A SURVEY OF THE REVERSE MATHEMATICS OF ORDINAL ARITHMETIC
JEFFRY L. HIRST
Abstract. This article surveys theorems of reverse mathematics concerning the comparability, addition, multiplication and exponentiation of countable well orderings.
In [13], Simpson points out that ATR0 is “strong enough to accommodate a good theory of countable ordinal numbers, encoded by countable well order- ings.” This paper provides a substantial body of empirical evidence supporting Simpson’s claim. With a few very interesting exceptions, most theorems of or- dinal arithmetic are provable in RCA0 or are equivalent to ATR0. Consequently, up to equivalence over RCA0, Friedman’s early result [2] on the equivalence of ATR0 and comparability of well orderings encapsulates most of countable or- dinal arithmetic. This paper is divided into sections on de nitions and alternative de nitions of well orderings, comparability and upper bounds, addition, multiplication, exponentiation, and other topics. The last section addresses Cantor’s normal form theorem, trans nite induction schemes, and indecomposable well order- ings, and concludes with a list of some omitted topics. Whenever possible, references to proofs are provided, rather than the actual proofs. Throughout, arbitrary sets are denoted by capital roman letters, but well ordered sets are denoted by lower-case greek letters. This notation emphasizes the parallels between the encoded theory and the usual development of ordinal arithmetic.
§1. De nitions of well orderings. First, we will de ne linear orderings and well orderings.
Definition. (RCA0) Let X be a set of pairs. We will write x y if (x, y) ∈ X. We say that X is a (countable) linear ordering, denoted LO(X), if 1. x y → (x x ∧ y y), 2. (x y ∧ y z) → x z, 3. (x y ∧ y x) → x = y, and 4. (x x ∧ y y) → (x y ∨ y x).
Received by the editors October 30, 2000. 1991 Mathematics Subject Classi cation. 03B30, 03E10, 03F35. Accepted for inclusion in Reverse Mathematics 2001, edited by S. Simpson. For a de- scription of this volume, see www.math.psu.edu/simpson/revmath/. Meeting c 1000, Association for Symbolic Logic 1 2 J. HIRST
The eld of X is the set {x ∈ N | x x}. We say that X is a (countable) well ordering, denoted WO(X), if for every nonempty Y eld(X) there is an element y0 ∈ Y such that y ∈ Y implies y0 y. That is, WO(X)ifevery nonempty subset of X has a least element. A well ordered set with a largest element is called a successor. A well ordered set with no largest element is called a limit. Some papers (for example [3]) de ne well orderings as linear orderings with no in nite descending sequences. This de nition is equivalent to the preceding one, and the equivalence is provable in RCA0.
Theorem 1. (RCA0) Let X be a linear ordering. The following are equiva- lent: 1. X is well ordered. That is, every nonempty subset of X has a least element. 2. X contains no in nite descending sequences. Proof. Theorem 2 of [5]. a Cantor’s original de nition of “well ordered aggregate” was closer to clause 3 of the following theorem. His de nition is equivalent to the usual one, but the proof of the equivalence requires ACA0.
Theorem 2. (RCA0) The following are equivalent:
1. ACA0. 2. If is a well ordering, then every subset of with an upper bound has a least upper bound. 3. Suppose X is a linear ordering. Then X is well ordered if and only if every subset of X with a strict upper bound has a least strict upper bound.
Proof. The equivalence of clause 2 and ACA0 is Theorem 3 of [5]. The equivalence of clause 3 and ACA0 is shown in Corollary 6 of [5]. a
§2. Comparability and upper bounds. In this section, after presenting a long list of equivalent versions of comparability, we will look at descending 1 sequences, strict inequality, comparisons to ω, suprema, and bounds for 1 classes. We begin by de ning two forms of comparability of well orderings.
