5.2 The R′-strategy

The R′-requirement says that:

A (1) for any set A, if ≤ is an A-computable order on G which is neither ≤G nor ∗ ≤G , then A computes an inﬁnite path on T ;

(2) if f ∈ [T ], then f computes an order on G which is not computable.

In order to satisfy (1) of the R′-requirement, we will initiate an A-effective process for searching nodes in 2<ω@ which approximates an infinite path on T lexicographically form below and above. Our method is to add dependence relations of the form qη@ = rqρ@, where r > 0, link(η@, ρ@) holds, and ρ@ 0, and then no new expressions are assigned to qτ@. Since an element, x say, is enumerated into Gs for first time is always of the form

··· x0qσ0@ + + xmqσm@ of a Q-linear sum of active base-elements at stage s, there is a large enough stage ≤ ≤ after which no more expressions are assigned to all qσi@ with 0 i m. Then x will arrive at a ﬁnal expression as a Q-linear sum of active base-elements.

75 Chapter 4. Group-order-computable degrees and PA degrees

When adding qη@ = rqρ@ at stage s + 1, we need to ensure the following two conditions:

(1) the assignment of Q-linear sums to elements in Gs remains one-to-one (i.e., the dependence relation is compatible with the P-strategy of constructing G );

(2) the sign of elements in Gs does not change (i.e., the dependence relation is

compatible with the Q-strategy of constructing ≤G ).

Adding dependence relations.

Since link is a c.e. equivalence relation on 2<ω@, just ﬁx a computable enumeration links of link such that at each stage s there is at most one pair of the form

(σ@, τ@) with |σ|, |τ| < s enumerating into links.

At stage 0. link0 = ∅. q@ is the only active base-element at stage 0.

Assume that we have already obtained Bs, the set of active base-elements at stage s. Each element in Bs is of the form qτ@ with |τ| ≤ s. Each g ∈ Gs has a unique Q-linear sum of elements in Bs. By convention, the unique Q-linear sum for

0G is just the empty sum.

At stage s+1.

Case 1. If links = links+1, or (σ@, τ@) ∈ links+1 − links, where at least one of qσ@ and qτ@ is non-active at stage s, then no dependence relations are added at stage s + 1. Just let Bs+1 = Bs ∪ {qσ@ : |σ| = s + 1}.

Case 2. Suppose that there is a pair (σ@, τ@) of active base-elements at stage s enumerating into links+1. We will add dependence relations between qσ@ and qτ@.

Without loss of generality, we assume that σ@

First, ﬁnd all qη@ ∈ Bs such that

(1) σ@ ≤lex η@ ≤lex τ@;

(2) there is an element g ∈ Gs and a nonzero coeﬃcient r such that rqη@ appears

in the unique expression of g as a Q-linear sum of elements in Bs.

76 Chapter 4. Group-order-computable degrees and PA degrees

Second, list all such η@s under the lexicographical order as follows:

σ@ = ξ0@

Since link(σ@, τ@) holds, for all i, j ≤ n, link(ξi@, ξj@) holds.

We start to add dependence relations between qξ0@ and qξ1@. Now each element of Gs has a unique Q-linear sum in which no active base-elements qζ@ with ξ0@

ζ@

Let δ0@ be the length-lexicographically less element between ξ0@ and ξ1@, and

γ0@ be the other element. In module D(0), we will add a relation of the form qγ0@ = c0qδ0@ with proper c0 > 0 such that after substituting this relation, the new Q − { } expressions of elements in Gs as -linear sums of elements in Bs qγ0@ remain one-to-one and the signs of elements in Gs computed according to new expressions remain unchanged.

Module D(0).

∈ Q 1. List all nonzero r where either rqδ0@ or rqγ0@ occurs in the unique expres-

sion of an element in Gs as a Q-linear combination of active base-elements at stage s as follows:

r1, r2, ··· , rl.

+ ′ ′ 2. Set r0 := 0. Choose a large enough k ∈ N such that for all i, i , j, j ≤ l with

rj − rj′ ≠ 0, |r − r ′ | k > i i . |rj − rj′ |

2.1. If δ0@

··· ··· g = + rδ0 qδ0@ + rγ0 qγ0@ + .

1 (1) Add a new dependence relation qγ0@ = k qδ0@. Declare qγ0@ is non-active at stage s + 1.

77 Chapter 4. Group-order-computable degrees and PA degrees

1 (2) We now show that adding qγ0@ = k qδ0@ does not change the sign of

elements in Gs.

1 By substituting qγ0@ = k qδ0@, g has a new expression r g = ··· + (r + γ0 )q + 0q + ··· . δ0 k δ0@ γ0@

rγ0 When rδ0 = 0. Since k is positive, the sign of rγ0 equals the sign of k .

r ̸ γ0 When rδ0 = 0. We show that the sign of rδ0 equals the sign of rδ0 + k .

r ≥ γ0 (i) rδ0 > 0. Since k > 0, if rγ0 0, then rδ0 + k > 0. If rγ0 < 0, by the − rγ0 rγ0 choice of k, k > , that is, rδ0 + > 0. rδ0 k

r ≤ γ0 (ii) rδ0 < 0. Since k > 0, if rγ0 0, then rδ0 + k < 0. If rγ0 > 0, since

rγ0 rγ0 k > − , rδ0 + < 0. rδ0 k

By definition, the current sign of nonzero g ∈ Gs is the sign of the first nonzero rational coefficient in this new expression as a Q-linear sum of current active base-elements. Hence, after adding the dependence relation 1 ∈ qγ0@ = k qδ0@, the current sign of g Gs is the same as before.

1 (3) We now show that adding qγ0@ = k qδ0@ ensures that the assignment of

Q-linear sums to elements in Gs remains one-to-one.

′ ∈ ··· ··· For x, x Gs, suppose that at stage s, x = + aδ0 qδ0@ + aγ0 qγ0@ + , and x′ = ··· + a′ q + a′ q + ··· . If the new expression of x and x′ δ0 δ0@ γ0 γ0@ is the same, then the coeﬃcient of active base-elements at stage s other

than qδ0@, qγ0@ is the same and

a a′ a + γ0 = a′ + γ0 , δ0 k δ0 k

i.e., (a − a′ )k = (a′ − a ). δ0 δ0 γ0 γ0

(i) If a = a′ , then a′ = a . In this case, the old expression of x, x′ δ0 δ0 γ0 γ0 at stage s is the same.

78 Chapter 4. Group-order-computable degrees and PA degrees

a′ −a ′ γ0 γ0 ′ (ii) If aδ ≠ a , then k = ′ . In this case, the old expression of x, x 0 δ0 aδ −a 0 δ0 |a′ −a | γ0 γ0 at stage s is diﬀerent. However, by the choice of k, k > ′ . So |aδ −a | 0 δ0 this case is impossible.

′ Hence, if x, x ∈ Gs have diﬀerent expression as Q-linear sums of active base-elements at stage s, then the new expression of them as Q-linear sums of current active base-elements is still diﬀerent.

2.2. If γ0@

··· ··· g = + rγ0 qγ0@ + rδ0 qδ0@ + .

(1) Add a new dependence relation qγ0@ = kqδ0@. Declare qγ0@ is non-active at stage s + 1.

(2) We ﬁrst show that adding qγ0@ = kqδ0@ does not change the sign of

elements in Gs.

By substituting qγ0@ = kqδ0@, g has a new expression

··· ··· g = + 0qγ0@ + (rδ0 + rγ0 k)qδ0@ + .

If rγ0 = 0, then the expression of g does not change. ̸ Suppose rγ0 = 0. There are two cases: −r ≥ δ0 (i) rγ0 > 0. If rδ0 0, then rδ0 + rγ0 k > 0. If rδ0 < 0, since k > , rγ0

rδ0 + rγ0 k > 0.

r ≤ δ0 (ii) rγ0 < 0. If rδ0 0, then rδ0 + rγ0 k < 0. If rδ0 > 0, since k > − , rγ0

rδ0 + rγ0 k < 0.

The current sign of g is calculated as the sign of the first nonzero rational coefficient in this new expression of g if such a nonzero coefficient exists.

Hence, the current sign of g ∈ Gs is the same as before.

79 Chapter 4. Group-order-computable degrees and PA degrees

(3) We now show that adding qγ0@ = kqδ0@ ensures that the assignment of

Q-linear sums to elements in Gs remains one-to-one.

′ ∈ ··· ··· For x, x Gs with diﬀerent expression x = + aγ0 qγ0@ + aδ0 qδ0@ + and x′ = ··· + a′ q + a′ q + ··· . Suppose that the new expression γ0 γ0@ δ0 δ0@ ′ of x and x by substituting qγ0@ = kqδ0@ is the same.

Then the coeﬃcient of active base-elements at stage s other than qγ0@, qδ0@ is the same and ′ ′ aδ0 + aγ0 k = aδ0 + aγ0 k,

i.e., (a − a′ ) = (a′ − a )k. δ0 δ0 γ0 γ0

(i) If a = a′ , then a′ = a . In this case, the old expression of x, x′ δ0 δ0 γ0 γ0 at stage s is the same, contracting with the assumption on x, x′.

′ aδ −a ′ 0 δ0 (ii) If a ≠ a , then k = ′ . However, by the choice of k, k > δ0 δ0 a −a γ0 γ0 ′ |aδ −a | 0 δ0 |a′ −a | . So this case is impossible. γ0 γ0

′ Hence, the new expression of x, x ∈ Gs as Q-linear sums of elements in − { } Bs qγ0@ is diﬀerent.

3. If n > 1, run module D(1). Otherwise, n = 1 and ξ1 = τ, stop adding dependence relations.

This completes the description of module D(0).

At the end of module D(0), each element in Gs has a unique Q-linear sum of − { } elements in Bs qγ0@ . Now there is no current active base-elements of the form qζ@ with δ0@

δi−1@

80 Chapter 4. Group-order-computable degrees and PA degrees

Let δi@ be the length-lexicographically smaller element in {δi−1@, ξi+1@} and γi@ be the other element. We will add a dependence relation of the form qγi@ = ciqδi@ in module D(i).

Module D(i) with 1 ≤ i ≤ n − 1.

1. List all nonzero rational numbers which are coeﬃcients of qδi@ or qγi@ in the

unique expression of an element in Gs as a Q-linear sum of elements in Bs − { ≤ − } qγj @ : j i 1 as follows:

r1, r2, ··· , rl.

+ ′ ′ 2. Set r0 := 0. Choose a large enough k ∈ N such that for all i, i , j, j ≤ l with

rj − rj′ ≠ 0, |r − r ′ | k > i i . |rj − rj′ |

1 If δi@

add a new dependence relation qγi@ = kqδi@. Declare qγi@ is non-active at stage s + 1.

By substituting the added dependence relation in the expression of g ∈ Gs Q − { ≤ − } as a -linear sum of elements in Bs qγj @ : j i 1 , we obtain a new Q − { ≤ } expression of g as a -linear sum of elements in Bs qγj @ : j i . One can show as in module D(0) that the sign of g determined by this new expression is

unchanged and that the new expression of elements in Gs remains one-to-one.

3. When n = i + 1, ξi+1 = τ, stop adding dependence relations. Otherwise, n > i + 1, continue to run module D(i + 1).

This completes the description of module D(i).

Suppose that we have added dependence relations qγi@ = ciqδi@ in module D(i) ≤ − for all i n 1. By a direct checking, one can show that qγi@ = aiqδn−1@ for some ai > 0. Finally, the set of active base-elements at stage s + 1 is

− { ≤ ≤ − } ∪ { | | } Bs+1 = (Bs qγi@ : 0 i n 1 ) qσ@ : σ = s + 1 .

81 Chapter 4. Group-order-computable degrees and PA degrees

This ends adding dependence relations.

We will show in the following sections 4.6.3 and 4.6.4 that for any set A, G admits exactly two A-computable orders iﬀ A is not PA-complete.

4.6 Construction and veriﬁcation

4.6.1 Construction

Fix an eﬀective enumeration links of link (a c.e. equivalence relation deﬁned on <ω 2 @) such that at most one pair is enumerated into links at each stage s. Assume that |σ| < s and |τ| < s when (σ@, τ@) ∈ links.

At stage 0. link0 = ∅. Let G0 = {0G , q@}, where 0G is the zero of the constructed group (its unique Q-linear sum is just the empty sum), q@ is the active base-element at stage 0. Set sign≤G (0G ) = 0 and sign≤G (q@) = +1.

At stage s + 1.

Step 1. If a new pair of active base-elements at stage s is enumerated into links+1, then add dependence relations according to the basic R-strategy, goto step 2. Otherwise, no dependence relations are added at stage s + 1, goto step 3.

Step 2. Build Gs+1 and sign≤G on Gs+1 respectively according to the basic P-strategy and the basic Q-strategy with dependence relations added.

Step 3. Build Gs+1 and sign≤G on Gs+1 respectively according to the basic P-strategy and the basic Q-strategy with no dependence relations added.

This ends the stage s + 1 of the construction.

In section 4.6.2, we will show that P and Q are both satisﬁed. In sections 4.6.3 and 4.6.4, we will show that R is satisﬁed.

4.6.2 G is a computable group possessing computable orders ⊕ Fix an eﬀective enumeration Hs of H := Qqσ@ such that for all s, |Hs+1 − σ∈2<ω Hs| ≤ 1. During the stage s of the construction, we enumerate qσ@ with |σ| = s into Gs, and obtain active base-elements Bs. We may enumerate one more element,

82 Chapter 4. Group-order-computable degrees and PA degrees

say x, into Gs; in this case, there is an element y in Hs such that x is a unique

Q-linear sum of elements in Bs obtained by replacing non-active base-elements in y by active ones with corresponding scaling, and no elements enumerated into G so far has an expression the same as x.

For each g ∈ Gs, it can be expressed as a unique Q-linear combination of active base-elements at stage s. Now we show that the expressions of g only change ﬁnitely many times after stage s. Then each element in G has a unique Q-linear combination of active base-elements at end.

We only need to show that each original base-element, say qσ@, has ﬁnitely many expressions (all such expressions are elements of H , they represent the same element of G ). qσ@ is ﬁrst enumerated into G as an active base-element at stage |σ|, it has a new expression later only if there is a τ@

(in this case, qσ@ will be non-active forever). Then qσ@ = aqτ@ for some a > 0. <ω Since ≤len−lex is a well ordering on 2 @, there is a length-lexicographically <ω least element, say ρ@, in 2 @ such that link(σ@, ρ@) holds. Then qσ@ has a last expression of the form bqρ@ for some b > 0. In this case, qρ@ is active forever. ∈ For each g Gs, we also deﬁne the sign sign≤G (g) such that if g has a new expression later, then the sign computed according to this new expression is the same as before. Now for each element, g say, in G , it may have several expressions H in terms of elements in ; however, sign≤G (g) is just the sign computed according to the ﬁrst expression of g.

