Cobham Recursive Set Functions and Weak Set Theories

March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 55

COBHAM RECURSIVE SET FUNCTIONS AND WEAK SET THEORIES

Arnold Beckmann Department of Computer Science Swansea University Swansea SA2 8PP, UK [email protected]

Sam Bussa Department of Mathematics University of California, San Diego La Jolla, California 92130-0112, USA [email protected]

Sy-David Friedmanb Kurt G¨odelResearch Center Universit¨atWien A-1090 Vienna, Austria [email protected]

Moritz Müller Kurt GödelResearch Center UniversitätWien A-1090 Vienna, Austria [email protected]

aSupported in part by NSF grants DMS-1101228 and CCR-1213151, and by the Simons Foundation, award 306202. bSupported in part by FWF (Austrian Science Fund) through Project P24654.

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56 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

Neil Thapenc Institute of Mathematics Czech Academy of Sciences 115 67 Prague, Czech Republic [email protected]

The Cobham recursive set functions (CRSF) provide a notion of polynomial time computation over general sets. In this chapter, we determine u a subtheory KP1 of Kripke-Platek set theory whose ⌃1-definable functions are precisely CRSF. The theory KPu is based on the -induction 1 2 scheme for ⌃1-formulas whose leading existential quantifier satisfies certain boundedness and uniqueness conditions. Dropping the uniqueness 4 condition and adding the axiom of global choice results in a theory KPC1 C whose ⌃1-definable functions are CRSF , that is, CRSF relative to a global choice function C. We further show that the addition of global choice is conservative over certain local choice principles.

1. Introduction Barwise begins his chapter on admissible set recursion theory with: “There are many equivalent definitions of the class of recursive functions on the natural numbers. [. . . ] As the various definitions are lifted to domains other than the integers (e.g., admissible sets) some of the equivalences break down. This break-down provides us with a laboratory for the study of recursion theory.” ([5, p.153]) Let us informally distinguish two types of characterization of the computable functions or subsets thereof, namely, recursion theoretic and definability theoretic ones. Recursion theoretically, the computable functions on ! are those obtainable from certain simple initial functions by means of composition, primitive recursion and the µ-operator. As a second example, the primitive recursive functions are similarly defined but without the µ-operator. A third example is the recursion theoretic definition of the polynomial time functions by Cobham recursion [13] or by Bellantoni- Cook safe-normal recursion [9]. Definability theoretically, the computable functions are those that are ⌃1-definable in the true theory of arithmetic. A more relevant example of a definability theoretic definition is the clas-

cResearch leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC grant agreement 339691. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO:67985840. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 57

Cobham Recursive Set Functions and Weak Set Theories 57

sic theorem of Parsons and Takeuti (see [12]) that the primitive recursive functions are those that are ⌃1-definable in the theory I⌃1; namely, one ad- ditionally requires that this theory proves the totality and functionality of the defining ⌃1-formula. Analogously, the polynomial time functions have b 1 a definability theoretic definition as the ⌃1-definable functions of S2 [11]. For more definability theoretic definitions of weak subrecursive classes, see Cook-Nguyen [15]. Admissible set recursion theory provides a definability theoretic generalization of computability: one considers functions which are ⌃1-definable (in the language of set theory) in an admissible set, that is, a transitive standard model of Kripke-Platek set theory KP. Recall that KP consists of the axioms for Extensionality, Union, Pair, 0-Separation, 0-Collection and -Induction for all formulas '(x,w ~ ): 2 y ( u y'(u,w ~ ) '(y,w ~ )) '(x,w ~ ). 8 8 2 ! ! To some extent this generalization of computability extends to the recursion theoretic view. By the ⌃-Recursion Theorem ([5, Chapter I, The- orem 6.4]) the ⌃ -definable functions of KP are closed under -recursion. 1 2 This implies that the primitive recursive set functions (PRSF) of [20] are all ⌃1-definable in KP. By definition, a function on the universe of sets is in PRSF if it is obtained from certain simple initial functions by means of composition and -recursion. Hence, PRSF is a recursion theoretic gener- 2 alization of the primitive recursive functions. Paralleling Parson’s theorem, Rathjen [22] showed that this generalization extends to the definability theoretic view: PRSF contains precisely those functions that are ⌃1-definable in KP , the fragment of KP where -Induction is adopted for ⌃ -formulas 1 2 1 only. One can thus view PRSF as a reasonable generalization of primitive recursive computation to the universe of sets. It is natural to wonder whether one can find a similarly good analogue of polynomial time computation on the universe of sets. In [7] we proposed such an analogue, following Cobham’s [13] characterization of the polynomial time computable functions on ! as those obtained from certain simple initial functions, including the smash function #, by means of composition and limited recursion on notation. Limited recursion on notation restricts both the depth of the recursion and the size of values. Namely, a recursion on notation on x has depth roughly log x; being limited means that all values are required to be bounded by some smash term x# #x. ··· In [7] a smash function for sets is introduced. The role of recursion on notation is taken by -recursion, and being limited is taken to mean being 2 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 58