Definition. (RCA0)If and are well orderings, then is strongly less than or equal to , denoted s , if there is an order preserving map of onto an initial segment of . If the initial segment is itself, we also write s .If +1 s , then we write
Definition. (RCA0)If and are well orderings, then is weakly less than or equal to , denoted w , if there is an order preserving map of into .If w and w , then we write w .If +1 w , then we write Sometimes, the same result will hold for both s and w. We will drop the subscripts and write and < whenever statements hold for both forms of comparability. A few properties of s and w are provable in RCA0. For example, RCA0 can prove that s and w are both transitive relations. Here is another example. Theorem 3. (RCA0) If is a proper initial segment of a well ordering , then 6 w . Proof. Lemma 2.3 of [3]. a Additionally, the fact that s implies w is provable in RCA0, although the converse requires ATR0, as shown in the next theorem. The next theorem also shows the equivalence of a wide variety of statements on comparability. The terminology has all been de ned with one exception. A well ordering is indecomposable if for every nal segment = {a ∈ | b Theorem 4. (RCA0) Suppose that and denote well orderings and that h i|i ∈ Ni denotes a sequence of well orderings. Then (the universal closures of) the following are equivalent: 1. ATR0. 2. (Strong comparability of well orderings.) s or s . 3. (Weak comparability of well orderings.) w or w . 4. (Weak comparability of indecomposable well orderings.) If and are indecomposable, then w or w . 5. (The class of well orderings has no in nite antichains.) For some distinct i and j, i w j. 6. For some distinct i and j, i s j. 7. (WO is wqo: the class of well orderings is well-quasi-ordered.) For some i Proof. The equivalence of ATR0 and Clause 2 was announced by Friedman in [2]. The result appears as Theorem V.6.8 in Simpson’s book [13]. Clause 3 follows from Theorem 3.21 of [3] and clause 4 follows from Theorem 4.4 of [6]. Clauses 5 and 7 are due to Shore [11]. The much easier proofs for 6 and 8 appear as Theorem 5.4 of [3]. The equivalence of clause 9 is Theorem 2 of [7]. Clause 10 follows from Theorem 5.2 of [3] and clause 11 follows from Theorem 2.9 of [3]. a Using clause 7 of the preceding theorem, given any sequence of well orderings (comparable or not) one can locate a pair that is ordered in accordance with 4 J. HIRST their subscripts. This is strictly stronger than asserting that there is no in - nite strictly descending sequence of well orderings, as shown by the following theorem. Theorem 5. (ACA0) The following are equivalent: 1 1. 1 AC0. 2. There is no sequence h i|i ∈ Ni of well orderings such that i+1 Theorem 6. (ACA0) The following are equivalent: 1 1 1. (Bounded 1 AC0) For any 1 formula and any b, (∀k Theorem 7. (RCA0) The following are equivalent: 1. ACA0. 2. If and are well orderings with s and 6 s , then Theorem 8. (RCA0) The following are equivalent: 1. ATR0. 2. If and are well orderings such that w and 6 w , then Theorem 9. (RCA0) The following are equivalent: 1. ACA0. SURVEY OF ORDINAL ARITHMETIC 5 2. If is a well ordering such that ω w and w ω, then s ω. 3. If is a well ordering such that ω w and 6 w ω, then ω Theorem 10. (RCA0) The following are equivalent: 1. ATR0. 1 2. For any 1 formula (X) we have ∀X ( (X) → WO(X)) →∃ (WO( ) ∧∀X( (X) → X s )) . Proof. Theorem V.6.9 of [13]. a ATR0 is necessary and su cient to prove the existence of unique suprema of sequences of ordinals. The following theorem holds for both s and w. Theorem 11. (RCA0) The following are equivalent: 1. ATR0. 