G is computable. For two original elements, say g, h, in H , one can effectively check whether they represent the same element in G (i.e. g equals h in G ) as follows: (1) find the stage t such that g − h is first enumerated into H ; (2) at stage t, we will consider g − h and update it as a Q-linear sum, say x, of elements in Bt; (3) g − h = 0G iff all coefficients in x are 0. Hence, this equality relation on H is computable. G can be viewed as the quotient group of H modulo this equality relation. Thus, G is computable. ≤ G G is computable. Now sign≤G (g) is a computable function on . Moreover, { ∈ G } ∪ { } P := g : sign≤G (g) = +1 0G is a computable set which is closed under addition, containing exactly one of g and −g when g ≠ 0G . That is, P is a positive cone on G . So P determines a computable order, namely ≤G , on G .

83 Chapter 4. Group-order-computable degrees and PA degrees

G 0 Q∞ is ∆2-isomorphic to . Since there are infinitely many infinite paths on T , the so defined link-relation on 2<ω@ has infinitely many equivalence classes, that { ∈ <ω } 0 is, B := qσ@ : σ 2 , qσ@ is active is infinite. Moreover, B forms a Π1-basis for G G ⊕ Q 0 G 0 , and is isomorphic to b via a ∆2-isomorphism. Hence, is ∆2-isomorphic b∈B to Q∞.

4.6.3 A-computable orders on G which are not computable

A A A ∗ Suppose that ≤ is an A-computable order on G such that ≤ ≠ ≤G and ≤ ≠ ≤G . We will show that A computes an infinite path on the co-c.e. binary tree T which is PA-complete. Then A is also PA-complete. Note that G is just H modulo the so defined computable equality relation, denoted by ∼. For any order on G with corresponding sign function, say φ, we can indeed extend φ to H by setting φ(h) = φ(g) where g ∈ G such that g ∼ h. In A particular, we will talk about the sign of elements in H under ≤ or ≤G through this way. A <ω We start by comparing ≤ with ≤G on {qσ@ : σ ∈ 2 }, the set of original base-elements. The sign of each qσ@ under ≤G is +1. For the sign determined by A ≤ or simply called A-sign, either there are two base-elements, say qσ@ and qτ@, having different A-sign; or all base-elements have the same A-sign.

Case 1. Suppose that there are qσ@ and qτ@ such that their A-sign are diﬀerent. Since each base-element is nonzero in G , its A-sign is not 0. Suppose that the A-sign of qσ@ is -1, and that the A-sign of qτ@ is +1. Claim. link(σ@, τ@) fails.

Proof. Suppose otherwise. We will add a dependence relation of the form qσ@ = ′ ′ rqτ@ with r > 0, τ@ 0, σ@

τ@. In both cases, qτ@ and qσ@ have the same A-sign. This ends the proof.

Recall that σ@ and τ@ are not linked means that there is at least one infinite path on T such that it is lexicographically between σ@ and τ@. With the help of ≤A, we will initiate an A-effective process which approximates lexicographically form below and above an infinite path of T . Without loss of generality, assume that τ@

84 Chapter 4. Group-order-computable degrees and PA degrees

Let ξ0@ = τ@, ζ0@ = σ@. By assumption, qξ0@ and qζ0@ have A-sign +1 and

-1 respectively. Let δ0 be the longest common initial segment of ξ0, ζ0, denoted by ξ0 ∩ ζ0. Then any inﬁnite binary string lexicographically between ξ0@ and ζ0@ extends δ0. Start module C(0) below, where one can determine a strictly extension

δ1 of δ0 such that δ1 is a preﬁx of the desired inﬁnite path (we are searching for) on T .

In general module C(i) with i ≥ 0, we deﬁne δi+1 ⊃ δi such that δi+1 can be extended into an inﬁnite path on T . C(i) proceeds as follows.

Module C(i) with i ≥ 0.

<ω 1. Pick the length-lexicographically least element η ∈ 2 with ξi@

ζi@, denoted by ηi.

2. If qηi@ has A-sign -1, then update ξi+1@ = ξi@, ζi+1@ = ηi@.

If qηi@ has A-sign +1, then update ξi+1@ = ηi@, ζi+1@ = ζi@.

In both cases, qξi+1@ and qζi+1@ have diﬀerent A-sign. So there is an inﬁnite

path on T which is lexicographically between ξi+1@ and ζi+1@.

3. Now deﬁne δi+1 ⊃ δi such that there is an inﬁnite binary string on T extending

δi+1. There are three cases.

3.1. If ξi@

(i) If ζi+1 = ηi, then any inﬁnite binary string lexicographically between

ξi+1@ and ζi+1@ extends δi0. Just set δi+1 = δi0.

(ii) If ξi+1 = ηi, then any inﬁnite binary string lexicographically between

ξi+1@ and ζi+1@ extends δi1. Just set δi+1 = δi1.

3.2. If δi@ ≤lex ξi@, then δi1 ⊆ ηi. In this case, any inﬁnite binary string

lexicographically between ξi+1@ and ζi+1@ extends δi1. So set δi+1 = δi1.

85 Chapter 4. Group-order-computable degrees and PA degrees

3.3. If ζi@ ≤lex δi@, then δi0 ⊆ ηi. In this case, any inﬁnite binary string

lexicographically between ξi+1@ and ζi+1@ extends δi0. So set δi+1 = δi0.

As link(ξi+1@, ζi+1@) fails, there is an inﬁnite path, say f, on T which is

lexicographically between ξi+1@ and ζi+1@. Since any inﬁnite binary string

lexicographically between ξi+1@ and ζi+1@ extends δi+1, f extends δi+1.

4. Start module C(i + 1).

This completes module C(i).

We obtain a strictly increasing sequence {δi : i ∈ N} of binary strings such that each δi can be extended into an inﬁnite path on T . Let g = ∪ δi, i.e., g(n) = δn+1(n) i∈N (note that |δn+1| ≥ n + 1). Then g is the desired inﬁnite path on T which is computable from ≤A, thus g is A-computable. Since g is PA-complete, A is also PA-complete.

Case 2. Suppose that each original base-element has the same A-sign under ≤A. Suppose that this A-sign is +1, we will show that ≤A can compute an infinite path of T [otherwise, we can deal with the reversal order of ≤A, and then show that this reversal order can compute an infinite path of T ]. ≤A̸ ≤ Now for each qσ@, its A-sign is the same as sign≤G (qσ@). Since = G , there is ∈ an element, say g G, such that sign≤G (g) = +1, but the A-sign of g is -1. g has Q ··· a final -linear sum of active base-elements, denoted by g = a0qσ0@ + + anqσn@ with ai ≠ 0 for each i = 0, ··· , n and σ0@

As sign≤G (g) = +1, a0 > 0. We may assume a0 = 1 [otherwise we deal with 1 g]. Now write g as a0

− ··· − ··· g = [qσ0@ + (a1 1)qσ1@] + + [qσm−1@ + (am 1)qσm@] + + qσn@.

≤ ≤ − For each 1 m n, sign≤G (qσm−1@ +(am 1)qσm@) = +1, and sign≤G (qσn@) = +1. ≤ Note that qσn@ has A-sign +1. Since the A-sign of g is -1, there is some m n − − such that qσm−1@ + (am 1)qσm@ has A-sign -1. In this case, am 1 < 0; otherwise, − ≥ − am 1 0, then qσm−1@ + (am 1)qσm@ has A-sign +1, a contradiction.

86 Chapter 4. Group-order-computable degrees and PA degrees

Hence, there are σ@, τ@ with σ@ 0 such that qσ@ − rqτ@ has A sign +1 under ≤G and -1 under ≤ respectively.

Claim. For such σ@, τ@, link(σ@, τ@) fails. Proof. Suppose otherwise. There are two cases:

If τ@ 0 will be added such that 0qσ@ + (a − r)qτ@ has sign +1 under ≤G . So a − r > 0.

In this case, since the A-sign of qτ@ is +1, the A-sign of 0qσ@ + (a − r)qτ@ is +1. A As qσ@ −rqτ@ = 0qσ@ +(a−r)qτ@ in G , if ≤ is an order on G , by our convention, the

A-sign of qσ@ −rqτ@ and 0qσ@ +(a−r)qτ@ is the same, contradicting the assumption that qσ@ − rqτ@ has A-sign -1.

If σ@ 0 such that sign≤G ((1 rb)qσ@ + 0qτ@) = +1. So 1 rb > 0. Then

(1 − rb)qσ@ + 0qτ@ has A-sign +1, a contradiction. This ends the proof.

Similar to case 1, we will initiate an A-eﬀective process which approximates an inﬁnite path on T lexicographically form below and above.

Find (σ@, τ@) with σ@ 0 such that qσ@−rqτ@ has sign +1 under A ≤G and sign -1 under ≤ . Let ξ0@ = σ@, ζ0@ = τ@ and r0 = r. Note that any inﬁnite binary string lexicographically between ξ0@ and ζ0@ extends δ0 := ξ0 ∩ ζ0. b First start module C(0), where we will determine the successor, say j, of δ0 such that there is an inﬁnite path on T which extends δ0j.

In general, for i ≥ 1, suppose that we have already obtained ξi@ 0, and δi in module C(i 1), where sign≤G (qξi@ riqζi@) = +1, the A-sign of − qξi@ riqζi@ is -1, and there is an infinite binary string on T which extends δi. Then b we will start module C(i) to search for longer approximation δi+1 of an infinite path on T . We point out here that for i ≥ 1, δi may not be ξi ∩ ζi, but the point is that any infinite binary string lexicographically between ξi@ and ζi@ extends δi.

Module C(b i) with i ≥ 0.

<ω 1. Pick the length-lexicographically least element η ∈ 2 such that ξi@

η@

87 Chapter 4. Group-order-computable degrees and PA degrees

′ − ′ − ri 2. Search until a positive r is found such that qξi@ r qηi@ and qηi@ r′ qζi@ have ′ r ≤G − − i the same sign +1 under ; and either qξi@ r qηi@ or qηi@ r′ qζi@ has sign -1 under ≤A.

′ We now show that such a r exists. Indeed, since link(ξi@, ζi@) fails, there is

an inﬁnite binary string, say f, on T which is lexicographically between ξi@

and ζi@. There are three cases:

2.1. If ξi@

(i) ηi@ 0, and r r a q − i q = (1 − i )q + 0q . ηi@ r′ ζi@ r′ ηi@ ζi@

′ − ria One can choose a positive r such that 1 r′ > 0.

(ii) ζi@ 0, and r r q − i q = 0q + (b − i )q . ηi@ r′ ζi@ ηi@ r′ ζi@

′ − ri One can choose a positive r such that b r′ > 0.

r A − i ≤G ≤ In both cases, qηi@ r′ qζi@ has sign +1 under both and .

Since link(ξi@, ηi@) fails (no relation will be added between qξi@ and − ′ ≤ qηi@), qξi@ r qηi@ has sign +1 under G .

− − ri As the A-sign of qξi@ riqηi@ is -1, the A-sign of qηi@ r′ qζi@ is +1, and r q − r q = (q − r′q ) + r′(q − i q ), ξi@ i ηi@ ξi@ ηi@ ηi@ r′ ζi@ − ′ the A-sign of qξi@ r qηi@ is -1.

2.2. If ηi@

(i) ξi@ 0, and

− ′ − ′ qξi@ r qηi@ = (1 r a)qξi@ + 0qηi@.

One can choose a positive r′ such that 1 − r′a > 0.

88 Chapter 4. Group-order-computable degrees and PA degrees

(ii) ηi@ 0, and − ′ − ′ qξi@ r qηi@ = 0qξi@ + (b r )qηi@.

One can choose a positive r′ such that b − r′ > 0.

− ′ − ri Similarly, one can show that qξi@ r qηi@ and qηi@ r′ qζi@ have sign +1 ′ r ≤G − − i under ; and that qξi@ r qηi@ and qηi@ r′ qζi@ have A-sign +1 and -1 respectively.

− 2.3. If link(ξi@, ηi@) and link(ηi@, ζi@) both fail. Clearly, qξi@ qηi@ and − ≤ − qηi@ riqζi@ have sign +1 under G . Since the A-sign of qξi@ riqζi@ is -1, and − − − qξi@ riqζi@ = (qξi@ qηi@) + (qηi@ riqζi@), − − at least one of qξi@ qηi@ and qηi@ riqζi@ has A-sign -1. In this case, just pick r′ = 1.

− ′ ′ 3. If the A-sign of qξi@ r qηi@ is -1, update ξi+1@ = ξi@, ζi+1@ = ηi@, ri+1 = r .

− ri ri If the A-sign of qηi@ r′ qζi@ is -1, update ξi+1@ = ηi@, ζi+1@ = ζi@, ri+1 = r′ . − − In both cases, sign≤G (qξi+1@ ri+1qζi+1@) = +1, but the A-sign of qξi+1@

ri+1qζi+1@ is -1. Hence, link(ξi+1@, ζi+1@) fails. There is an inﬁnite path on

T which is lexicographically between ξi+1@ and ζi+1@.

4. Deﬁne δi+1 ⊃ δi such that there is an inﬁnite path on T extending δi+1 as in the step 3 of module C(i) in case 1.

5. Start module C(b i + 1).

This completes module C(b i).

After running all modules C(b i) with i ≥ 0, we arrive at an increasing sequence

{δi : i ∈ N} such that there is an inﬁnite path on T extending each δi. Let g = ∪ δi. i∈N Then g is an A-computable path on T , thus, A is PA-complete. In sum, we have showed that every order on G is either computable or computable form a PA-complete set. Hence, if A is not PA-complete, then every A-computable on G is computable.

89 Chapter 4. Group-order-computable degrees and PA degrees

4.6.4 Orders on G which are computable from PA-complete sets

Recall that the set of infinite paths of the co-c.e. tree T is just the set of {0, 1}- valued diagonally nonrecursive functions. Let A be any PA-complete set. Then A computes an infinite path on T . In order to prove that there is an A-computable order on G which is not computable, we only need to show that any infinite path on T computes a noncomputable order on G . Let f ∈ [T ]. Now build a f-sign on G which determines a f-computable order, f ≤ say, on G . For each original base-element qσ@,

(1) if σ@

(2) if f

f We then build the f-sign of ≤ by a similar algorithm as in building the sign of ≤G .

For instance, assume that x ∈ Gs has a unique Q-linear sum of active base- elements at stage s as follows: ··· ··· x = a0qσ0@ + + aiqσi@ + b0qτ0@ + + bjqτj @, where bj ≠ 0, σ0@

(1) If there is a k ≤ i such that ak ≠ 0, then ﬁnd such a least k, and set the f-sign

of x to be the sign of ak.