58 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

in a certain sense embeddable into some #-term. In this way, [7] defines the class of Cobham recursive set functions (CRSF), a recursion theoretic generalization of polynomial time computation from ! to the universe of sets. This chapter extends the analogy to the definability theoretic view. A definability theoretic characterization of polynomial time on ! has been given by Buss (cf. [12]). It is analogous to Parsons’ theorem, with 1 b I⌃1 replaced by S2 and ⌃1 replaced by a class ⌃1 of “bounded” ⌃1-formulas. 1 The theory S2 has a language including the smash function # and is based b on a restricted form of induction scheme for ⌃1-formulas, in which the depth of an induction is similar to the depth of a recursion on notation. 1 Both directions in Buss’ characterization hold in a strong form. First, S2 defines polynomial time functions in the sense that one can conservatively b add ⌃1-defined symbols and prove Cook’s PV1 [14], a theory based on the equations that can arise from derivations of functions in Cobham’s calculus. 1 Second, polynomial time functions witness simple theorems of S2. More precisely, if S1 proves y'(y,~x)with' in b , that is, a “bounded” - 2 9 0 0 formula, then ~x' (f(~x ),~x) is true for some polynomial time computable f, 8 1 even provably in PV1 and S2. In this chapter, we analogously replace Rathjen’s theory KP1 and ⌃1- 4 4 4 definability with a theory KP1 and ⌃1 -definability; here ⌃1 -formulas are “bounded” ⌃1-formulas, defined using set smash and the embeddability 4 notion 4 of [7]. The theory KP1 has a finite language containing, along with , some basic CRSF functions including the set smash and has - 2 4 2 Induction restricted to ⌃1 -formulas. 4 As we shall see, KP1 defines CRSF analogously to the first part of Buss’ characterization. An analogy of the second part, the witnessing theorem, would state that whenever KP4 proves y'(y,~x) for a -formula ',then 1 9 0 '(f(~x ),~x) is provable in ZFC (or ideally in a much weaker theory) for some f in CRSF. But this fails: a function witnessing y (x =0 y x) would 9 6 !4 2 be a global choice function, and this is not available in KP1 . We discuss two ways around this obstacle. If we add the axiom of global 4 choice we get a theory KPC1 and indeed can prove a witnessing theorem as 4 desired (Theorem 6.13). The functions ⌃1-definable in KPC1 are precisely those that are CRSF with a global choice function as an additional initial 1 function (Corollary 6.2). Thus, Buss’ theorems for S2 and polynomial time on ! have full analogues on universes of sets equipped with global choice, if we consider the global choice function as a feasible function in such a universe. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 59

Cobham Recursive Set Functions and Weak Set Theories 59

Our second way around the obstacle is to further weaken the induction scheme, the crucial restriction being that the witness to the existential 4 quantifier in a ⌃1 -formula is required to be unique. The resulting theory u KP1 still defines CRSF in the strong sense that one can conservatively 4 add ⌃1 -defined function symbols and prove Tcrsf , an analogue of PV1 containing the equations coming from derivations in the CRSF calculus (The- u orem 5.2). We prove a weak form of witnessing (Theorem 6.10): if KP1 proves y'(y,~x) for ' a -formula, then T proves y f(~x ) '(y,~x) for 9 0 crsf 9 2 some f in CRSF. This suces to infer a definability theoretic characterization of CRSF on an arbitrary universe of sets: the ⌃1-definable functions of u KP1 are precisely those in CRSF (Corollary 6.1). We do not know whether 4 this holds for KP1 . 4 In conclusion, we address the question on how much stronger KPC1 4 is compared to KP1 . We show that the di↵erence can be encapsulated in certain local choice principles.

The outline of the chapter is as follows. Section 2 recalls the development of CRSF from [7]. For the formalizations in this chapter we will use a slightly di↵erent, but equivalent, definition of CRSF which we describe in 4 u Proposition 2.9. Section 3 defines the three theories KP1 ,KP1 and Tcrsf mentioned above. They extend a base theory T0 that we “bootstrap” in Section 4, in particular deriving various lemmas which allow us to manip- ulate embedding bounds. Section 5 proves the Definability Theorem 5.2 u u 4 for KP1 . A technical diculty is that KP1 is too weak to eliminate ⌃1 - defined function symbols in the way this is usually done in developments of 1 KP or S2 (see Section 5.2). Section 6 proves the Witnessing Theorems 6.10 u 4 and 6.13 for KP1 and KPC1 , and the Corollaries 6.1 and 6.2 on definability theoretic characterizations of CRSF. This is done via a modified version of Avigad’s model-theoretic approach to witnessing [4] (see Section 6.1). Our proof gives some insight about CRSF: roughly, its definition can be given using only a certain simple form of embedding (see Section 6.4). Section 7 proves the conservativity of global choice over certain local choice principles (Theorem 7.2). Here, we use a class forcing as in [17] to construct a generic expansion of any (possibly non-standard) model of our set theory. Some extra care is needed since our set theory is rather weak.