2. Suppose h x | x ∈ i is a well ordered sequence of well orderings. Then suph x | x ∈ i exists. That is, there is a well ordering unique up to order isomorphism satisfying ∀x ∈ ( x ), and ∀ ( < →∃x ∈ ( x 6 )). Proof. Theorem 7 of [5]. a §3. Addition. This section explores the properties of ordinal addition. The section begins with a de nition of addition and a veri cation that ad- dition preserves well orderings. Then two theorems describing the interaction of addition and comparability are presented. The section concludes with two theorems on triangular numbers. Definition. h | ∈ i PLet b b be a well ordered sequence of well orderings. { | ∈ ∧ ∈ } The notation b∈ b denotes the set (b, a) b a b ordered by the relation (b0,a0) < (b1,a1) if and only if b0 contains an in nite descending sequence in bi , yielding another contradiction and completing the proof. a Weak comparability and strong comparability behave di erently with re- spect to addition, as shown by the next two theorems. Theorem 13. (RCA0) If h b | b ∈ i and h b | b ∈ i are well ordered se- h | ∈ i quences of well orderings, and fb b P is a sequenceP of functions witnessing ∈ that b w b for each b , then b∈ b w b∈ b. In particular, if 0 w 0 and 1 w 1 are well orderings, then 0 + 1 w 0 + 1. P P Proof. The order preserving injection witnessing b∈ b w b∈ b can be directly constructed from those witnessing b w b for each b. a Theorem 14. (RCA0) The following are equivalent: 1. ATR0. h | ∈ i h | ∈ i 2. If b b and b b are well orderedP sequencesP of well orderings ∈ such that b s b for each b , then b∈ b s b∈ b. 3. If 0 s 0 and 1 s 1 are well orderings, then 0 + 1 s 0 + 1. Proof. First we will prove clause 2 using ATR0. Let h b | b ∈ i and h | ∈ i b b be well ordered sequences of well orderingsP such that b Ps b for ∈ each b . As a notational convenience, let = b∈ b and = b∈ b. By Theorem 12, both and are well ordered. By strong comparability of well orderings (Theorem 4 clause 2), either s or s .If s , then we are done. Suppose that f witnesses that s .Iff maps onto a proper initial segment of , then there must be a least b ∈ such that f maps the least element of b into c where c Theorem 16. (RCA0) The following are equivalent: SURVEY OF ORDINAL ARITHMETIC 7 1. ATR0. 2. ( -lemma) Suppose that h b | b ∈ ω i is a well ordered sequence of well 0 0 6 orderings such that b §4. Multiplication. This section begins with a de nition of ordinal mul- tiplication and veri cation that products are well ordered and multiplication is associative. This is followed by results on comparability and multiplication, distributive laws, division algorithms, right factors and prime factors. Definition. If and are well orderings, the product is the set {(a, b) | a ∈ ∧ b ∈ }, ordered by the relation (a1,b1) < (a2,b2)ifand only if (b1 Theorem 17. (RCA0) Suppose that , and are well ordered. Then is well ordered, and ( ) s ( ) . Proof. Working in RCA0, one can show that if contains an in nite descending sequence, then either or must contain an in nite descending sequence. The associative law is proved by direct construction of a bijection between the orderings. a As with addition, weak comparability and strong comparability interact in di erent fashions with multiplication. Theorem 18. (RCA0) If 0, 1, 0, and 1 are well orderings such that 0 w 1 and 0 w 1, then 0 0 w 1 1. Proof. Lemma 6 of [7]. a Theorem 19. (RCA0) The following are equivalent: 1. ATR0. 2. If 0, 1, 0, and 1 are well orderings such that 0 s 1 and 0 s 1, then 0 0 s 1 1. Proof. Theorem 7 of [7]. a In ordinal arithmetic, multiplication on the right by a successor ordinal preserves strict inequalities. This statement varies in strength depending on the form of comparability used. Recall that a well ordering with a largest element is called a successor. Theorem 20. (RCA0) If 0, 1 and are wellorderings, is a successor, and 0 Proof. Lemma 11 of [7]. a Theorem 21. (RCA0) The following are equivalent: 1. ATR0. 2. If 0, 1 and are well orderings, is a successor, and 0 The statements in the following omnibus theorem can be proved in RCA0 for both types of comparability. Theorem 22. (RCA0) The following statements hold for all well orderings. 1. If =06 , then . Furthermore, ( )+1 . 2. If =06 and 0 1, then 0 1. 3. If 0 < 1, then 0 1. 4. If ( 0) < 1, then 0 < 1. 5. If ( 0 ) < 1 , then 0 < 1. 6. If is a successor and 0 1 , then 0 1. The left distributive law for ordinal multiplication over ordinal addition is provable in RCA0. The corresponding right distributive law fails. Sherman’s inequality, which is a weak version of the right distributive law is equivalent to ATR0. These results hold for both forms of comparability and are stated as the next two theorems. Theorem 23. (RCA0) For well orderings , , and , ( + ) + . Proof. The bijection can be constructed using only recursive comprehen- sion and the de nitions of the arithmetical operations. a Theorem 24. (RCA0) The following are equivalent: 1. ATR0. 