′ (2) Otherwise, let k ≤ j be the least number with bk′ ≠ 0, and set the f-sign of

x to be the sign of −bk′ . Furthermore, if the expression of x changes later because some base-elements become non-active, then the algorithm preserves the f-sign of x. This ensures that this sign function is well-deﬁned on G . Clearly, the sign function is f-computable, so the corresponding order ≤f is f- <ω computable. Since there are σ@, τ@ ∈ 2 @ such that σ@

90 Chapter 5

Group orders and c.e. degrees

In this chapter, we will show that for any nonzero c.e. degree a, there is a nonzero c.e. degree c < a and a computable group G such that G has orders of degree ≥ a and c is group-order-computable via G , which implies that G has no incomputable orders of degree ≤ c.

Theorem 5.1 For every noncomputable c.e. set A, there is a noncomputable c.e. set C ≤T A and a computable torsion-free abelian group G with inﬁnite rank such that

(1) G has exactly two computable orders;

(2) G has an A-computable basis;

(3) every C-computable order on G is computable.

For a computable torsion-free abelian group A with inﬁnite rank, if A has a basis B, then the set O(A ) of degrees of orders on A contains {d : d ≥ deg(B)}.

Hence, our method is to control the degree of the constructed basis B := {bi : i ∈ N} of G . If deg(B) ≤ a, then the constructed G also satisﬁes

{d : d ≥ a} ⊆ {d : d ≥ deg(B)} ⊆ O(G ).

5.1 Requirements and strategies

Let {Hi : i ∈ N} be a uniformly computable sequence of subgroups with rank one, and let H = ⊕ Hi be the direct sum of Hi(i ∈ N). As in [3], we will build a i∈N computable group G isomorphic to H which admits computable orders.

91 Chapter 5. Group orders and c.e. degrees

To prove Theorem 5.1, we will build a computable group G , a computable group order ≤G on G , and a c.e. set C ≤T A to satisfy the following requirements.

Pe : We ≠ C.

Q : G has an A-computable basis B = {bi : i ∈ N}. ∼ Q0 : G = H .

∗ Q1 : ≤G and ≤G are exactly two computable orders on G .

N ≤C G ≤C ≤ ≤C ≤∗ e : If e is a C-computable group order on , then either e = G or e = G .

Where {We : e ∈ N} is an enumeration of c.e. subsets of natural numbers, and 1 {Φe : e ∈ N} is an enumeration of partial computable functionals defined on G ×G , ≤C C is the complement of C. The symbol e stands for the C-computable binary G C ∈ G ≤C relation on computed by Φe ; specifically, we view for any x, y , x e y iff C ↓ C ↓ C Φe (x, y) = 1. If Φe (x, y) , then let ϕe (x, y) denote the use for the computation C Φe (x, y).

To ensure C ≤T A, we will add an A-permitting to the standard Pe-strategy.

The Q-strategy ensures that G has an A-computable basis. The Q0-strategy and the Q1-strategy are similar to corresponding strategies in Theorem 1.17. We will develop new N -strategies.

5.1.1 A Pe-strategy with permitting

Pe is a standard Friedberg-Muchnik strategy with permitting. So we use cycles, we start with cycle 0 and cycle k with k ≥ 0 proceeds as follows (assume y−1 = 0).

(1) Choose a big witness yk > yk−1.

(2) Wait for yk enumerating into We at some stage, say sk.

̸ (3) Open cycle k + 1 and simultaneously, wait for Ask yk = As yk at some stage

s ≥ sk.

1 Here, we explain Φe in more detail. We first fix an effective numbering, say ι, of G (i.e., ι is a computably bijective mapping from N to G ). Let {Θe : e ∈ N} be an enumeration of N × N ⊆ N ∈ G X ≃ partial computable functionals defined on . Then for any X , and x, y ,Φe (x, y) X −1 −1 Θe (ι (x), ι (y)).

92 Chapter 5. Group orders and c.e. degrees

(4) Put yk into Cs, and close all cycles.

Suppose that the Pe-strategy opens infinitely many cycles. Then each cycle k is waiting at (3) forever, that is, we find witnesses yk ∈ We and stages sk such that ∈ N Ask yk = A yk . Then we can compute A as follows: let x , to decide whether it is in A or not, find a least cycle k such that y > x. Then x ∈ A ⇔ x ∈ A . x kx skx Hence, by the noncomputability of A, there are only finitely many cycles opened during the construction. If there is a first cycle k reaching at (4), then all cycles are closed and yk ∈ C ∩ We. Otherwise, no cycles are closed, and there is a least cycle k waiting at (2) forever; in this case, cycle k is the largest cycle opened, yk ∈/ We and also yk ∈/ C. In both cases, Pe is satisfied.

We can compute C by A as follows: let x ∈ N, ﬁnd a least stage sx such that { ∈ ≤ } ∈ ⇔ ∈ Asx x= A x (= y A : y x ), then x C x Csx .

5.1.2 The Q0-strategy

The Q0-strategy is a global strategy developed in [3]. To build the group G isomor- H { s ··· s} { s ∈ N} phic to , we maintain an approximation b0, , bs of a basis bi = limbi : i s G ⊕ G { s ··· s} { s ∈ N} of = i and an approximation N0 , ,Ns of a basis Ni = limNi : i i∈N s of H = ⊕ Hi at each stage s such that i∈N

(1) {bi} and {Ni} is respectively a basis of rank one groups Gi and Hi;

(2) the isomorphism Ω : G = (G, 0G , +G ) → H of abelian groups is determined

by sending each bi to Ni.

As each computable torsion-free abelian group of rank one can be eﬀectively em- bedded into Q, we view the subgroup Hi of H as a subgroup of additive group of s ∈ H ⊆ Q ∈ Q rational numbers. So Ni i and Ni .

Construction of G .

{ 0} G { 0} At stage 0. G0 = 0G , b0 , where 0G is the zero element of and b0 is the 0 approximation basis corresponding to N0 = 1.

At the end of stage s, Gs is the set containing all elements of G enumerated by ∈ Q s s ··· s s stage s, and each nonzero g Gs has assigned a unique -linear sum q0b0+ +qnbn,

93 Chapter 5. Group orders and c.e. degrees

s ̸ ≤ s ≤ Q s s ∈ H where qn = 0(n s), each qi (i n) in satisfying qi Ni i at stage s. By convention, 0G is assigned the empty sum.

At stage s + 1.

Case 1. If no dependence relations are added among approximation basis ele- N ≤ s+1 s s+1 s ≤ ments by any e-strategy (e s), bi = bi and Ni = Ni for all i s. For each ∈ s s ··· s s g Gs, it has the same expression as before at stage s+1. That is, if q0b0 + +qnbn Q s+1 s is the -linear sum of approximation basis elements at stage s, then let qi = qi ≤ Q s+1 s+1 ··· s+1 s+1 for all i n and g has the -linear sum q0 b0 + + qn bn at stage s + 1. s+1 s+1 Let Ns+1 = 1 and add a new element bs+1 into approximation basis. Find the

first tuple < x0, ··· , xm > of rationals (under a fixed effective numbering of the set of all finite sequences of rationals) with xm ≠ 0 and m ≤ s such that for all i ≤ m, s+1 ∈ H Q s+1 ··· s+1 xiNi i at stage s + 1, and the -linear sum x0b0 + + xmbm is not assigned to elements of the constructed group. If such a tuple exist, then enumerate s+1 ··· s+1 x0b0 + + xmbm into Gs+1. N s s Case 2. If some e-strategy adds a dependence relation of the form bl = qbk at stage s with l an odd number, k an even number (e < k < l), q a rational number. To ensure the uniqueness of Q-linear sums at each stage, q also satisfies additional conditions. We require that for any two elements in Gs, if their Q-linear sums of approximation basis elements at stage s are different, then after substituting the s s Q relation bl = qbk, their new -linear sums are still different. Q s s We only need to consider -linear sums involving basis elements bk and bl .

• s s ′ s ′ s ̸ ′ For example, assume that g = xbk + ybl , h = x bk + y bl , and either x = x ̸ ′ s s s ′ ′ s or y = y . If now after substituting bl = qbk,(x + yq)bk = (x + y q)bk, i.e., ′ ′ ̸ ′ x′−x x + yq = x + y q. Then y = y and q = y−y′ .

Thus, for any two different elements f, g in Gs, if after substituting the relation s s bl = qbk, the new expression of f, g at stage s+1 is the same, then one can compute a unique q(f, g) such that q = q(f, g). As Gs is a finite set, there are finitely many such q(f, g). We will choose the desired q different from all q(f, g) with f ≠ g in Gs s s but f, g has the same expression at stage s + 1 when adding bl = q(f, g)bk. Then for any two different elements in Gs, their assigned sums remain different at stage s + 1.

94 Chapter 5. Group orders and c.e. degrees

s+1 s ≤ Now we update basis approximation elements. Let bi = bi for all i s with ̸ s+1 s ≤ ̸ s+1 s+1 i = l, and Ni = Ni for all i s with i = k. Add two new elements bl and bs+1 s+1 s+1 into approximation basis. Set Ns+1 = 1. We now deﬁne Nk .

Let g ∈ Gs.

• s s ··· s s ··· s s ··· s s Assume that g = q0b0 + + qkbk + + ql bl + + qnbn at stage s. Now g changes to

s s ··· s s s ··· s ··· s s q0b0 + + (qk + ql q)bk + + 0bl + + qnbn.

s+1 s ≤ ∈ { } s+1 s s s+1 Let qi = qi for all i n with i / k, l , qk = qk + ql q, ql = 0. Then at s+1 s+1 ··· s+1 s+1 stage s + 1, g = q0 b0 + + qn bn .

To ensure the Q-linear sum of g is well-deﬁned after substituting the dependence s s s+1 s+1 ∈ H ≤ ∈ { } s+1 s+1 relation bl = qbk, we need qi Ni i for all i n. For i / k, l , qi Ni = s s ∈ H s+1 s+1 ∈ H qi Ni i; note that ql Nl = 0 l is already true. So we only require s+1 s+1 ∈ H qk Nk k. s+1 s Now deﬁne Nk = dqdNk , where dq is the denominator of q and d is the product s s s of all d s such that d s is the denominator of nonzero q , and q b appears in the ql ql l l l Q ∈ Z s ∈ Z -linear sum of an element in Gs. Then qdq and ql d , so

s+1 s+1 s s s+1 s s s s ∈ H qk Nk = (qk + ql q)Nk = dqdqkNk + ql dqdqNk k.

··· Q s+1 Finally, search the ﬁrst tuple < x0, , xm > and add the -linear sum x0b0 + ··· s+1 + xmbm into Gs+1 as in case 1. N s Case 3. If some e-strategy adds a dependence relation of the form bl = s − s m1bj m2bk at stage s with l an odd number, l > j, k an even number (e < k < l), m1, m2 integers.

The ﬁrst condition of m1, m2 is to keep the assignment of sums of elements in s s − s Gs remaining one-to-one after substituting the relation bl = m1bj m2bk.

• s s s ′ s ′ s ′ s ̸ For example, assume that g = xbj + ybk + zbl , h = x bj + y bk + z bl , and g = h ′ ′ ′ in Gs, that is, x ≠ x or y ≠ y or z ≠ z . If

s − s ′ ′ s ′ − ′ s (x + zm1)bj + (y zm2)bk = (x + z m1)bj + (y z m2)bk,

′ ′ − ′ − ′ ̸ ′ x′−x then x + zm1 = x + z m1 and y zm2 = y z m2. So z = z , m1 = z−z′ , and y−y′ m2 = z−z′ .

95 Chapter 5. Group orders and c.e. degrees

In general, for f ≠ g in Gs, so their expressions as Q-linear sum of approximation basis elements are different at stage s. If f, g has the same expression at stage s + 1 s s − s after substituting the relation bl = m1bj m2bk, then for i = 1, 2, we can compute a unique mi(f, g) such that mi = mi(f, g). Note that there are finitely many such mi(f, g), the desired mi will be different from all those mi(f, g) with f ≠ g in Gs.

Then the Q-linear sum of elements in Gs remains one-to-one at stage s + 1. s+1 s ≤ ̸ s+1 s ≤ Let bi = bi for all i s with i = l and Ni = Ni for all i s. Add two new s+1 s+1 s+1 elements bl and bs+1 into approximation basis, and set Ns+1 = 1.

The second conditions on m1, m2 are to ensure that the Q-linear sums are well- s s − s deﬁned after substituting the dependence relation bl = m1bj m2bk. That is, for any g ∈ Gs,

• s s ··· s s ··· s s ··· s s ··· s s assume that g = q0b0 + + qj bj + + qkbk + + ql bl + + qnbn. Now rewrite g as

s s ··· s s s ··· s − s s ··· s ··· s s q0b0 + + (qj + ql m1)bj + + (qk ql m2)bk + + 0bl + + qnbn.

s+1 s ≤ ∈ { } s+1 s s s+1 s − s Let qi = qi for all i n with i / j, k, l , qj = qj +ql m1, qk = qk ql m2, s+1 ql = 0. Then s+1 s+1 ··· s+1 s+1 g = q0 b0 + + qn bn .

s+1 s+1 ∈ H ≤ We need to ensure qi Ni i for all i n. s+1 s+1 ∈ H ∈ { } s+1 s+1 ∈ H Note that qi Ni i for i / k, j, l and ql Nl = 0 l. We only s+1 s+1 ∈ H s+1 s+1 ∈ H require qj Nj j and qk Nk k.

s+1 s+1 s s s+1 s s s s qj Nj = (qj + ql m1)Nj = qj Nj + ql m1Nj .

Choose m to be the multiple of the product d of all d s such that d s is the denom- 1 ql ql s s ̸ s s Q inator of ql , where ql = 0 and ql bl appears in the -linear sum of an element in Gs. s ∈ Z s+1 s+1 ∈ H | Then ql m1 and qj Nj j. Similarly, choose m2 with d m2, we have

s+1 s+1 s − s s+1 s s − s s ∈ H qk Nk = (qk ql m2)Nk = qkNk ql m2Nk k.

At last, ﬁnd the ﬁrst tuple < x0, ··· , xm > of rational numbers and add the Q s+1 ··· s+1 -linear sum x0b0 + + xmbm into Gs+1 as in case 1.

96 Chapter 5. Group orders and c.e. degrees

For f, g, h ∈ Gs, deﬁne f +s g = h if the addition of Q-linear sums of f and g is equal to the Q-linear sum of h. By linearity of additions, when updating expressions of f, g, h at stage s + 1, we still have f +s+1 g = h.

This completes the construction of G at stage s + 1.