Several related recursion theoretic notions of polynomial time set functions have been described earlier by other authors. The characterization of polynomial time by Turing machines has been generalized in Hamkins and Lewis [18] to allow binary input strings of length !. We refer to [8] for March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 60

60 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

some comparison with CRSF. Yet another characterization of polynomial time comes from the Immerman-Vardi Theorem from descriptive complexity theory (cf. [16]). Following this, Sazonov [23] gives a theory operating with terms allowing for least fixed-point constructs to capture polynomial time computations on (binary encodings of) Mostowski graphs of hereditarily finite sets. Not all of Sazonov’s set functions are CRSF [7]. But under a suitable encoding of binary strings by hereditarily finite sets, CRSF does capture polynomial time [7, Theorems 30, 31]. Arai [2] gives a di↵erent such class of functions. His Predicatively Com- putable Set Functions (PCSF) form a subclass of the Safe Recursive Set Functions (SRSF) from [6]. SRSF is defined in analogy to Bellantoni and Cook’s recursion theoretic characterization of polynomial time [9], di↵erent from Cobham’s. Bellantoni and Cook’s functions have two sorts of arguments, called “normal” and “safe”, and recursion on notation is allowed to recurse only on normal arguments, while values obtained by such recur- sions are safe. Similarly, SRSF and PCSF contain two-sorted functions. It is shown in [7] that CRSF coincides with the functions having only normal arguments in PCSF+ (from [2]), a slight strengthening of PCSF. In a re- cent manuscript [1], Arai gives a definability theoretic characterization of PCSF◆, a class of set functions intermediate between PCSF and PCSF+. He proves a weak form of witnessing akin to ours. He uses two-sorted set- theoretic proof systems whose normal sort ranges over a transitive substructure of the universe, and which contains an inference rule ensuring u closure of this substructure under certain definable functions. Like KP1 , these systems contain a form of “unique” ⌃1-Induction. As in our setting, eliminating defined function symbols is problematic; the final system in [1] is a union of a hierarchy of systems, each level introducing infinitely many function symbols. Thus, dealing with similar problems, Arai’s solution is quite di↵erent from the one presented here; as is his proof, which is based on cut-elimination.

2. Cobham Recursive Set Functions In this section we review some deﬁnitions and results from [7]. In later sections, many of these results will be formalized in suitable fragments of KP. As mentioned in the introduction, [7] generalizes Cobham’s recursion theoretic characterization of polynomial time to arbitrary sets. We recall x y Cobham’s characterization. On ! the smash x#y is deﬁned as 2| |·| | where March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 61

Cobham Recursive Set Functions and Weak Set Theories 61

x := log(x+1) is the length (of the binary representation) of x.Wehave | | d e successor functions s0(x):=2x and s1(x)=2x + 1 which add respectively 0 and 1 to the end of the binary representation of x.

Theorem 2.1: (Cobham 1965) The polynomial time functions on ! are r obtained from initial functions, namely, projections ⇡j (x1,...,xr):=xj, constant 0, successors s0,s1 and the smash #, by composition and limited recursion on notation:ifh(~x ), g0(y, z,~x), g1(y, z,~x) and t(y,~x) are polynomial time, then so is the function f(y,~x) given by f(0,~x)=h(~x ), f(s (y),~x)=g (y, f(y,~x),~x) for b 0, 1 and s (y) =0, b b 2{ } b 6 provided that f(y,~x) 6 t(y,~x) holds for all y,~x. One can equivalently ask t to be built by composition from only projections, 1 and #; or just demand f(y, x ,...,x ) p( y , x ,..., x ) for some | 1 k | 6 | | | 1| | k| polynomial p. We move to some fixed universe of sets, that is, a model of ZFC. The analogue of smash defined in [7] is best understood in terms of Mostowski graphs. The Mostowski graph of a set x has as vertices the elements of the transitive closure tc+(x):=tc( x ) and has a directed edge from u to v if { } u v. Every such graph has a unique source and a unique sink. 2 The set smash x#y replaces each vertex of x by (a copy of the graph of) y with incoming edges now going to the source of y and outgoing edges now leaving the sink of y. It can be defined using set composition x y, which places a copy of x above y and identifies the source of x with the sink of y. Writing 0 for , ; y if x =0, x y := u y : u x otherwise ⇢ { 2 } x#y := y u#y : u x . { 2 } The Mostowski graph of x#y is isomorphic to the graph with vertices + + tc (x) tc (y) and directed edges from u0,v0 to u, v if either u0 = u ⇥ h i h i ^ v0 v or u0 u v0 = y v = 0 (see [7, Section 2]). An isomorphism is 2 2 ^ ^ given by (u, v):=v u0#y : u0 u . x,y { 2 } A#-term is built by composition from projections, #, and the constant 1 = 0 . Such terms serve as analogues of polynomial length bounds, { } with the bounding relation 4 defined as follows: x 4 y means that there is a (multi-valued) embedding that maps vertices u tc(x) to pairwise dis- 2 joint non-empty sets V tc(y) such that whenever u0 u and v V , u ✓ 2 2 u March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 62

62 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

then there exists v0 V tc(v). The notation ⌧( ,w~ ):x y means that 2 u0 \ · 4 u ⌧(u,w ~ ) is such an embedding. Then [7] generalizes Cobham’s deﬁnition 7! as follows.