2. (Sherman’s Inequality) If , , and are well orderings, then ( + ) + . Proof. Theorem 5.4 of [6]. a Some special instances of the right distributive law do hold. The following example is useful in manipulating ordinals expressed in Cantor normal form. Ordinal exponentiation is de ned in the next section. Theorem 25. (RCA0) If m0,m1, ...mk > 0, >0 is a well ordering, and 0, 1, ... k are initial segments of a well ordering that satisfy i+1 0 1 k 0+ (ω m0 + ω m1 + + ω mk)ω s ω . Proof. Lemma 3 of [9]. a SURVEY OF ORDINAL ARITHMETIC 9 The next two theorems give three versions of the division algorithm. The version in the rst theorem is a strong comparability analog of clause 2 in the second theorem. Theorem 26. (RCA0) If , and are well orderings satisfying Theorem 27. (RCA0) The following are equivalent: 1. ATR0. 2. If , and are well orderings satisfying Theorem 28. (RCA0) If =06 and =06 are well orderings, then w . Proof. Lemma 8 of [7]. a Theorem 29. (RCA0) The following are equivalent: 1. ATR0. 2. If =06 and =06 are well orderings, then s . Proof. Theorem 9 of [7]. a A ordinal is said to be prime if whenever = , either =1or = . The notion of prime can be formalized using either weak or strong compara- bility. Using either formalization, the following theorem holds. Theorem 30. (RCA0) The following are equivalent: 1. ATR0. 2. Every well ordering has a prime right factor. That is, if is a well ordering, then there are well orderings and such that and is prime. 3. If is a well ordering, then for some k ∈ N, there are prime well orderings 1, 2... k such that k k 1 1. Proof. Theorem 6 and Corollary 7 of [9]. a 10 J. HIRST §5. Exponentiation. This section begins with the de nition of ordinal exponentiation, discusses closure and basic properties of exponentiation, and concludes with a theorem on the existence of ordinal logarithms. Definition. Let and be well orderings. The set exp( , ) is the col- lection of all nite sequences of the form h(b0,a0), (b1,a1),...,(bn,an)i such that (1) for all i n, bi ∈ and 0 =6 ai ∈ , and (2) whenever i Theorem 31. (RCA0) The following are equivalent: 1. ACA0. 2. If and are well ordered, then so is . 3. If is well ordered, then so is 2 . Proof. This result is included in a larger equivalence theorem proved by Girard in [4]. For a direct proof, see Theorem 2.6 of [6]. a RCA0 su ces to prove numerous basic properties of exponentiation. The next theorem holds with either strong or weak of comparability. Theorem 32. (RCA0) Suppose that , and are well orderings. The following hold: 1. + . 2. ( ) ( ). 3. implies . 4. implies . 5. 2ω ω. 6. ω 2(ω ). Proof. Theorems 2.3, 2.4, and 2.5 of [6] a The following useful partial converse to clause 4 of Theorem 32 is provable in ACA0. Theorem 33. (ACA0) If and are well orderings and ω w ω , then w . SURVEY OF ORDINAL ARITHMETIC 11 Proof. Lemma 4.3 of [6] a To conclude this section, we list two results on the existence and uniqueness of logarithms. The next result holds with both strong and weak comparability. Theorem 34. (RCA0) The following are equivalent: 1. ATR0. 2. (Existence of logarithms) If >1 and are well orderings, then there is a well ordering such that < +1. Proof. Theorem 2.7 of [6]. a Theorem 35. (ATR0) (Uniqueness of logarithms) If , , and are well +1 +1 orderings such that w §6. Other topics. This section contains formalized versions of Cantor’s normal form theorem, several results about indecomposable well orderings, and some trans nite induction schemes. In the last paragraph, we list some related topics in reverse mathematics which were not included in this survey. We will begin with two normal form results. Theorem 36. (RCA0) Let >1 and be well orderings. Fix an element of , x0 = h(b0,a0),...,(bn,an)i.Let = {x ∈ | x 0 1 n s 0 + 1 + + n. Proof. Lemma 5.1 of [6]. a The next theorem holds with both forms of comparability. Recall that the notation < is used to abbreviate +1 . Theorem 37. (RCA0) The following are equivalent: 1. ATR0. 