→ H s s Let Ωs : Gs be the linear extension of mapping which sends each bi to Ni . 0 G H Then Ω = limΩs is a ∆2-isomorphism from to such that Ω(bi) = Ni for all i. s

5.1.3 The Q1-strategy

The Q1-strategy is also a global strategy developed in [3]. Similar to the Q0-strategy, { s ··· s} R at each stage s, we maintain approximation elements r0, , rs in which is Q s s s ≤ ≤ linearly independent over , and map bi to ri . We choose each ri (1 i s) as a √ positive rational multiple of pi, where pi is the i-th prime number for i ≥ 1. ≥ s s For i 1, ri = limri . r0 = r0 = 1R for all s. s

(1) Let Θ be the linear extension of the map Ψ : G → (R, 0R, +R); bi 7→ ri;

(2) Let Θ(G ) be the image of G under Θ, then G is isomorphic to the subgroup

Θ(G ) of (R, 0R, +R) and {ri : i ∈ N} is a basis of Θ(G ).

The linear order ≤G will satisfy x ≤G y ⇔ Θ(x) ≤R Θ(y) for any x, y ∈ G , this deﬁnition does not ensure the computability of ≤G . To make ≤G computable, we will approximate ≤G via ≤s at each stage s, and keep x ≤s y implies x ≤t y for all stages t ≥ s.

s ∈ H ⊆ Q Ni 8 i qqq Ωs qq qqq qqq s ∈ G bi i NNN NNΘ NNsN NN& s ∈ R ri s ≤ s R At each stage s, each approximation basis bi (i s) will be mapped to ri in , and we have an approximation Θs of Θ which is a linear extension of mapping on 0 G R approximation basis elements. Θ = limΘs is a ∆2 map from to . Although at s

97 Chapter 5. Group orders and c.e. degrees

limit level, the linear order ≤G is deﬁned by Θ which is a linear extension on basis elements of G , we cannot simply deﬁne the approximation ≤s of ≤G at stage s via

Θs as: g ≤s h ⇔ Θs(g) ≤R Θs(h).

The obstacle is when some Ne-strategy adding dependence relations of the form s s s s − s ≤ bl = qbk or bl = m1bj m2bk at stage s + 1, if s+1 is really deﬁned by Θs+1, note s s s s s s Θs(bl ) = rl , this means that bl maps to new positions Θs+1(bl ) = qrk or Θs+1(bl ) = s − s R s+1 s s+1 s m1rj m2rk in at stage s + 1 (we will see rk = rk and rj = rj ). We may change the relative positions of two elements in Gs, that is, maybe Θs(x)

s s ··· s s s ≥ s bs (i) ag = x0σ(a0) + + xnσ(an) with σ(ai ) = ai if xi R 0, σ(ai ) = ai if xi

bs bs ··· bs bs bs ≥ bs s (ii) ag = x0τ(a0) + + xnτ(an) with τ(ai ) = ai if xi R 0, τ(ai ) = ai if xi

Now deﬁne ≤s via assigned intervals as follows.

(1) Let g ≤s g for each g ∈ Gs.

∈ s bs ∩ s bs ∅ bs s (2) For each g, h Gs, if (ag, ag) (ah, ah) = and ag

In order to make ≤G computable, we choose approximation intervals such that

s+1 ∈ s+1 bs+1 ⊆ s bs ri (ai , ai ) (ai , ai )

bs+1 − s+1 1 ∈ s+1 bs+1 ⊂ s bs and ai ai

Construction of ≤G .

98 Chapter 5. Group orders and c.e. degrees

{ 0} 0 At stage 0. G0 = 0G , b0 , 0G sends to 0R, and set 0G <0 b0 by deﬁning r0 = 1R. { s ··· s} Assume that we have deﬁned r0, , rs by stage s.

At stage s + 1.

Case 1. If no dependence relations are added via any Ne-strategy. For each ≤ s+1 s s+1 bs+1 s+1 ∈ s+1 bs+1 ⊆ i s, ri = ri . Choose rational numbers ai , ai such that ri (ai , ai ) s bs bs+1 − s+1 1 bs+1 1 (ai , ai ) and ai ai

s s s+1 bs+1 s (1) if bl = qbk, choose q such that the new interval (qak , qak ) for bl at stage s bs s + 1 is contained in (al , al );

s s − s ∈ Z+ (2) if bl = m1bj m2bk with m1, m2 , choose m1, m2 such that the new s+1 − bs+1 bs+1 − s+1 s interval (m1aj m2ak , m1aj m2ak ) for bl at stage s + 1 is contained s bs in (al , al );

s s − s ∈ Z+ (3) if bl = m1bk m2bj with m1, m2 , choose m1, m2 such that the new s+1 − bs+1 bs+1 − s+1 s interval (m1ak m2aj , m1ak m2aj ) for bl at stage s + 1 is contained s bs in (al , al ). √ s+1 s+1 Let rl = pl, corresponding to new bl . Choose new rational numbers s+1 bs+1 s+1 ∈ s+1 bs+1 bs+1 − s+1 1 al , al such that rl (al , al ) and al al

This completes the construction of ≤G at stage s + 1.

5.1.4 The Q-strategy

In order to satisfy the Q-requirement, we modify the Ne-strategy developed in [3] s s through the following way: we add dependence relations of the form bl = qbk or

99 Chapter 5. Group orders and c.e. degrees

s s s bl = mbk + nbj for some odd l at stage w only when permitted by A, that is, ̸ t Aw−1 l= Aw l. Then for any stage t, At l= A l implies bl = bl.

The main idea is that for same active witness of Ne, we may search for multiple diagonalization witnesses until permitted by A, and then add dependence relations

[here, active witness and diagonalization witness will be defined in the basic Ne- strategy]. As A is assumed to be incomputable, such a permitting occurs; otherwise, we can initiate an effective process for computing A. { s ∈ N} ≤ ∈ G Hence, B = bi = limbi : i T A. Indeed, for an arbitrary g , let sg s G sg sg be the first stage when g enumerates into . If g is neither bsg nor bl with odd ∈ sg sg l < sg, then g / B. Otherwise, g = bsg or g = bl for some odd l (in this case, we sg−1 sg add dependence relations on bl , and bl is the new approximation for bl),

sg sg ∈ (1) if g = bsg with sg even or g = bl for some odd l < sg, then g B;

sg ≥ (2) if g = bsg with sg odd, ﬁnd a least stage t sg such that At sg = A sg , then ∈ t g B iﬀ g = bsg = bsg .

Ne still acts ﬁnitely often, so the whole construction is still a ﬁnite injury argument.

5.1.5 A Ne-strategy

≤C ̸ ≤ ≤C ̸ ≤∗ ≤C We need to ensure that if e = G and e = G , then e is not a group order on

G . A basic Ne-strategy works as follows.

s s s s (A1) Wait for an active witness (bj, n, bk) at some stage s, namely, (bj, n, bk) satisfying (a1), (a2), (a3) or (a1), (a2), (a4).

(a1) k is an even number larger than e.

s s s (a2) 0G ~~e 0G and either bj e (n + 1)bk. s C ∗ s C ∗ s s C ∗ s (a4) bk>e 0G and either bje (n + 1)bk.~~

≤C G ≤C ̸ ≤ By Lemma 3.9 [3], as long as e is a group order on satisfying e = G ≤C ̸ ≤∗ ≤C and e = G , we can ﬁnd active witnesses for e . That is, if no active witnesses ≤C G appeared during the construction, then either e is not a group order on or ≤C ≤ ≤C ≤∗ N e = G or e = G , and hence e is satisﬁed.

~~100 Chapter 5. Group orders and c.e. degrees~~

~~C s s (A2) We ﬁrst set a restraint r2 (bj, n, bk) on C to preserve the computations or- ≤C N derings in (a3) or (a4) for e at stage s. Say that e has active witnesses s s (bj, n, bk) now.~~

s s For the active witness (bj, n, bk), assume that we are in case (a3), we deal with (a4) ≤C ∗ similarly by considering the reveral order e . s s C s s As long as (bj, n, bk) is not initialized, that is, the C-restraint r2 (bj, n, bk) still s exist and bj is not canceled, we use the mechanism of cycles to assign multiple s s diagonalization witnesses for (bj, n, bk). Cycles can be opened or closed, and cycle i can open cycle i + 1. Assume that we have opened cycles n < i. During cycle i with i ≥ 0, proceed as follows:

~~(D1) At the least odd stage ti > ti−1 ≥ s (set t−1 = s), declare that the new~~

~~ti s s approximate basis element bti is a diagonalization witness for (bj, n, bk) at~~

~~ti cycle i. Order bti as in (b1) or (b2) when (b0) holds.~~

~~v s (b0) For all stages v with s ≤ v ≤ ti, b = b , Cs C s s = Ct C s s . j j r2 (bj ,n,bk) i r2 (bj ,n,bk)~~

~~s C s s ti s (b1) If we have bj~~

~~s C s s ti s (b2) If we have bj >e (n + 1)bk in (a3), set bj~~

~~ti s s We call (bti , bj, n, bk) a pre-potentially permanent witness at cycle i of the active s s N witnesses (bj, n, bk) of e. Proceed as follows to add dependence relations.~~

~~(D2) Wait for a potentially permanent witness (bu , bs, n, bs ) at some stage u > t , ti j k i namely, (bu , bs, n, bs ) satisfying (c1), (c2) or (c1), (c3) or (c1), (c4). ti j k~~

~~(c1) For all stages v with t ≤ v ≤ u, bv = bti , for all stages v with s ≤ v ≤ u, i ti ti v s b = b , and Cs C s s = Cu C s s . j j r2 (bj ,n,bk) r2 (bj ,n,bk)~~

~~101 Chapter 5. Group orders and c.e. degrees~~

~~(c2) bu~~

~~≤C When (c1) holds, if (c2) or (c3) or (c4) never appears, then e does not satisfy totality of linear orders, we have done.~~

~~(D3) We ﬁrst set a restraint rC (bu , n, bs ) ≥ rC (bs, n, bs ) on C to preserve ≤C - 4 ti k 2 j k e~~

orderings in (c2) or (c3) or (c4). Now Ne has a potentially permanent witness (bu , bs, n, bs ). Open cycle i + 1 to look for new diagonalization witnesses for ti j k s s the active witness (bj, n, bk). (D4) Wait for A ti to change. ≥ (D5) Suppose that A ti changes at some least stage w u > ti. That is, w is the

least stage ≥ u at which some element x ≤ ti enumerates into A. If (d0) holds, s s stop all cycles for the active witnesses (bj, n, bk), and proceed as in (D5.1) or (D5.2) to add dependence relations.

~~(d0) For all stage v with t ≤ v ≤ w−1, bv = bti , for all stage v with s ≤ v ≤ w, i ti ti v s bj = bj, and Cu rC (bu ,n,bs )= Cw rC (bu ,n,bs ). 4 ti k 4 ti k~~

~~w−1 C s s C w−1 (D5.1) If bti~~

~~also satisﬁes the conditions asserted by the Q0-strategy and the Q1-strategy. We now state the conditions of q.~~

∈ ≤C G ∈ (d1) q (n, n + 1). Then if e is a group order on , as q (n, n + 1) and C C w−1 C G ≤ ≤ bk >e 0 , we have nbk e bti = qbk e (n + 1)bk, contradicting with ≤C w−1 s s the assumption for e -orderings on bti , nbk and (n + 1)bk. w−1 w w bw w (d2) If bti = qbk , we need to ﬁnd a rational-endpoint interval (ak , ak ) for bk Q w bw ⊆ under the 1-strategy. Then we need to choose q such that (qak , qak ) w−1 w−1 w−1 w−1 w−1 w−1 b − − R (ati , ati ). As nbk

~~102 Chapter 5. Group orders and c.e. degrees~~

~~(d3) q satisﬁes the conditions under the Q0-strategy, that is, q ≠ q(f, g) for~~

~~any f, g ∈ Gw−1, such a q(f, g) turns an inequality f ≠ g in Gw−1 to be w−1 w an equality f = g in Gw when adding bti = q(f, g)bk . Note that there~~

are only finitely many such q(f, g) as Gw−1 is a finite set. (d4) Now we have showed that there are infinitely many q satisfying (d1), (d2), (d3). Our final q will be the least one of them such that the numbering of q (we fix an effective numbering of Q) is larger than the current stage w.

~~s C w−1 C s (D5.2) If nbk w 1 bj [i.e., we are in (b2)]. Remember that our target is still to diagonalize axioms of group orders~~

~~and to make the diagonalization to be compatible with the Q0-strategy and~~

~~the Q1-strategy.~~

~~w−1 w−1 C − ≤G ≤ (D5.2.1) If bti~~

~~w−1 w−1 w−1 w−1 ≤G : nb < − b < − b < − (n + 1)b k w 1 ti w 1 j w 1 k ≤C : bw−1~~

~~w−1 w w − N Add dependence relations of the form bti = m1bj m2bk with m1 > 1. ≤ w w−1 On the one hand, in order to be compatible with G , (note that bj = bj , w w−1 bk = bk ), we need~~

~~w−1 w−1 − w−1 w−1 w−1 nbk~~

~~w−1 Note that bk >G 0G . Then~~

~~− w−1 − w−1 w−1 (m1 1)nbk~~

~~and thus (m1 − 1)n~~

~~w−1 C w−1 C w−1 − w−1 C w−1 bj~~

~~103 Chapter 5. Group orders and c.e. degrees~~

~~w−1 C Note that bk >e 0G . Then~~

~~w−1 C − w−1 C − w−1 m2bk~~

~~and thus m2~~

~~Before looking for additional conditions on m1, m2, we recall the following important lemma which will be used.~~

~~+ Lemma 5.1 [3] If {r1, r2} ⊆ R is linearly independent over Q, then for any~~

~~two rational numbers q1 < q2, and any integer n ≥ 1, there are inﬁnitely many~~

~~positive integers m1, m2 such that n|m1, n|m2 and m1r1 − m2r2 ∈ (q1, q2).~~

~~+ The desired m1, m2 ∈ N will satisfy the following conditions:~~

~~∈ N | | ≤ m2 (f1) Choose m1, m2 such that nd m1, nd m2 and 1~~

~~w w−1 ∈ w bw ⊆ w−1 bw−1 w w−1 ∈ w bw ⊆ w−1 bw−1 rk = rk (ak , ak ) (ak , ak ), rj = rj (aj , aj ) (aj , aj )~~

~~bw − w 1 bw − w 1 where ak ak~~

~~(♣) m rw − m rw ∈ (m aw − m baw, m baw − m aw) ⊆ (aw−1, baw−1) 1 j 2 k 1 j 2 k 1 j 2 k ti ti~~

By Lemma 5.1, there are inﬁnitely many natural numbers m1, m2 with | | w − w ∈ w−1 bw−1 nd m1, nd m2 such that m1rj m2rk (ati , ati ). Then for such w bw w bw ♣ m1, m2, we can ﬁnd proper ak , ak , aj and aj satisfying ( ). Moreover, ≤ m2 the calculation above shows that m1 N n .