Deﬁnition 2.2: The Cobham recursive set functions (CRSF) are obtained from initial functions, namely projections, constant 0 := , pair x, y , ; { } union x, set smash x#y, and the conditional

S x if u v cond (x, y, u, v):= 2 2 y otherwise, ⇢ by composition and Cobham recursion:ifg(x, z,w ~ ), ⌧(u, x,w ~ ) and t(x,w ~ ) are CRSF, then so is the function f(x,w ~ ) given by

f(x,w ~ )=g(x, f(y,w ~ ):y x ,w~ ) { 2 } provided that ⌧( ,x,w~ ):f(x,w ~ ) t(x,w ~ ) holds for all x,w ~ . · 4 Here the embedding proviso ⌧( ,x,w~ ):f(x,w ~ ) t(x,w ~ )ensures,in- · 4 tuitively, that a deﬁnition by recursion is allowed only provided that we can already bound the “structural complexity” of the deﬁned function f. A relation is CRSF if its characteristic function is. Direct arguments show (see [7, Theorem 13]):

Proposition 2.3:

(a) (Separation) If g(u,w ~ ) is in CRSF, then so is f(x,w ~ ):= u x : { 2 g(u,w ~ ) =0 . 6 } (b) The CRSF relations contain x y and x = y, are closed under Boolean 2 combinations and -bounded quantiﬁcations u x and u x . 2 9 2 8 2 It is then not hard to show that transitive closure tc(x), set composition x y, the isomorphism (u, v) and its inverses ⇡ (z),⇡ (z) x,y 1,x,y 2,x,y are CRSF [7, Theorem 13]. In particular, #-terms are CRSF. Further, one can derive the following central lemma [7, Lemma 20]. It says that 4 is a pre-order and that #-terms enjoy some monotonicity properties one would expect from a reasonable analogue of “polynomial length bounds”.

Lemma 2.4: Below if ⌧0 and ⌧1 are in CRSF then can also be chosen in CRSF.

(a) (Transitivity) If ⌧ ( ,x,y,w~ ):x y and ⌧ ( ,y,z,w~ ):y z, then 0 · 4 1 · 4 there exists (u, x, y, z,w ~ ) such that ( ,x,y,z,w~ ):x z. · 4 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 63

Cobham Recursive Set Functions and Weak Set Theories 63

(b) (Monotonocity) Let t(x,w ~ ) be a #-term. If ⌧ ( ,x,z,w~ ):z t(x,w ~ ) 0 · 4 and ⌧ ( ,x,y,w~ ):x y, then there exists (u, x, y, z,w ~ ) such that 1 · 4 ( ,x,y,z,w~ ):z t(y,w ~ ). · 4 Based on this lemma, a straightforward induction on the length of a derivation of a CRSF function shows [7, Theorem 17]:

Theorem 2.5: (Bounding) For every f(~x ) in CRSF there are a #-term t(~x ) and a CRSF function ⌧(u, ~x) such that ⌧( ,~x):f(~x ) t(~x ). · 4 In fact, in the deﬁnition of Cobham recursion one can equivalently require the function t in the embedding proviso to be a #-term [7, Theorem 21]. Using the Bounding Theorem and the Monotonicity Lemma one can obtain, similarly to Theorems 23, 29 and 30 of [7]:

Theorem 2.6:

(a) (Replacement) If f(y,w ~ ) is CRSF, then so is

f”(x,w ~ )= f(y,w ~ ):y x . { 2 } (b) (Course of values recursion) If g(x, z,w ~ ), ⌧(u, x,w ~ ) and t(x,w ~ ) are CRSF, then so is

f(x,w ~ ):=g(x, u, f(u,w ~ ) : u tc(x) ,w~ ) {h i 2 } provided ⌧( ,x,w~ ):f(x,w ~ ) t(x,w ~ ) holds for all x,w ~ . · 4 (c) (Impredicative Cobham recursion) If g(x, z,w ~ ), ⌧(u, y, x,w ~ ) and t(x,w ~ ) are CRSF, then so is

f(x,w ~ )=g(x, f”(x,w ~ ),w~ )

provided ⌧( ,f(x,w ~ ),x,w~ ):f(x,w ~ ) t(x,w ~ ) holds for all x,w ~ . · 4 Closure under replacement (a) implies that x y is CRSF ⇥ [7, Theorem 14]. Impredicative Cobham recursion (c) is, intuitively, some- what circular in that the embedding ⌧ may use as a parameter the set f(x,w ~ ) whose existence it is supposed to justify. We introduce a variant deﬁnition of CRSF that uses syntactic Cobham recursion. The name “syntactic” indicates that it does not have an embedding proviso, but rather constructs a new function from any CRSF functions g, ⌧ and #-term t. We also allow the bound to be impredicative in the sense of (c) above. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 64

64 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

Deﬁnition 2.7: Let g(x, z,w ~ ) and ⌧(u, y, x,w ~ ) be functions and t(x,w ~ )a #-term. Then syntactic Cobham recursion gives the function f(x,w ~ )de- ﬁned by g(x, f”(x,w ~ ),w~ )if⌧ is an embedding into t at x,w ~ f(x,w ~ )= (2.1) 0 otherwise ⇢ where the condition “⌧ is an embedding into t at x,w ~ ” stands for ⌧( ,g(x, f”(x,w ~ ),w~ ),x,w~ ):g(x, f”(x,w ~ ),w~ ) t(x,w ~ ). (2.2) · 4 Note that Proposition 2.3 implies that the condition (2.2) is a CRSF relation, cf. (3.2) in Section 3.2. Proposition 2.8: The CRSF functions are precisely those obtained from the initial functions by composition and syntactic Cobham recursion.