2. If >1 and are well orderings then there are nite sequences of well orderings 0 > 1 > > n and 0, 1,..., n such that 0 < i < for each i n and 0 1 n 0 + 1 + + n. Furthermore, this representation is unique in the following sense. If 0 0 0 0 0 0 0 1 m 0 + 1 + + m is a similar representation of , then m = n 0 0 and for every i, i i and i i. 3. (Cantor’s normal form theorem) If is a well ordering then there is a nite sequence 0 > 1 > > n of well orderings and a nite collection d0,...dn of positive integers such that 0 n ω d0 + + ω dn. Proof. Theorem 5.2 of [6]. a 12 J. HIRST Recall that a well ordering is indecomposable if for every nal segment of , we have w . The next theorem lists useful properties of indecomposable well orderings that are provable in RCA0. Clause 1 is a particularly handy fact. Note that in clause 5, the hypothesis must include the requirement that ω is well ordered. For some well ordered sets , ACA0 may be required to prove that ω is well ordered, as shown in Theorem 31. Theorem 38. (RCA0) Suppose that , , and are well orderings. 1. If is indecomposable and Theorem 40. For j 2 and k 1, RCA0 proves that the following are equivalent: 1. ACA0. 2. ATI0. 0 3. k TI0. 0 4. k TLE0. 0 5. j TI0. 0 6. j TLE0. Proof. Simpson proves that ACA0 implies ATI0 in Lemma V.2.1 of [13]. The least element schemes can be deduced from ATI0 by the usual arguments. Corollary 3 and Corollary 4 of [8] provide the reversals. a SURVEY OF ORDINAL ARITHMETIC 13 This survey only addresses well orderings and the formalized operations of ordinal arithmetic. It does not include references to a substantial body of literature related to this topic. For example, the formalization of ordinal notations in reverse mathematics, proof theoretic ordinals for the subsystems, and various applications of ordinal arithmetic to algebra, analysis, topology, graph theory and quasi-ordering theory have all been omitted. As always, [13] is strongly recommended as a source for further reading and references. The interested reader will be able to discover numerous open questions on the reverse mathematics of ordinal arithmetic. Most of the results in this survey were motivated by Cantor’s introductory articles [1]. Many topics in Chapter XIV of Sierpi nski’s book [12] could be formalized in second order arithmetic and analyzed. Sometimes even a single exercise can motivate a substantial number of new results. REFERENCES [1] Georg Cantor, Beitr age zur Begr undung der trans niten Mengenlehre, Math. Ann., vol. 49 (1897), pp. 207–246, English translation by P. Jourdain published as Contri- butions to the founding of the theory of trans nite numbers, Dover, New York, 1955. [2] Harvey M. Friedman, Systems of second order arithmetic with restricted induction, I, II (abstracts), J. Symbolic Logic, vol. 41 (1976), no. 2, pp. 557–559. [3] Harvey M. Friedman and Jeffry L. Hirst, Weak comparability of well orderings and reverse mathematics, Ann. Pure Appl. Logic, vol. 47 (1990), no. 1, pp. 11–29. [4] Jean-Yves Girard, Proof theory and logical complexity, Bibliopolis, Naples, 1987. [5] Jeffry L. Hirst, Reverse mathematics and ordinal suprema, elsewhere in this vol- ume. [6] , Reverse mathematics and ordinal exponentiation, Ann. Pure Appl. Logic, vol. 66 (1994), no. 1, pp. 1–18. [7] , Reverse mathematics and ordinal multiplication, MLQ Math. Log. Q., vol. 44 (1998), no. 4, pp. 459–464. [8] , Ordinal inequalities, trans nite induction, and reverse mathematics, J. Symbolic Logic, vol. 64 (1999), no. 2, pp. 769–774. [9] , Reverse mathematics of prime factorization of ordinals, Arch. Math. Logic, vol. 38 (1999), no. 3, pp. 195–201. [10] , Reverse mathematics and rank functions for directed graphs, Arch. Math. Logic, vol. 39 (2000), to appear. [11] Richard A. Shore, On the strength of Fra ss e’s conjecture, Logical methods (Ithaca, NY, 1992), Birkh auser Boston, Boston, MA, 1993, pp. 782–813. [12] Wac law Sierpinski , Cardinal and ordinal numbers, Polska Akademia Nauk, Monogra e Matematyczne, Pa nstwowe Wydawnictwo Naukowe, Warszawa, 1958. [13] Stephen G. Simpson, Subsystems of second order arithmetic, Springer-Verlag, Berlin, 1999. DEPARTMENT OF MATHEMATICAL SCIENCES APPALACHIAN STATE UNIVERSITY BOONE, NC 28608 USA E-mail: [email protected] URL: www.mathsci.appstate.edu/ jlh