~~(f3) To make sure the Q-linear sums of elements in Gw−1 remaining one-to-~~

~~one, we also require that for i = 1, 2, mi ≠ mi(f, g) for any f, g ∈ Gw−1, w−1 w − w where adding bti = m1(f, g)bj m2(f, g)bk entails f = g at stage w,~~

~~but f ≠ g at stage w − 1. There are only ﬁnitely many such mi(f, g), so~~

~~we can ﬁnd inﬁnitely many desired m1, m2 satisfying (f1), (f2), (f3).~~

~~104 Chapter 5. Group orders and c.e. degrees~~

~~(f4) The ﬁnal m1, m2 will satisfy (f1), (f2), (f3) with m1, m2 > w and ⟨m1, m2⟩ least.~~

~~w−1 w−1 C − ≤G ≤ (D5.2.2) If bti >w 1 bj . Our environment for and e is~~

~~w−1 w−1 w−1 w−1 ≤G : nb < − b < − b < − (n + 1)b k w 1 j w 1 ti w 1 k ≤C : nbw−1~~

~~w−1 w − w Add dependence relations of the form bti = m1bk m2bj . We choose m1, m2 such that substituting the relation into ≤G holds, that is,~~

~~w−1 w−1 w−1 − w−1 w−1 nbk~~

~~Then~~

~~− w−1 w−1 (m2 + 1)[m1 (n + 1)]bk~~

~~So (m2 + 1)[m1 − (n + 1)]~~

~~| m1 Now under the conditions m1~~

~~C w−1 C w−1 w−1 w−1 0G < nb < b = m b − m b e k e ti 1 k 2 j m ≤C m bw−1 − 1 bw−1 e 1 k n + 1 j m1 w−1 w−1 C = ((n + 1)b − b ) < 0G n + 1 k j e~~

~~We now determine m1, m2:~~

~~∈ N | | m1 ≤ (g1) Choose m1, m2 such that (n + 1)d m1,(n + 1)d m2 and (n+1) N m2, where d is the same as in (f1).~~

w−1 w − w (g2) Assume that bti = m1bk m2bj . We need to assign rational-endpoint w−1 w bw w bw intervals for bti at stage w, that is, ﬁnd ak , ak , aj and aj the same as in (f2) but with the condition (♣) changed to

~~(♠) m rw − m rw ∈ (m aw − m baw, m baw − m aw) ⊆ (aw−1, baw−1) 1 k 2 j 1 k 2 j 1 k 2 j ti ti~~

~~105 Chapter 5. Group orders and c.e. degrees~~

According to Lemma 5.1, there are inﬁnitely many natural numbers | | w− w ∈ w−1 bw−1 m1, m2 with (n+1)d m1,(n+1)d m2 such that m1rk m2rj (ati , ati ). w bw ⊆ w−1 bw−1 w bw ⊆ Then we can choose proper (ak , ak ) (ak , ak ), and proper (aj , aj ) − − w 1 bw 1 ♠ m1 ≤ (aj , aj ) satisfying ( ). Then (n+1) N m2 automatically holds.

~~(g3) The same as in (f3), we require that the choice of m1 and m2 preserves~~

~~the one-to-one assignment of Q-linear sums to elements in Gw−1.~~

~~(g4) The ﬁnal m1 and m2 satisfy (g1), (g2), (g3), m1 > w, m2 > w with~~

~~⟨m1, m2⟩ least.~~

~~This ends the description of the basic Ne-strategy.~~

The interactions between two N -strategies are finite as in [3]. They both set C-restraints, so there is no confliction in building C; the confliction is in choosing witnesses. N s s Suppose that e choose an active witness (bj, n, bk) at stage s, and then add a t diagonalization witness bt at an odd stage t > s. To diagonalize axioms of group N t orders, e may add dependence relations to change bt at stages > t. However, some N u ′ u u t i may find an active witness (bj′ , n , bk′ ) at stage u > t with bj′ = bt, and then add w a diagonalization witness bw at an odd stage w > u. To preserve its active witness, N t i requires bt keeping the same at stages > u. The solution is a priority argument. When e < i,

~~N t N (1) if e acts ﬁrst to add dependence relations, it changes bt to initialize i’s~~

~~active witnesses (Ne is satisﬁed and cannot injure Ni later, Ni will choose new active witnesses at later stages);~~

~~N w (2) if i acts ﬁrst, it adds dependence relations on bw and hence does not change u t bj′ (= bt), so no conﬂiction occurs.~~

~~When i < e,~~

• N u ′ u u t if i finds its active witness, say (bj′ , n , bk′ ), at stage u > t with bj′ = bt, it N t N cancels e’s diagonalization witness bt to avoid potential conflictions ( e will t′ ′ find its diagonalization witness bt′ with t big at later odd stages).

~~106 Chapter 5. Group orders and c.e. degrees~~

P-requirements are purely set-theoretic requirements to ensure the noncomputability of C. Each Pe is positive on C and acts at most once to enumerate a permitted witness y ∈ We into C. So Pe injures negative requirements with lower priority which may set restraints on C ﬁnitely often.

~~5.2 Construction~~

~~Q, Q0 and Q1 are global requirements. The priority of other requirements is as follows:~~

~~N0 ≺ P0 ≺ · · · Ne ≺ Pe ≺ · · · .~~

The construction mainly splits into two parts. One is set-theoretic constructions, that is, building set C. The other is algebraic constructions, that is, adding ≤C G dependence relations to defeat e to be a group order on , and then constructing

~~G and ≤G according to whether some dependence relations are added or not.~~

~~Construction.~~

~~Stage 0. Let C0 = ∅. Build G0 and ≤0 by the Q0-strategy in Section 5.1.2 and the Q1-strategy in Section 5.1.3 respectively.~~

~~Stage s > 0.~~

~~Step 1. Find the highest priority requirement, say Q, among~~

N0 ≺ P0 ≺ · · · Ns ≺ Ps which requires attention (such a requirement exists since Ps requires attention at stage s). Act correspondingly and initialize all strategies ≻ Q. Goto Step 2 if no dependence relations are added; goto Step 3 otherwise.

Say that Pe is satisﬁed at stage s if We,s ∩ Cs ≠ ∅, where We,s is the set of all elements enumerated into We by stage s. Say that Pe requires attention at stage s if it is unsatisﬁed and one of the following (P1) and (P2) happens. Act accordingly.

~~(P1) Pe has no cycles opened at stage s.~~

~~Action: Open cycle 0 by appointing a large witness y0, and start to wait for~~

~~y0 enumerating into We. Proceed as in the basic Pe-strategy.~~

~~107 Chapter 5. Group orders and c.e. degrees~~

P ∈ (P2) e has a previous witnesses yk We,sk , cycle k + 1 is opened at stage sk and cycle k started to wait for Ask yk -changes at stage sk. Now Ask yk = ̸ ≥ P As−1 yk = As yk and stage s sk is the ﬁrst stage at which some cycles of e

~~reach (4) of the basic Pe-strategy, and this cycle is just cycle k.~~

~~Action: Put yk into Cs, close all opened cycles of Pe. Declare that Pe is satisﬁed.~~

~~Say that Ne is satisﬁed if we add dependence relations between some active witness and some diagonalization witness. That is, we have ﬁnished the following actions.~~

~~s s (1) We ﬁnd some active witness, say (bj, n, bk), and assign C-restraints to protect this active witness.~~

~~s s ti (2) We arrive at some cycle i for (bj, n, bk) with some diagonalization witness bti and set C-restraints to protect potentially permanent witness (bu , bs, n, bs ) at ti j k~~

~~some stage u ≥ ti.~~

~~ (3) A ti changes after stage u, and we add dependence relations between diagonalization witness and active witness.~~

~~≤C G In this case, e is not an order on .~~

~~Say that Ne requires attention at stage s if it is not satisﬁed at stage s, and one of the following holds at stage s. Act correspondingly.~~

~~(N1) Some active witness appears at stage s.~~

~~Action: First, set C-restraints to protect this active witness. Second, if some~~

~~Ni-strategy (i > e) has a diagonalization witness in some cycle, cycle j say, of~~

~~some active witness which occurs in the active witness of Ne, then cancel the~~

~~diagonalization witness at cycle k, where k ≥ j, of Ni’s active witness.~~

~~(N2) Ne has an active witness and no cycles opened for this active witness at stage s.~~

~~Action: Open cycle 0 of the active witness to ﬁnd diagonalization witnesses.~~

~~108 Chapter 5. Group orders and c.e. degrees~~

~~(N3) Ne has some active witness, and some cycle i is opened for this active witness. No diagonalization witness is found in cycle i at stage s, and s is odd.~~

~~ti s Action: Set the diagonalization witness bti = bs at cycle i of the active witness, s N and order bs with the active witness in (b1) or (b2) as in the basic e-strategy.~~

~~(N4) Ne has some active witness, and some cycle i with uncanceled diagonalization~~

~~ti witness bti is opened for this active witness. Now one of the ordering relations~~

ti between the active witness and bti in (c2) or (c3) or (c4) appears at stage s. Action: Set C-restraints to protect the potentially permanent witnesses at cycle i. Open cycle i + 1 for the active witness to seek for new diagonalization witnesses and simultaneously, wait for A ti to change.

~~N ti (N5) e has a potentially permanent witnesses with diagonalization witness bti in~~

~~cycle i, and some x ≤ ti ﬁrst enumerates into A at stage s.~~

~~ti Action: For such a least cycle i, add dependence relations on bti according~~

~~to the basic Ne-strategy, and then stop all cycles of Ne. Declare that Ne is satisﬁed.~~

~~Step 2. Build Gs and ≤s respectively according to the Q0-strategy in Section~~

~~5.1.2 and the Q1-strategy in Section 5.1.3 with no dependence relations added.~~

~~Step 3. Build Gs and ≤s respectively according to the Q0-strategy in Section~~

~~5.1.2 and the Q1-strategy in Section 5.1.3 with the added dependence relation.~~

~~This ends the stage s of the construction.~~

~~5.3 Veriﬁcation~~

~~Lemma 5.2 For each requirement, Q say,~~

~~(1) Q can be initialized at most ﬁnitely often;~~

~~(2) Q acts at most ﬁnitely often and is satisﬁed;~~

~~(3) Q initializes strategies with lower priority ﬁnitely often.~~

~~109 Chapter 5. Group orders and c.e. degrees~~

Proof: We will show that for each e, (1)-(3) hold for all Ne, Pe. We will prove the lemma by induction. For e = 0. N0 has the highest priority, it cannot be initialized. (1) holds. We now show that it acts finitely often and is satisfied. ≤C If 0 has no active witnesses during the construction, then by Lemma 3.9 in [3], ≤C ≤C ≤ ≤C ≤∗ N either 0 is not a group order, or 0 = G or 0 = G . 0 is satisfied. s s N N Suppose that there is some active witness (bj, n, bk) for 0 at stage s. Then 0 s s acts to protect (bj, n, bk) by setting C-restraints, and initializes all strategies with lower priority.

~~Under the basic N0-strategy, it starts by opening cycle 0, that is, by waiting for~~

~~t0 a diagonalization witness bt0 at some odd stage t0 > s and then waiting for~~

~~♠ t0 C s ∨ s C t0 C s ∨ s C t0 ( 0): bt0 <0 nbk nbk <0 bt0 <0 (n + 1)bk (n + 1)bk <0 bt0 .~~

~~In general, cycle i > 0 works as follows.~~

~~t i − − (i) Wait for a diagonalization witness bti at some odd stage ti > ui 1 > ti 1 and then wait for~~

~~♠ ti C s ∨ s C ti C s ∨ s C ti ( i): bti <0 nbk nbk <0 bti <0 (n + 1)bk (n + 1)bk <0 bti .~~

~~♠ N ≤C If ( i) never happens, then 0 is satisﬁed because 0 does not satisfy the axioms of linear orders.~~

~~(ii) If (♠i) happens at some stage ui > ti. Open cycle i + 1 and simultaneously wait for A ti to change.~~

~~ – If A ti changes at some stage wi > ui, add dependence relations between ti s s s s N bti and (bj, n, bk). Close all cycles of (bj, n, bk), and 0 is satisﬁed.~~

If there is a cycle i which waits at (♠i) forever, then cycle i + 1 is never opened. ≤C N In this case, 0 is not a linear order and thus 0 is satisfied. So assume that no cycles i wait at (♠i) forever. Suppose that each cycle i is not closed, then each cycle i waits for A ti to change ≤ forever, that is, Aui ti = A ti . Note that s ti < ui < ti+1 for all i. Now we can ∈ ∈ compute A as follows: for any x, find the least ti with x < ti, then x A iff x Aui .

~~110 Chapter 5. Group orders and c.e. degrees~~

As A is assumed to be noncomputable, there is a cycle i which provides a diag- ti ′ ≥ onalization witness bti , and Aui ti changes later, say at stage t ui. Then we add ti s s ′ N dependence relations between bti and (bj, n, bk) at stage t , and 0 is satisﬁed.

~~Hence, in all cases above, N0 acts ﬁnitely many times and is satisﬁed at end. (2) and (3) hold.~~

Let s0 be the least stage such that N0 never acts later, then P0 is not injured after stage s0. Let t0 ≥ s0 be the least stage such that cycle 0 of P0 is opened. As long as P0 is not satisﬁed, the basic P0-strategy will open cycles, say k, by picking a new witness yk at cycle k and starting to wait for yk ∈ W0; when yk ∈ W0, yk is ̸ enumerated into C at some stage s only if As yk = As−1 yk . As A is noncomputable, if such witnesses eventually enumerate into W0, then this A-permitting happens at certain cycles. So P0-strategy can only open ﬁnitely many cycles. That is, P0 acts

~~ﬁnitely often after stage s0.~~

~~As long as P0 is not satisﬁed automatically, there is a cycle k such that either yk is never put into C in which case yk ∈/ W0 or yk is put into C in which case yk ∈ W0.~~

~~In both cases, W0 ≠ C. So P0 is satisﬁed after ﬁnitely many actions, thus (2) and (3) hold.~~

~~Now assume that the lemma holds for all Ni, Pi with i < e. By induction hypothesis, there is a stage se such that Ni, Pi with i < e are satisﬁed at stage se.~~

~~Then Ne has the highest priority after stage se. Similar to case e = 0, (1)-(3) hold for Ne and Pe.~~

~~Lemma 5.3 C ≤T A.~~

~~Proof: During the construction, only permitted elements can be enumerated into~~

C. That is, x ∈ Cs only if there is some small element z ≤ x such that z ∈ As −As−1. To decide whether x ∈ C or not, by using A, find a least stage s such that for all z ≤ x, z ∈ A ⇔ z ∈ As. Then x ∈ C ⇔ x ∈ Cs. The differences between Theorem 5.1 and Theorem 1.17 are that elements enumerated into C are permitted by A and that the diagonalization witnesses are per- s mitted by A. We now show that for each i, the approximation basis element bi only s Q Q changes finitely often. Then limbi exists, and thus we can satisfy 0 and 1 the s same as in Theorem 1.17.