Proof: Since the condition (2.2) is a CRSF relation, (2.1) can be written f(x,w ~ )=g0(x, f”(x,w ~ ),w~ ) for some g0 in CRSF. The embedding proviso ⌧( ,f(x,w ~ ),x,w~ ):f(x,w ~ ) t(x,w ~ ) holds for all x,w ~ , since either (2.2) · 4 holds, in which case f(x,w ~ )=g(x, f”(x,w ~ ),w~ ) so (2.2) gives us the embedding, or f(x,w ~ ) = 0, in which case any function is an embedding of f(x,w ~ )intot(x,w ~ ). We thus get that f is CRSF by impredicative Cobham recursion. Conversely, assume f(x,w ~ ) is obtained from g, ⌧, t by Cobham recursion, and in particular that the embedding proviso is satisfied for all x,w ~ .Thenf satisfies (2.1), so f can be obtained by syntactic Cobham recursion. By [7, Theorem 21], we can assume that t is a #-term. ⇤ The next proposition describes the definition of CRSF that we will formalize with the theory Tcrsf in Section 3.4. Closure under replacement and the extra initial functions are included to help with the formalization. Proposition 2.9: The CRSF functions are precisely those obtained from projection, zero, pair, union, conditional, transitive closure, cartesian product, set composition and set smash functions by composition, replacement and syntactic Cobham recursion. Remark 2.10: For an arbitrary function g(~x ), let CRSFg be defined as CRSF but with g(~x ) as additional initial function. This class might be interpreted as a set-theoretic analogue of polynomial time computations with an oracle function g(~x ). If there is ⌧(u, ~x)inCRSFg such that ⌧( ,~x):g(~x ) t(~x ) for some #-term t(~x ), then all results mentioned in · 4 this section “relativize”, that is, hold true with CRSF replaced by CRSFg. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 65

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3. Theories for CRSF

3.1. The language L0 and theory T0 The language L contains the relation symbol and symbols for the fol- 0 2 lowing CRSF functions: 0, 1, x, x, y ,x y, tc(x),x y, x#y. { } ⇥ The meaning of theseS symbols is given by their defining axioms: symbol defining axiom 0 u/0 2 1 u 1 u =0 2 $ x u x y x (u y) 2 $9 2 2 x, y u x, y u = x u = y { S } 2S{ }$ _ x y u x y x0 x y0 y (u = x0,y0 ) ⇥ 2 ⇥ $9 2 9 2 h i tc(x) u tc(x) u x y x (u tc(y)) 2 $ 2 _9 2 2 x y 0 y = y (x =0 x y = z y : z x ) ^ 6 ! { 2 } x#y w tc+(x#y)(w = z#y : z x (x#y = y w)) 9 2 { 2 }^ The table above uses some special notations: as usual, x stands for the { } term x, x , x, y for the term x , x, y , x y for the term x, y , and { } h i {{ } { }} [ { } x y for the formula u x (u y). We write tc+(x) for the term tc( x ). ✓ 8 2 2 S { } The final two lines of the table use “replacement terms”. More generally, we use three types of comprehension terms:

Deﬁnition 3.1: The following notations are used for comprehension terms: - Proper comprehension terms: for a formula '(u, ~x), we write z = u x : '(u, ~x) { 2 } for u z (u x '(u, ~x)) u x ('(u, ~x) u z). 8 2 2 ^ ^8 2 ! 2 - Collection terms: for a formula '(u, v, ~x), we write z = v : u x'(u, v, ~x) { 9 2 } for v z u x'(u, v, ~x) v (( u x'(u, v, ~x)) v z). (3.1) 8 2 9 2 ^8 9 2 ! 2 - Replacement terms: for a term t(u, ~x), we write z = t(u, ~x):u x { 2 } March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 66

66 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

for z = v : u x (v = t(u, ~x)) . { 9 2 } Such terms may not be used as arguments to functions.

We use collection terms only in contexts where we have u x !v'(u, v, ~x), 8 2 9 so that (3.1) is equivalent to v z u x'(u, v, ~x) u x v z'(u, v, ~x). 8 2 9 2 ^8 2 9 2 In particular, this is the case for replacement terms. These restrictions en- sure that our formulas are 0(L0)whenever' is, provided that the comprehension terms have the uniqueness property. As usual, we write !y'(y,~x) 9 for 61y' y',where 61y'stands for 9 ^9 9 y, y0 ('(y,~x) '(y0,~x) y = y0). 8 ^ ! Instead of 0 etc. we use more precise notation, making the language explicit:

Definition 3.2: Let L be a language containing L0.Then0(L)istheset of L-formulas ' in which all quantifiers are -bounded, that is, of the form 2 x t or x t where t is an L-term not involving x.Werefertot as an 9 2 8 2 -bounding term in '. 2 The classes ⌃1(L) and ⇧1(L) contain the formulas obtained from 0(L)- formulas by respectively existential and universal quantification, and ⇧2(L) contains those obtained from ⌃1(L)-formulas by universal quantification. We define our basic theory, which the other theories we consider will extend.

Deﬁnition 3.3: The theory T0 consists of

- the deﬁning axioms for the symbols in L0 - the Extensionality axiom: x = y u x (u/y) u y (u/x) 6 !9 2 2 _9 2 2 - the Set Foundation axiom: x =0 y x u y (u/x) 6 !9 2 8 2 2 - the tc-Transitivity axiom: y tc(x) y tc(x) 2 ! ✓ - the (L )-Separation scheme: z (z = u x : '(u,w ~ ) ) for ' 0 0 9 { 2 } 2 0(L0).