~~111 Chapter 5. Group orders and c.e. degrees~~

~~s Lemma 5.4 For each i, bi = limbi exists. s~~

i i Proof: For each i, consider bi. It may change at later stages only if bi is chosen as a diagonalization witness in some cycle of some active witnesses for some N - i s requirements. When bi is not a diagonalization witness, it never changes. So bi = bi for all s ≥ i. i i Assume that bi is a diagonalization witness. bi may be changed at most once when s−1 i ̸ adding dependence relations on bi = bi at some large stage s with As−1 i= As i. s s Then we will deﬁne a new basis approximation element bi , and bi = bi never changes s later since bi is not a diagonalization witness any more. So the approximations s ∈ N s bi (s ) of bi change at most once. Thus, bi = limbi exists. s

~~Lemma 5.5 B = {bi : i ∈ N} ≤T A.~~

∈ ∪ G s ≥ Proof: For any g G = Gs, the domain of . If it is not bi with s i when s ∈ s ﬁrst added into G, then g / B. Otherwise, assume that g = bi is ﬁrst added at stage s. i i ∈ Case 1. i = s. If i is even, then bi never changes later and g = bi = bi B. If i i N ∈ is odd and g = bi is not a diagonalization witness for -requirements, then g B.

If g is a diagonalization witness: first, by using A, find the least stage t0 such that i At0 i= A i; second, check whether we add dependence relations on bi during stages in (i, t0]. If we add dependence relations and redefine approximations for bi, then ̸ ∈ i t0 i g = bi and g / B. If no dependence relations are added on bi, that is, bi = bi, then g = bi ∈ B. s−1 s Case 2. i < s. Then i is odd and we change bi at stage s; in this case, g = bi is never changed later. So g = bi ∈ B.

By Lemma 5.2, all Pe, Ne hold. Then C is not computable; and for all e, either ≤C G ≤C ≤ ≤C ≤∗ e is not a group order on or e = G or e = G , that is, every C-computable order on G is computable. So G has no incomputable orders of degree ≤ deg(C).

~~By Lemma 5.3, C ≤T A. By Lemma 5.5, G has an A-computable basis, so G has an order of degree ≥ deg(A). Moreover, as G has no orders of degree C, C~~

~~112 Part III~~

~~Reverse mathematics, abelian groups and modules~~

~~113 Chapter 6~~

~~Reverse mathematics and divisible abelian groups~~

In this chapter, we will examine the proof-theoretic strength of several classical theorems related to countable divisible abelian groups in the program of reverse mathematics. For more background on abelian groups, refer to [29].

~~Deﬁnition 6.1 (RCA0) Let G be an abelian group. G is divisible if for any g ∈ G and nonzero n ∈ N, there is a g′ ∈ G such that g = ng′, and we say g is divisible by n in G.~~

~~In this chapter, groups always mean abelian groups.~~

~~6.1 Divisible subgroups~~

Deﬁnition 6.2 (RCA0) Let I be an abelian group. I is called injective if for any monomorphism µ : A → B and any homomorphism φ : A → I, there exists a homomorphism ψ : B → I such that φ = ψµ.

~~We now state a theorem related to injective abelian groups in reverse mathematics, as we will use it later to prove Theorem 6.2.~~

~~Theorem 6.1 [26] The following are equivalent over RCA0.~~

~~(1) ACA0.~~

~~(2) Every divisible abelian group is injective.~~

~~114 Chapter 6. Reverse mathematics and divisible abelian groups~~

Friedman, Simpson and Smith [26] showed that ACA0 proves the statement that “every divisible subgroup of an abelian group G is a direct summand of G”. We are interested in the reversal part of this statement. We will complete the work above of Friedman, Simpson and Smith, by proving the following theorem.

~~Theorem 6.2 Over RCA0, the following are equivalent:~~

~~(1) ACA0.~~

~~(2) Every divisible subgroup of an abelian group is a direct summand.~~

~~Proof: (1) ⇒ (2) was proved by Friedman, Simpson and Smith in their paper [26] (Lemma 6.2). For completeness, we provide a proof here.~~

Let G be an abelian group and D be a divisible subgroup of G. As ACA0 proves that every divisible group is injective, D is injective. Consider the following diagram with 1D : D → D the identity homomorphism and µ : D → G the identical embedding: / µ / 0 D ÒG Ò 1D Ò ÑÒ ψ D

~~Then there is surjective homomorphism ψ with 1D = ψµ, and the kernel of ψ, ker(ψ), is {g ∈ G : ψ(g) = 0}, and G = ker(ψ) ⊕ im(ψ) = ker(ψ) ⊕ D. This completes the proof of (1) ⇒ (2).~~

(2) ⇒ (1). Fix a one-to-one function f : N → N. Let G be the divisible ⊕ abelian group Qxn, where elements of G are ﬁnite Q-linear sums of elements n∈N ⊕ { ∈ N} ∈ N ′ Q ′ from xn : n . For m , let xm = x2f(m) + mx2f(m)+1. Then xm is a m∈N divisible subgroup of G, denoted by D. D exists in RCA0. By our assumption (2), D is a direct summand of G. So there is a subgroup K of G such that G = D ⊕ K. For any n, x2n and x2n+1 now can be uniquely written as

~~• x2n = y2n + z2n,~~

~~• x2n+1 = y2n+1 + z2n+1,~~

~~115 Chapter 6. Reverse mathematics and divisible abelian groups~~

where y2n, y2n+1 ∈ D and z2n, z2n+1 ∈ K. Recall that any element in D can be expressed as a unique Q-linear sum of { ′ ∈ N} elements in xm : m . Let In be the ﬁnite set consisting of all indices m where ′ xm appears in the expression of y2n or y2n+1. Then

~~n ∈ range(f) ⇔ ∃m ∈ In[f(m) = n].~~

~~We only need to show that if n = f(m0), then m0 ∈ In. As~~

~~′ xm0 = x2n + m0x2n+1 = (y2n + m0y2n+1) + (z2n + m0z2n+1)~~

{ ′ ∈ and the expressions of elements in D as linear combinations of elements in xm : m N} ′ ′ are unique, we have xm0 = y2n + m0y2n+1. Then xm0 appears in the expression of y2n or y2n+1, i.e., m0 ∈ In. 0 Now range(f) exists, by the ∆1-comprehension.

Classically, every abelian group has the unique largest divisible subgroup which 1 is the union of all divisible subgroups. This deﬁnition is Σ1. We will show that the existence of the largest divisible subgroup of an abelian group is equivalent to 1 Π1-CA0 over RCA0. In order to prove the reversal part, we need the following well- known theorem of Friedman, Simpson and Smith. Recall that an abelian group is reduced if it has no nontrivial divisible subgroups.

~~Theorem 6.3 [26] Over RCA0, the following are equivalent.~~

~~1 (1) Π1-CA0.~~

~~(2) Every abelian group is a direct sum of a divisible subgroup and a reduced subgroup.~~

However, for the special case of torsion-free abelian groups, as shown in Theorem 6.4 below, we show that the existence of the largest divisible subgroup is equivalent to the arithmetical comprehension.

~~Theorem 6.4 Over RCA0, the following are equivalent:~~

~~(1) ACA0.~~

~~116 Chapter 6. Reverse mathematics and divisible abelian groups~~

~~(2) Every torsion-free abelian group has the largest divisible subgroup.~~

~~Proof: (1) ⇒ (2). Let G be a countable abelian group. Let~~

~~D = {g ∈ G : ∀n > 0∃g′ ∈ G[g = ng′]}.~~

~~D exists by arithmetical comprehension.~~

We will show that D is the largest divisible subgroup of G. Clearly, the zero 0G − of G is in D. For any m, if m divides gn1 and gn2 in G, then m divides gn1 gn2 in G. Thus, D is a subgroup of G. We now show that D is divisible. For any g ∈ D, g is divisible by all nonzero natural numbers in G. So for any m ≥ 1, there is a hm ∈ G such that g = mhm.

~~We claim that hm ∈ D. Then g is also divisible by all nonzero natural numbers in D. Thus, D is a divisible subgroup of G.~~

~~Claim. hm ∈ D. Proof of the claim.~~

Fix k ≥ 1. As g = mhm = mkhmk, m(hm − khmk) = 0G. Here, the index mk of hmk is the multiplication of m and k. As m ≠ 0 and G is torsion-free, hm − khmk = 0G, i.e., hm = khmk. So hm is divisible by k in G. We have showed that hm is divisible by all nonzero natural numbers in G. By deﬁnition, hm ∈ D. This ends the proof of the claim.

Now we show that D is the largest divisible subgroup of G. Take any divisible subgroup A of G. For any a ∈ A and n > 0, there is an a′ ∈ A such that a = na′. Since a′ ∈ G, a ∈ D. Thus A ⊆ D.

(2) ⇒ (1). Let f : N → N be a one-to-one function. Consider the torsion-free ⊕ abelian group Qxn, denoted by G. Again, elements of G are just ﬁnite Q-linear n∈N sums of elements in {xn : n ∈ N}. ∑ Let H be the set of all elements anxn of G such that for each n, if the n

~~D′ = {g ∈ H : ∀n > 0∃g′ ∈ H[g = ng′]}.~~

~~117 Chapter 6. Reverse mathematics and divisible abelian groups~~

~~From the proof of the direction (1) ⇒ (2), we see that D′ is a divisible subgroup of H and that D ⊆ D′. Then by the choice of D, D′ = D.~~

~~We now show n ∈ range(f) ⇔ xn ∈/ D.~~

~~• If n = f(m), then 1 x ∈/ H. Note that the only element y in G with x = p y pm n n m is 1 x . Thus no elements y in H satisfy x = p y, and x ∈/ D. pm n n m n~~

~~• ∈ 1 ∈ ≥ ∈ If n / range(f), then k xn H for any k 1. Thus xn D.~~

~~0 By ∆1-comprehension, range(f) exists.~~

~~Corollary 6.1 Over RCA0, the following are equivalent:~~

~~1 (1) Π1-CA0.~~

~~(2) Every abelian group has the largest divisible subgroup.~~

~~(3) Every abelian group is a direct sum of a divisible subgroup and a reduced subgroup.~~

Proof: By deﬁnition, (1) ⇒ (2). By Theorem 6.3 of Friedman, Simpson and Smith, (3) ⇒ (1). We now show (2) ⇒ (3). Assume (2). Then every torsion-free abelian group has the largest divisible subgroup. By Theorem 6.4, this implies ACA0. Now let G be an abelian group.

~~Again, by (2), it has the largest divisible subgroup, say D. By Theorem 6.2, ACA0 proves G = D ⊕ R for some subgroup R. As D is the largest divisible subgroup, R is reduced. Hence, (3) holds.~~

~~6.2 Two decomposition theorems~~

~~Deﬁnition 6.3 (RCA0) An abelian group is primary if there is a prime p such that its elements have orders a power of p.~~

~~For each i ∈ N, pi is the i-th prime number, starting with p0 = 2. Let P be the set of all prime numbers.~~

~~Theorem 6.5 Over RCA0, the following are equivalent:~~

~~118 Chapter 6. Reverse mathematics and divisible abelian groups~~

~~(1) ACA0.~~

(2) Every nontrivial torsion group, say T , is a direct sum of primary subgroups. ⊕ That is, there is a set I ⊆ P such that T = Tp, where for each p ∈ P , p∈I n Tp = {g ∈ T : ∃n[o(g) = p ]}, and for p ∈ I, Tp ≠ {0T }.

Proof: (1) ⇒ (2). We use classical proof in [29]. Let T be a countable nontrivial torsion group, i.e., T ≠ {0T }. Let o(g) be the order of g in T . Because T is torsion, o(g) is the least nonzero natural number n such that ng = 0T . The function o exists 0 in RCA0 (o is indeed Σ0). n For every prime p, Tp = {g ∈ T : ∃n[o(g) = p ]} exists in RCA0. As g ∈ Tp iﬀ n ≤ 0 o(g) = p for some n o(g). Tp is also Σ0. We now show that any element of T can be uniquely written as a Z-linear sum of elements in Tp for various primes p. ∈ r1 ··· rk Let g T with order o(g). Factorize o(g) = q1 qk as prime powers. Let o(g) ··· ≤ ≤ ni = ri for all i = 1, , k. Then ni, 1 i k, are coprime, so there are integers qi ai for all 1 ≤ i ≤ k such that a1n1 + ··· + aknk = 1. Then,

g = (a1n1 + ··· + aknk)g = a1n1g + ··· + aknkg. ∑ o(g) ri Because o(nig) = = q , each nig ∈ Tq . Thus, g ∈ Tq . ni i i i 1≤i≤k ∈ We now show that the sum of g is unique. Suppose that there are xi, yi Tqi ··· ··· − ∈ ≤ ≤ such that g = x1 + + xk = y1 + + yk. Then xi yi Tqi for all 1 i k, and

~~x1 − y1 = (y2 − x2) + ··· + (yk − xk).~~

~~The order of x1 − y1 is a power of q1, but the order of right hand side is a multiplication of powers of qi with 2 ≤ i ≤ k. Hence, x1 − y1 = 0. Now consider~~

~~x2 + ··· + xk = y2 + ··· + yk.~~

In a similar way, we can show that x2 = y2, ··· , xi = yi. Thus, xi = yi for all i = 1, ··· , k. This proves the uniqueness of sums. ⊕ Hence, T = Tp, and we may have Tp = {0T } for some primes p. p∈P

~~119 Chapter 6. Reverse mathematics and divisible abelian groups~~

{ ∈ ∃ ∈ − { } ∃ n } 0 Now let I = p P : g T 0T n [o(g) = p ] . Then I is Σ1, so it exists in ACA . Moreover, T ≠ {0 } ⇔ p ∈ I. Hence, I is the desired set such that ⊕ 0 p T T = Tp with Tp ≠ {0T } for each p ∈ I. p∈I

(2) ⇒ (1). Let f : N → N be a one-to-one function. Let G be an abelian group generated by {x : n ∈ N} with relations p x = 0 for each m ∈ N. Then ⊕ n f(m) m Z ∈ N+ Z Z Z Z G = pf(m) xm, where for any n , n = /n . We view elements of n as m∈N the least nonnegative representatives under relation x ∼ y ⇔ x − y ∈ nZ = {nk : k ∈ Z}, i.e., for n ≥ 1, Z = {i ∈ N : i ≤ n − 1}. n ∑ ∈ Z ≤ Elements of G are of the form aixi for some ai pf(i) with each i n. So G i≤n exists in RCA . Clearly, G is torsion. 0 ⊕ By assumption (2), there is a set I ⊆ P such that G = Gp, where for each p∈I n prime p, Gp = {g ∈ G : ∃n[o(g) = p ]}, and for each p ∈ I, Gp ≠ {0G}.

By deﬁnition of G, each element has order a product of pf(m) for various m. If ∈ { } ∈ n / range(f), Gpn = 0G . If n = f(m) for some m, then pnxm = 0G, so xm Gpn , and thus G ≠ {0 }. So pn G ⊕

G = Gpn . n∈range(f) ∈ ⇔ ∈ 0 Then n range(f) pn I. By ∆1-comprehension, range(f) exists. We will consider the decomposition theorem of divisible abelian groups. Before proving that, we recall a few reverse mathematical results about countable algebra.

~~Theorem 6.6 [26] Over RCA0, the following are equivalent:~~

~~(1) ACA0.~~

~~(2) Every abelian group has a torsion subgroup T containing all torsion elements.~~

~~(3) Every vector space has a basis.~~

~~Deﬁnition 6.4 (RCA0) ⊕ (1) For 1 ≤ k < ∞, Gk denotes the direct sum of k many Gs, that is, Gk = G. 1≤i≤k ⊕ (2) G∞ denotes the direct sum of inﬁnitely many Gs, i.e., G∞ = G. k∈N~~

~~120 Chapter 6. Reverse mathematics and divisible abelian groups~~

Proposition 6.1 (RCA0) ⊕ (1) For 1 ≤ k < ∞, a direct sum Gi of abelian groups is divisible iﬀ for all 1≤i≤k 1 ≤ i ≤ k, Gi is divisible. ⊕ (2) A direct sum Gi of abelian groups is divisible iﬀ for all i ∈ N, Gi is divisible. i∈N { } n For a prime number p, the p-quasicyclic group Z(p∞) = : n, m ∈ Z /Z pm exists in RCA0.

~~Theorem 6.7 Over RCA0, the following are equivalent:~~

~~(1) ACA0.~~

~~(2) Every divisible abelian group is isomorphic to a direct sum of the form ⊕ ⊕ ∞ Z ln ⊕ Q ( (pn )) n∈I n∈J~~

~~for some I,J ⊆ N, and if I ≠ ∅, then for each n ∈ I, 1 ≤ ln ≤ ∞.~~

Proof: (1) ⇒ (2). This was already mentioned by Friedman, Simpson and Smith in paper [26], we now provide a complete proof. Let G be a countable divisible abelian group. ACA0 proves the existence of torsion subgroup T of G. We ﬁrst show that T is a divisible subgroup of G. For g ∈ T and n ≥ 1, as G is divisible, there is a g′ ∈ G such that g = ng′. Let the order of g be m. Then mng′ = 0, so the order of g′ is also ﬁnite. Thus g′ ∈ T .

By Theorem 6.2, ACA0 proves T is a direct summand of G. Let G = T ⊕ F . F is a divisible torsion-free subgroup of G. For each x ∈ F and nonzero n ∈ Z, ∈ 1 there is a unique y F with x = ny and we write y = n x. F is a vector space over the rational ﬁeld. As ACA0 proves that every vector space has a basis, let X ⊆ F ⊕ ∼ be a basis of F . Then F = Qx, and each Qx = Q as abelian groups. x∈X Now consider the torsion subgroup T . Suppose that T is nontrivial (otherwise, we have done). By Theorem 6.5, ACA proves that T is a direct sum of nontrivial 0 ⊕ primary subgroups. That is, there is some Z ⊆ P such that T = Tp, where p∈Z

~~121 Chapter 6. Reverse mathematics and divisible abelian groups~~

~~n for each p ∈ Z, Tp = {g ∈ T : ∃n[o(g) = p ]}= ̸ {0G}. Since T is divisible, by~~

~~Proposition 6.1 in this section, each Tp is divisible.~~

~~Fix p ∈ Z, we now decompose Tp.~~

Let P0 = {g ∈ Tp : pg = 0}. g ∈ P0 iff g = 0 or o(g) = p, where o(g) is the order 0 of g in Tp. P0 is Σ0 with parameter Tp, and thus exists in ACA0. Another view is that P = {g ∈ T : pg = 0} is a Z -module, where Z is the finite field of size p. 0 p p p ⊕ Similarly, ACA0 proves P0 has a basis Y0 as vector spaces over Zp. So P0 = Zpy. y∈Y0 List Y0 as y0,1, y0,2, ··· without repetitions. Let Kp = {i ∈ N : y0,i ∈ Y0}.

For each y0,i ∈ Y0, since Tp is divisible, for each n ≥ 1, there are yn,i ∈ Tp such n that y0,i = p yn,i. Let Yn := {yn,i : i ∈ Kp}. 2 Consider P1 = {g ∈ Tp : o(g) = p }. Then pP1 = {pg : g ∈ P1} ⊆ P0. We now show that any element in P1 is a unique sum of elements in Y0 ∪ Y1. Let g ∈ P1, ∈ ··· ∈ ··· ∈ Z then pg P0. There are y0,i1 , , y0,ik Y0, a1, , ak p such that

~~··· pg = a1y0,i1 + + aky0,ik~~

~~in P0 and the expression of Zp-linear sums is unique. ··· ∈ As y0,il = py1,il for each l = 1, , k and y1,il Y1, we have~~

~~··· pg = p(a1y1,i1 + + aky1,ik ).~~

− ··· ∈ − ··· So g (a1y1,i1 + + aky1,ik ) P0. Then g (a1y1,i1 + + aky1,ik ) can be written as a unique Z -linear combination of elements in Y , and thus, g can be written as p 0 ⊕ a unique Zp-linear combination of elements in Y0 ∪ Y1. So P0 ∪ P1 ⊆ Zpy. y∈Y0∪Y1 In general, let P = {g ∈ T : o(g) = pn+1}. Any element in P is a unique n ∪p ⊕ n Zp-linear sum of elements in Yj. Thus P0 ∪ · · · ∪ Pn ⊆ Zpy. Then 0≤j≤n y∈Y0∪···∪Yn ∪ ⊕ ⊕ Tp ⊆ Pn ⊆ Zpy ∈N ∈N ∈ n ⊕n y⊕Yn = Zpyn,i ⊆ Tp

~~n∈N i∈Kp ⊕ ⊕ ∼ ⊕ ⊕ Now Tp = Zpyn,i = Zpyn,i. n∈N i∈Kp i∈Kp n∈N~~

~~122 Chapter 6. Reverse mathematics and divisible abelian groups~~

For each i ∈ K , the subgroup generated by {y : n ∈ N} is isomorphic to p ⊕ ⊕ ⊕ n,i ∞ ∼ ∼ ∞ ∞ |Kp| Z(p ). Hence, Tp = Zpyn,i = Z(p ) = (Z(p )) , where |Kp| is the i∈Kp n∈N i∈Kp size of Kp if Kp is ﬁnite, and ∞ otherwise.

~~This completes the decomposition of Tp for a ﬁxed p ∈ Z.~~

~~Finally, we have ⊕ ∼ ∼ G = T ⊕ F = Tp ⊕ F ⊕p∈Z ⊕ ∼ ∞ |K | = (Z(p )) p ⊕ Q p∈Z x∈X~~

~~Now let I = {n ∈ N : p ∈ Z}, J = X, and for each n ∈ I, l = |K |. Then ⊕ ⊕ n n pn ∞ ∼ Z ln ⊕ Q G = ( (pn )) . n∈I n∈J~~

(2) ⇒ (1). Let f : N → N be a one-to-one function. Consider abelian group ⊕ ∑ Z ∞ (pf(m))xm, denoted by G. Elements of G are of the form aixi for some m∈N i≤n ∈ Z ∞ ≤ ai (pf(i)) with each i n. G exists in RCA0, and is divisible. By (2), G is a direct sum of subgroups isomorphic to Q or Z(p∞) for various ⊕ ⊕ ∞ ∼ Z ln ⊕ Q ⊆ N ̸ ∅ primes p. Let G = ( (pn )) for some I,J , and when I = , n∈I n∈J for each n ∈ I, 1 ≤ ln ≤ ∞.

~~Then I = range(f), J = ∅, and ln = 1 for each n ∈ I. Hence, n ∈ range(f) iﬀ n ∈ I, so range(f) exists.~~

~~123 Chapter 7~~

~~Reverse mathematics and modules~~

~~7.1 Projective modules~~

~~Recall that a R-module P is projective if for every surjective homomorphism ε : B → C of R-modules and every homomorphism g : P → C, there exists a homomorphism f : P → B such that g = εf.~~

~~7.1.1 Direct sums of projective modules ⊕ Proposition 7.1 (RCA0) A direct sum Pi is projective iﬀ Pi is projective for 0≤i≤n all 0 ≤ i ≤ n.~~

However, for the inﬁnite case, we have the following theorem. ⊕ Theorem 7.1 (1) (RCA0) If a direct sum P = Pi is projective, then Pi is i∈N projective for all i ∈ N. ⊕ 1 ∈ N (2) (Π1-CA0) If for all i , Pi is projective, then the direct sum P = Pi is i∈N projective.

~~Proof: We ﬁrst prove (1) within RCA0. Suppose that P is projective. We will prove that each Pi is projective. Given any diagram of the form~~

~~gi B ε / C / 0~~

~~124 Chapter 7. Reverse mathematics and modules~~

~~Deﬁne g : P → C as g(x) = gi(xi) where xi is the i-th column of x ∈ P . As P is projective, there is a f : P → B such that g = εf. For each z ∈ Pi, we have~~

~~εf(⟨0, ··· , 0, z, 0, ···⟩) = g(⟨0, ··· , 0, z, 0, ···⟩) = gi(z).~~

So there is a homomorphism fi : Pi → B with fi(z) = f(⟨0, ··· , 0, z, 0, ···⟩) such that εfi = gi. 1 We now prove (2) within Π1-CA0. Suppose that Pi is projective for all i, and we will prove that P is also projective. Give a surjective homomorphism ε : B → C and a homomorphism g : P → C, we need to ﬁnd a homomorphism f : P → B such that εf = g.

~~ÔP f Ô Ô g ÑÔ B ε / C / 0~~

~~For each i, let gi : Pi → C be a homomorphism with fi(z) = g(⟨0, ··· , 0, z, 0, ···⟩).~~

~~As Pi is projective, there is a homomorphism fi : Pi → B such that~~

~~εfi(z) = gi(z) = g(⟨0, ··· , 0, z, 0, ···⟩).~~

Then for any x = ⟨x0, ··· , xn⟩ ∈ P , set f(x) = f0(x0) + ··· + fn(xn), which is in B, and we have ∑ ∑ εf(x) = εfi(xi) = g(⟨0, ··· , 0, xi, 0, ···⟩) = g(x). 0≤i≤n 0≤i≤n

1 ⟨ ∈ N⟩ So f exists in Π1-CA0. Note that to deﬁne f, we use fi : i . The following theorem has already been studied by Yamazaki in [22]. It provides several equivalence characterizations of projective modules, and such equivalence are provable over RCA0. For completeness, we provide a proof here.

~~Theorem 7.2 (Yamazaki [22]) (RCA0) For a R-module P , the following are equivalent:~~

~~(1) P is projective.~~

~~(2) If ε : B → P is a surjective homomorphism, then there exists a homomorphism~~

~~f : P → B such that εf = 1P , the identity homomorphism on P .~~

~~125 Chapter 7. Reverse mathematics and modules~~

~~∼ (3) If P = B/A, then P is isomorphic to a direct summand of B.~~

~~(4) P is isomorphic to a direct summand of a free module F .~~

~~Proof: (1) ⇒ (2). As P is projective, there is a homomorphism f : P → B such that εf = 1P . That is, the diagram~~

ÔP f Ô Ô 1P ÑÔ B ε / P / 0 commutes. (2) ⇒ (3). Let π : B → B/A be the canonical surjective homomorphism, and g : B/A → P be an isomorphism. Then ε = gπ is a surjective homomorphism from B to P . Let µ : A → B be the identical embedding, i.e., for any x ∈ A, µ(x) = x. µ Consider the short exact sequence 0 → A → B →ε P → 0. By (2), it splits with ∼ a splitting f. Then by Lemma 1.2, B = A ⊕ f(P ). As f is one-to-one, P = f(P ), which is a direct summand of B. ∼ (3) ⇒ (4). By Proposition 1.7, P = F/A for some free module F . By (3), P is isomorphic to a direct summand of F . (4) ⇒ (1). By (4), P is isomorphic to a submodule Q of F via some isomorphism f, and F = Q ⊕ A for some submodule A. As free modules are projective, F is projective, and hence, by Proposition 7.1, Q is projective. Now we show that P is projective. Give a surjective homomorphism ε : B → C and a homomorphism g : P → C, we need to ﬁnd a homomorphism h : P → B such that the diagram

~~ÔP Ô hÔ g ÑÔ B ε / C / 0 commutes. Consider the homomorphism gf −1 : Q → C. Since Q is projective, there is a homomorphism h′ : Q → B such that gf −1 = εh′. Let h = h′f. Then~~

~~εh = ε(h′f) = (εh′)f = (gf −1)f = g(f −1f) = g.~~

~~126 Chapter 7. Reverse mathematics and modules~~

~~0 7.1.2 Free modules and projective modules over Σ1-PIDs~~

Free modules are projective, but generally, projective modules may not be free. 0 Now we restrict to Σ1-PIDs, because over such special rings, projective modules are 0 always free. Within RCA0, deﬁne an integral domain R to be a Σ1-PID if every 0 Σ1-ideal of R is generated by one element.

~~0 Lemma 7.1 (ACA0) Let R be a Σ1-PID. A submodule of a free R-module is free.~~

Proof: We reason in ACA . As we only consider countable modules, we assume 0 ⊕ ⊕ that a free R-module F is either R for some n ∈ N, or R. We prove it for ⊕ 0≤i≤n i∈N the case F = R. The following proof adopts ideas of a classical proof in [30]. i∈N Let N be a submodule of F . Fix a code for N. Now elements of N are coded as ⊕ natural numbers. For all i, let Ni = N ∩ R, and deﬁne a map fi : Ni → R with 0≤j≤i fi(⟨x0, ··· , xi−1, xi⟩) = xi. Then fi is a R-homomorphism, and im(fi) = Ni/Ni−1 is an ideal of R. Again, we view elements in im(fi) as representatives of equivalence classes under the equivalence relation x ∼ y ⇔ x − y ∈ Ni−1 such that their codes are least as natural numbers. im(fi) exists in RCA0. 0 As R is a Σ1-PID, for each i, there exists a bi such that (bi) = im(fi). Deﬁne a function g by setting g(i) = bi. Since g(i) = bi if bi ∈ im(fi) and

~~∀a ∈ im(fi)∃c ∈ R[a = cbi],~~

~~0 g is Π2, thus exists in ACA0.~~

Let J = {j ∈ N : g(j) ≠ 0}. List J as j0 < ··· jn < jn+1 < ··· . For each jn ∈ J, ∈ define cjn by primitive recursion as follows. Let cj0 Nj0 be the element with least ≤ − code such that fj0 (cj0 ) = bj0 . Assume that we have defined cjl for all l k 1, to ∈ define cjk , we let rjk Njk be the element with least code such that fjk (rjk ) = bjk , and check whether the code of rjk is bigger than or equal to the code of cjk−1 or not.