Lemma 3.4: The theory T0 proves the 0(L0)-Induction scheme y ( u y'(u,w ~ ) '(y,w ~ )) '(x,w ~ ) for ' (L ). 8 8 2 ! ! 2 0 0 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 67

Cobham Recursive Set Functions and Weak Set Theories 67

Proof: 0(L0)-Induction is logically equivalent to 0(L0)-Foundation '(x,w ~ ) y ('(y,w ~ ) u y '(u,w ~ )) !9 ^8 2 ¬

for ' in 0(L0)whichisderivedinT0 as follows. Assume '(x,w ~ ) and use (L )-Separation to get the set z = y tc+(x):'(y,w ~ ) .Then 0 0 { 2 } x tc+(x) by the deﬁning axiom for tc, so x z = 0. Choose y as the 2 2 6 -minimal element in z according to Set Foundation. Then '(y,w ~ ) and, 2 if u y,thenu/z, and thus '(u,w ~ ) because u y tc+(x)bytc- 2 2 ¬ 2 ✓ Transitivity. ⇤

Remark 3.5: It is for the sake of the previous lemma that the tc- Transitivity axiom is included in T0. In fact, this axiom is equivalent to 0(L0)-Induction with respect to the remaining axioms of T0.

3.2. Embeddings An embedding of a set x into a set y is an injective multifunction ⌧ from tc(x)totc(y)whichrespectsthe -ordering on tc(x) in a certain sense. 2 There are several variants of embeddings, depending on how ⌧ is deﬁned.

Deﬁnition 3.6: A function symbol ⌧(u,w ~ )isastrongly uniform embedding (with parameters w~ ) of x in y if the following (L ⌧ )-formula holds 0 0 [{ } (where for the sake of readability we suppress the parameters w~ ):

u tc(x)(⌧(u) tc(y)) 8 2 ✓ u tc(x)(⌧(u) = 0) ^82 6 (3.2) u, u0 tc(x)(u = u0 (⌧(u) and ⌧(u0) are disjoint)) ^8 2 6 ! u tc(x) u0 u v ⌧(u) v0 ⌧(u0)(v0 tc(v)). ^82 8 2 8 2 9 2 2 The last conjunct is read as “for all u, u0 tc(x), if u0 u then for every v 2 2 in the image of u there is some v0 in the image of u0 with v0 tc(v).” Note 2 that the “identity” multifunction u u is an embedding of x in x;we 7! { } will call an embedding of this form the identity embedding.

We abbreviate (3.2) by ⌧( ,w~ ):x y. We next introduce terminology · 4 for embeddings whose graphs are given by formulas and embeddings whose graphs are given by sets.

Deﬁnition 3.7: Given a formula "(u, v,w ~ ), we deﬁne "( , ,w~ ):x y to · · 4 be condition (3.2) with v ⌧(u,w ~ ) replaced by v tc(y) "(u, v,w ~ ). More 2 2 ^ March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 68

68 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

precisely, "( , ,w~ ):x y means: · · 4 u tc(x) v tc(y) "(u, v,w ~ ) 8 2 9 2 u, u0 tc(x) v tc(y)(u = u0 "(u, v,w ~ ) "(u0,v,w~ )) (3.3) ^8 2 8 2 6 !¬ _¬ u tc(x) u0 u v tc(y)("(u, v,w ~ ) v0 tc(v) "(u0,v0,w~ )). ^82 8 2 8 2 !9 2 This kind of embedding is called a weakly uniform embedding.

Deﬁnition 3.8: For a set e,wewritee : x y if e tc(x) tc(y) and 4 ✓ ⇥ "( , ,e):x y holds when "(u, v, e) is the formula u, v e.Wewrite · · 4 h i2 simply x y to abbreviate e (e : x y). This is called a nonuniform 4 9 4 embedding.

Note that e : x 4 y is 0(L0). More generally, if " is a 0(L)-formula in a language L L then "( , ,w~ ):x y is also (L). ◆ 0 · · 4 0 Definition 3.9: We say that a theory defines a 0(L0)-embedding x 4 y if there is a 0(L0)-formula "(u, v, x, y) such that the theory proves "( , ,x,y):x y. · · 4 The next lemma is useful for constructing embeddings. We state it for nonuniform embeddings, but there are analogous versions for strongly and weakly uniform embeddings. Say that two embeddings e : x 4 z and f : y 4 z are compatible if their union is still an injective multifunction, that is, if it satisfies the disjointness condition of (3.2). In particular, embeddings with disjoint ranges are automatically compatible.