If yes, just let cjk = rjk ; otherwise, we can have m big enough such that the code of mcjk−1 + rjk is bigger than the code of cjk−1 , and we let cjk = mcjk−1 + rjk . This ∈ N ensures that the sequence cjk , k , has increasing codes, and we still have

~~fjk (cjk ) = fjk (mcjk−1 + rjk ) = fjk (rjk ) = bjk .~~

~~127 Chapter 7. Reverse mathematics and modules~~

~~{ ∈ } 0 C = cj : j J is ∆1 in J and hence exists in ACA0.~~

Claim 1. C = {cj : j ∈ J} is linearly independent. Proof of claim 1: For any ﬁnite subset A ⊆ C, let A = {c , c , ··· , c } with ∑ i1 i2 in ··· ∈ i1 < i2 < < in. If dik cik = 0 for some dik R, then 0≤k≤n ∑ ∑

0 = fin ( dik cik ) = dik fin (cik ) = din bin . 0≤k≤n 0≤k≤n ∑ ̸ As bin = 0, din = 0, then dik cik = 0. By applying fin−1 , we obtain din−1 = 0. 0≤k≤n−1 ··· Continue this process, we arrive at din = din−1 = = di1 = 0. So A is linearly independent. This completes the proof of claim 1.

~~Claim 2. C generates N.~~

~~Proof of claim 2: We prove it by contradiction. Assume that there is a least n ∈ N such that some x = ⟨x0, ··· , xn⟩ ∈ Nn that cannot be written as linear combinations of elements of C.~~

~~By the minimality of n, xn ≠ 0. Then fn(x) = xn ∈ (bn) ≠ {0}, thus, bn ≠ 0, and n ∈ J. Let xn = dnbn for some dn ∈ R − {0}. Then~~

~~fn(x) = xn = dnbn = dnfn(cn) = fn(dncn), so x − dncn ∈ ker(fn) = Nn−1.~~

As x cannot be written as linear combinations of elements of C and cn ∈ C, so does x − dncn. But x − dncn ∈ Nn−1, contradicting the minimality of n. This completes the proof of claim 2. By claims 1 and 2, C is a basis of N, and thus N is free.

~~0 Theorem 7.3 (ACA0) Let R be a Σ1-PID. A projective R-module is free.~~

Proof: Since RCA0 proves that every projective module is isomorphic to a direct summand of a free module, it proves that every projective module is isomorphic to a submodule of a free module. Let M be a projective R-module. There is an isomorphism f : M → M ′, where M ′ is a submodule of a free R-module. By Lemma ′ ′ −1 −1 7.1, ACA0 proves M is free. Let C be a basis of M . Then f (C) := {f (c): c ∈ C} is a basis of M, i.e., M is free.

~~128 Chapter 7. Reverse mathematics and modules~~

~~0 Corollary 7.1 (ACA0) Let R be a Σ1-PID. A submodule of a projective R-module is projective.~~

~~7.2 Injective modules~~

The notion of injective modules is the dual of projective modules. A R-module I is injective if for any monomorphism µ : A → B and any homomorphism g : A → I, there exists a homomorphism f : B → I such that g = fµ.

~~7.2.1 Baer’s criterion for injective modules~~

~~The following classical lemma is known as Baer’s criterion for injective modules.~~

Lemma 7.2 (Baer’s Criterion) A module I over a commutative ring R with identity is injective iﬀ for every ideal J of R and every R-module homomorphism g : J → I, there is a homomorphism f : R → I such that f J = g.

By deﬁnition, it is easy to see that RCA0 proves one direction: a module I over a ring R is injective implies that for every ideal J of R and every R-module homomorphism g : J → I, there is a homomorphism f : R → I such that f J = g.

~~The other direction is equivalent to ACA0, due to Yamazaki in [22]. For completeness, we will provide a proof.~~

~~Theorem 7.4 (Yamazaki [22]) The following are equivalent over RCA0.~~

~~(1) ACA0.~~

~~(2) If for every ideal J of R and every R-module homomorphism g : J → I, there~~

~~is a homomorphism f : R → I such that f J = g, then I is injective.~~

~~Proof: (1) ⇒ (2). We reason in ACA0. Assume that for an ideal J of R, and a R-module homomorphism g : J → I, there is a homomorphism f : R → I such that the diagram~~

~~/ µ / 0 J ÔR Ô g Ô ÑÔ f I~~

129 Chapter 7. Reverse mathematics and modules commutes, where µ(x) = x for x ∈ J. We need to show that for any submodule A of a R-module B, the identical embedding µ : A → B, and a R-module homomorphism g : A → I, there is a homomorphism f : B → I making the following diagram commutative. / µ / 0 A ÔB Ô g Ô ÑÔ f I

~~List B as b0, b1, ··· , and let An be the R-module generated by A ∪ {b0, ··· , bn}. We extend g : A → I to B by stages.~~

~~Stage 0. If A0 = A, then let f0 : A0 → I be the same as g. Otherwise, let~~

S0 = {r ∈ R : rb0 ∈ A}. Since A is a R-module, S0 is an ideal of R. Deﬁne a homomorphism h0 : S0 → I by letting h0(r) = g(rb0). By assumption, there is a → ∈ ∈ homomorphism g0 : R I with g0 S0 = h0, g0(1R) I, and for r S0,

~~rg0(1R) = g0(r) = h0(r) = g(rb0).~~

~~/ S / 0 0 ÔR Ô h0 Ô ÑÔ g0 I~~

So we deﬁne f0 : A0 → I by letting f0(b0) = g0(1R), and for any x = a+rb0 ∈ A0 with a ∈ A, r ∈ R, f0(x) = g(a) + rf0(b0). We claim that f0 is a R-homomorphism ′ ′ ′ ′′ from A0 to I. Let x = a + rb0 and x = a + r b0 be two elements in A0, and r ∈ R. Then

~~′ ′ ′ f0(x + x ) = g(a + a ) + (r + r )f0(b0) ′ ′ = g(a) + g(a ) + rf0(b0) + r f0(b0) ′ = f0(x) + f0(x ), and~~

~~′′ ′′ ′′ f0(r x) = g(r a) + r rf0(b0) ′′ ′′ = r g(a) + r rf0(b0) ′′ = r f0(x).~~

~~130 Chapter 7. Reverse mathematics and modules~~

~~Moreover, f0 A= g. In fact, if rb0 ∈ A with r ≠ 0, then r ∈ S0, and~~

~~f0(rb0) = rf0(b0) = rg0(1R) = g0(r) = h0(r) = g(rb0).~~

Stage n + 1. Assume that we have fn : An → I extending fn−1. If bn+1 ∈ An, do nothing, let fn+1 = fn. Otherwise, let Sn+1 = {r ∈ R : rbn+1 ∈ An}, which is a proper ideal of R. Let hn+1 : Sn+1 → I by hn+1(r) = fn(rbn+1). Then there is a

~~R-module homomorphism gn+1 : R → I such that the diagram~~

~~/ S / 0 n+1 R hn+1 gn+1 I commutes. So for any nonzero rbn+1 ∈ An,~~

~~fn(rbn+1) = hn+1(r) = gn+1(r) = rgn+1(1R).~~

~~Now deﬁne fn+1 : An+1 → I by choosing~~

~~fn+1(bn+1) = gn+1(1R)~~

~~and extending it linearly to all An+1. That is, for x = a + rbn+1 ∈ An+1 with a ∈ An, r ∈ R, deﬁne~~

~~fn+1(x) = fn(a) + rfn+1(bn+1). It is easy to check that fn+1 is the homomorphism from An+1 to I with fn+1 An = fn.~~

~~This provides a mapping from B to I by having f(x) = fn(x) for x ∈ An. It is obvious that f is well-deﬁned and is a homomorphism, and that f is the limit of fn.~~

(2) ⇒ (1). We prove the reversal direction by showing that (2) implies the statement “every divisible abelian group is injective” which is equivalent to ACA0 over RCA0. Let D be a divisible abelian group, (n) be an ideal of Z, and g :(n) → D be a Z-module homomorphism. We will extend g to Z, i.e., deﬁne a homomorphism f : Z → D such that

~~f (n)= g.~~

~~131 Chapter 7. Reverse mathematics and modules~~

~~• If n = 0, just deﬁne f : Z → D to be the zero homomorphism.~~

~~• If n ≠ 0, g(n) ∈ D. By divisibility of D, there is a c ∈ D such that g(n) = nc. Then deﬁne a Z-module homomorphism f : Z → D by letting f(1) = c, and~~

~~f(n) = nf(1) = nc = g(n).~~

~~So for any x ∈ (n), let x = mxn, then~~

~~f(x) = mxf(n) = mxg(n) = g(mxn) = g(x).~~

~~Thus, the assumption of (2) is true for R = Z and I = D. Now by applying (2) to R = Z and I = D, D is injective.~~

~~0 7.2.2 Divisible modules and injective modules over Σ1-PIDs~~

Let R be an integral domain. Recall that a R-module D is divisible if for every d ∈ D and every nonzero r ∈ R, there is a c ∈ D such that d = rc. Classically, a module over a principal ideal domain is injective iﬀ it is divisible. By Proposition

~~1.11, RCA0 proves that injective modules over an integral domain are divisible.~~

~~However, to show the other direction, we require ACA0.~~

~~0 Lemma 7.3 (ACA0) Let R be a Σ1-PID. A divisible R-module D is injective.~~

Proof: We reason in ACA0. The proof is similar to the one for abelian groups. Let µ : A → B be a given monomorphism. Without loss of generality, we assume that A is a submodule of B. For a given homomorphism g : A → D, we need to ﬁnd a homomorphism f : B → D such that g = fµ.

~~/ µ / 0 A ÒB Ò g Ò ÑÒ f D~~

~~List B as b0, b1, ··· , and let An be the submodule of B generated by A ∪ { ··· } ∈ N 0 b0, , bn for n . An exists by Σ1-comprehension. We will extend g to B by stages.~~

~~132 Chapter 7. Reverse mathematics and modules~~

~~Stage 0. If b0 ∈ A, then A = A0, let g0 = g with domain A0. Otherwise, consider set~~

~~S0 = {r ∈ R : rb0 ∈ A}.~~

0 As A is a R-module, S0 is an ideal of R. Obviously, S0 is ∆1, so by assumption 0 ∈ that R is a Σ1-PID, there is a t0 R such that S0 = (t0). If t0 = 0, just choose g0(b0) as 0. If t0 ≠ 0, g(t0b0) ∈ D; by divisibility of D, there is a c0 ∈ D such that g(t0b0) = t0c0, and deﬁne

~~g0(b0) = c0.~~

~~Then for any x ∈ A0, x = a + rb0 for some a ∈ A and r ∈ R, set~~

~~g0(x) = g(a) + rg0(b0).~~

~~We can check g0 is a R-module homomorphism from A0 to D, as in Theorem~~

~~7.4. We now show that g0 extends g. If rb0 ∈ A with r ≠ 0, then g0(rb0) = rc0. As ′ ′ r ∈ S0, there is a r ∈ R such that r = r t0. Then~~

~~′ ′ ′ g0(rb0) = rc0 = r t0c0 = r g(t0b0) = g(r t0b0) = g(rb0).~~

~~Stage n + 1. Assume that we have already deﬁned gn : An → D extending gn−1.~~

~~Now consider bn+1.~~

~~If bn+1 ∈ An, then An+1 = An and we let gn+1 = gn. Otherwise, let~~

~~Sn+1 = {r ∈ R : rbn+1 ∈ An}.~~

~~0 Again, Sn+1 is an ideal of R, and Sn+1 is Σ1. By assumption that R is a Σ1-PID,~~

~~Sn+1 is a PID and we let Sn+1 = (tn+1). If tn+1 = 0, deﬁne gn+1(bn+1) = 0. If tn+1 ≠ 0, as tn+1bn+1 ∈ An, and hence gn(tn+1bn+1) ∈ D. As D is divisible, there is a cn+1 ∈ D such that~~

~~gn(tn+1bn+1) = tn+1cn+1.~~

~~We now deﬁne gn+1(bn+1) = cn+1, and extend it linearly to An+1 by letting~~

~~gn+1(a + rbn+1) = gn(a) + rgn+1(bn+1) for a ∈ An and r ∈ R. We can check that gn+1 is a R-module homomorphism with domain An+1 and that gn+1 An = gn.~~

~~133 Chapter 7. Reverse mathematics and modules~~

~~∪ Then B = An, and for all n, gn+1 extends gn. f : B → D deﬁned by n∈N~~

~~f(x) = gn(x)~~

for some n with x ∈ An is well-deﬁned and is a R-module homomorphism extending g. f is the desired homomorphism. In [26], it was proved that the statement “every divisible abelian group is injective” is equivalent to ACA0 over RCA0. Together with Lemma 7.3, we obtain:

~~Theorem 7.5 The following are equivalent over RCA0:~~

~~(1) ACA0.~~

~~0 (2) Every divisible module over a Σ1-PID is injective.~~

~~(3) Every divisible abelian group is injective.~~

~~Proposition 7.2 (RCA0)~~

~~0 (1) Every quotient of a divisible module over a Σ1-PID is divisible.~~

~~0 (2) Every quotient of an injective module over a Σ1-PID is divisible.~~

Proof: (1). Let D be a divisible R-module, and ε : D → B a surjective homomorphism. We show that B is divisible. For any nonzero b ∈ B, ﬁnd a d ∈ D with ε(d) = b. By divisibility of D, for any nonzero r ∈ R, there is a d′ ∈ D such that d = rd′. Then b = ε(d) = rε(d′) with ε(d′) ∈ B. 0 (2). Let M be an injective module over a Σ1-PID. RCA0 proves that M is 0 divisible and that every quotient of a divisible module over a Σ1-PID is divisible.

~~Hence, RCA0 proves that every quotient of M is divisible. 0 By Lemma 7.3, ACA0 proves that every divisible module over a Σ1-PID is injective, and we obtain the following corollary.~~

~~0 Corollary 7.2 (ACA0) Every quotient of an injective module over a Σ1-PID is injective.~~

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