Lemma 3.10: Provably in T0, if two embeddings e : x 4 z and f : y 4 z are compatible, then e f : x y z. [ [ 4 Proof: It follows from the axioms that u tc(x y) if and only if u tc(x) 2 [ 2 or u tc(y). The proof is then immediate. 2 ⇤

4 u 3.3. The theories KP1 and KP1 4 u This section defines theories KP1 and KP1 that, intuitively, are to Rath- 1 1 jen’s KP1 as S2 is to I⌃1. The role of “sharply bounded” quantification in S2 is now played by -bounded quantification. The analogue of a “bounded” 2 quantifier in our context is one where the quantified variable is embeddable in a #-term:

Deﬁnition 3.11: A#-term is a 1, , # -term. { } March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 69

Cobham Recursive Set Functions and Weak Set Theories 69

Saying that a set is embeddable in a #-term t(x) is analogous to saying that a number/string has length at most p( x ) for some polynomial p.When | | we write a #-term, we will use the convention that the # operation takes precedence over , and otherwise we omit right-associative parentheses. So for example 1 x#y z is read as 1 ((x#y) z). 4 Deﬁnition 3.12: Let L be a language containing L0. The class ⌃1 (L) consists of L-formulas of the form

x t(~x ) '(x, ~x) 9 4 where t is a #-term not involving x and ' is (L). Here x t' stands 0 9 4 for x (x t '). Recall that x y denotes a nonuniform embedding, i.e., 9 4 ^ 4 it stands for e (e : x y). Hence a ⌃4(L)-formula is also a ⌃ (L)-formula. 9 4 1 1 (See also Lemma 4.13.)

Note that the term 4-bounding the leading existential quantiﬁer in a ⌃4(L)-formula is required to be a #-term while the -bounding terms in 1 2 the 0(L)-part can be arbitrary L-terms.

4 Deﬁnition 3.13: The theory KP1 consists of T0 without tc-Transitivity, that is, the deﬁning axioms for the symbols in L0, Extensionality, Set Foun- dation and 0(L0)-Separation, together with the two schemes:

- 0(L0)-Collection: u x v'(u, v,w ~ ) y u x v y'(u, v,w ~ ) for ' (L ), 8 2 9 !9 8 2 9 2 2 0 0 4 - ⌃1 (L0)-Induction: y ( u y'(u,w ~ ) '(y,w ~ )) '(x,w ~ ) for ' ⌃4(L ). 8 8 2 ! ! 2 1 0 4 We omitted tc-Transitivity from the definition of KP1 because it is not one of the usual axioms for Kripke-Platek set theories. However, tc- Transitivity can be proven by 0(L0)-Induction from the rest of the ax- 4 ioms of T0.Since0(L0)-Induction is contained in KP1 , it follows that 4 4 tc-Transitivity is a consequence of KP1 .ThusKP1 contains T0. The same u holds for the theory KP1 defined next. 4 4 Our goal is to ⌃1 (L0)-define all CRSF functions in KP1 in the following sense. Fix a universe of sets V (a model of ZFC); of course, we may view V as interpreting L0. Let T be a theory and a set of formulas. A function f(~x ) over V is -definable in T if there is '(y,~x) such that V = ~x' (f(~x ),~x) 2 | 8 and T proves !y'(y,~x). 9 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 70

70 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

u In fact, we will show that an apparently weaker theory KP1 is sucient u 4 for this purpose. KP1 is deﬁned in the same way as KP1 , except that the 4 induction scheme is restricted to ⌃1 (L0)-formulas of a special form, where the witness to the leading existential quantiﬁer is required to be unique and uniformly embeddable into a #-term (hence the superscript u). We will see later (in Lemma 4.13) that it is only the uniqueness requirement 4 that distinguishes this from KP1 .

u Deﬁnition 3.14: The theory KP1 consists of T0 without tc-Transitivity, that is, the deﬁning axioms for the symbols in L0, Extensionality, Set Foun- dation and 0(L0)-Separation, together with 0(L0)-Collection and the scheme:

4 - Uniformly Bounded Unique ⌃1 (L0)-Induction u 61v'(u, v,w ~ ) y u y v'",t(u, v,w ~ ) v'",t(y, v,w ~ ) 8 9 ^8 8 2 9 !9 v'",t(x, v,w ~ ) !9 where '",t(u, v,w ~ ) abbreviates the formula '(u, v,w ~ ) "( , ,v,u,w~ ):v t(u,w ~ ) ^ · · 4 and the scheme ranges over 0(L0)-formulas ', " and #-terms t.

3.4. The language Lcrsf and theory Tcrsf

Our final main theory, Tcrsf , is an analogue of the bounded arithmetic theory PV1.Tcrsf has a function symbol for every CRSF function, and ⇧1 axioms describing how the CRSF functions are defined from each other. 4 u 1 By comparison, KP1 and KP1 are analogues of S2. One of our main re- u sults is Theorem 6.9, which states that a definitional expansion of KP1 is ⇧2(Lcrsf )-conservative over Tcrsf , that is, every ⇧2(Lcrsf )-sentence provable in the former theory is also provable in the latter. Theorem 6.12 states an 4 analogous result for KP1 , but only with the addition of a global choice function to Tcrsf .

Deﬁnition 3.15: The language L consists of and the function symbols crsf 2 listed below. The theory Tcrsf contains the axioms of Extensionality, Set Foundation and tc-Transitivity, together with a deﬁning axiom for each function symbol of Lcrsf , as follows.

- Lcrsf contains the function symbols from L0, and Tcrsf contains their deﬁning axioms. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 71

Cobham Recursive Set Functions and Weak Set Theories 71

n - Lcrsf contains the function symbols proji for 1 6 i 6 n and n cond (x, y, u, v) with deﬁning axioms proji (x1,...,xn)=xi and 2 x if u v cond (x, y, u, v)= 2 2 y otherwise. ⇢ - (Closure under composition) For all function symbols h, g ,...,g 1 k 2 Lcrsf of suitable arities, Lcrsf contains the function symbol fh,~g with deﬁning axiom

fh,~g(~x )=h(g1(~x ),...,gk(~x )). - (Closure under replacement) For all function symbols f L ,L 2 crsf crsf contains the function symbol f” with defining axiom f”(x, ~z)= f(y,~z):y x . { 2 } - (Closure under syntactic Cobham recursion) Suppose g, ⌧ are function symbols in Lcrsf and t is a #-term. Let us write “⌧ is an embedding into t at x,w ~ ” for the 0(Lcrsf )-formula ⌧( ,g(x, f”(x,w ~ ),w~ ),x,w~ ):g(x, f”(x,w ~ ),w~ ) t(x,w ~ ). · 4 Then Lcrsf contains the function symbol f = fg,⌧,t with defining axiom g(x, f”(x,w ~ ),w~ )if⌧ is an embedding into t at x,w ~ f(x,w ~ )= 0 otherwise. ⇢ Proposition 3.16: The universe V of sets can be expanded uniquely to a model of Tcrsf .TheLcrsf -function symbols then name exactly the CRSF functions as defined in Proposition 2.9.

Because of closure under composition, every Lcrsf -term is equivalent to an Lcrsf -function symbol, provably in Tcrsf . Hence we will not always be careful to distinguish between terms and function symbols in Lcrsf .

Lemma 3.17: For every function symbol f L , there is a function 2 crsf symbol g L such that T proves g(x,w ~ )= y x : f(y,w ~ ) =0 . 2 crsf crsf { 2 6 } Proof: Using cond we may construct a function symbol h(y,w ~ )which 2 takes the value y if f(y,w ~ ) / 0 and the value 0 otherwise. We put { } 2{} g(x,w ~ )= h”(x,w ~ ). ⇤ The nextS lemma is proved as in the development of CRSF in [7, Theorem 13]. Note that we do not need recursion to prove either Lemma 3.17 or Lemma 3.18. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 72

72 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

Lemma 3.18: Every 0(Lcrsf )-formula is provably equivalent in Tcrsf to a formula of the form f(~x ) =0for some L -function symbol f.Itfollows 6 crsf that the Lcrsf functions are closed under 0(Lcrsf )-Separation provably in Tcrsf ; that is, for every 0(Lcrsf )-formula '(y,w ~ ) there is an Lcrsf -function symbol f such that T proves f(x,w ~ )= y x : '(y,w ~ ) . crsf { 2 }

Corollary 3.19: The theory Tcrsf extends T0.

4. Bootstrapping 4.1. Bootstrapping the defining axioms We first derive some simple consequences of the defining axioms, namely basic properties of tc, a description of the Mostowski graph of x y,injec- tivity of in its first argument, and associativity of .

Lemma 4.1: The theory T0 proves (a) x y tc(x) tc(y), tc(x)=tc(tc(x)), ✓ ! ✓ (b) u tc(x y) (u tc(y) u0 tc(x)(u = u0 y)), 2 $ 2 _9 2 (c) x = x0 x y = x0 y, 6 ! 6 (d) x (y z)=(x y) z. Proof: We omit the proof of (a). For (b) argue in T0 as follows. If x = 0, the claim follows from the -axiom, so assume x = 0. We prove ( )by(L )-Induction (recall 6 ! 0 0 Lemma 3.4), so assume it to hold for all x0 x. Let u tc(x y). By the 2 2 tc, -axioms, either u tc(x0 y) for some x0 x or u x y.Inthe 2 2 2 ﬁrst case our claim follows by induction noting tc(x0) tc(x) by (a). In the ✓ second case, u = u0 y for some u0 x by the -axiom. 2 Conversely, we ﬁrst show u tc(y) u tc(x y). By (L )- 2 ! 2 0 0 Induction we can assume this holds for all z x. Assume u tc(y) and let 2 2 z x be arbitrary. Then u tc(z y) by induction. But z y x y,so 2 2 2 tc(z y) tc(x y) by (a). ✓ Finally, we show u tc(x) u y tc(x y). We assume this for 2 ! 2 all z x. Let u tc(x). If u x,thenu y x y tc(x y). 2 2 2 2 ✓ Otherwise u tc(z) for some z x.Thenu y tc(z y) by induction; 2 2 2 but tc(z y) tc(x y)byz y x y and (a). ✓ 2 For (c) argue in T0 as follows. Suppose there are y, x0,x0 such that + x = x0 and x y = x0 y. It is easy to derive x (x tc (x)), so the set 0 6 0 0 0 8 2 + + z := x tc (x ): x0 tc (x0 )(x = x0 x y = x0 y) { 2 0 9 2 0 6 ^ } March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 73

Cobham Recursive Set Functions and Weak Set Theories 73

is non-empty because it contains x0.Thesetexistsby0(L0)-Separation. + By Foundation, it contains an -minimal element x . Choose x0 tc (x0 ) 2 1 1 2 0 with x = x0 and x y = x0 y. 1 6 1 1 1 We claim that x1,x10 are both non-empty. Assume otherwise, say, x10 =0 and hence x = 0. An easy (L )-Induction shows x/tc(x) and (x = 1 6 0 0 2 6 0 0 tc(x)) for all x.Theny/y = x0 y and y =0 y tc(x y) ! 2 2 1 2 1 by (b). This contradicts x y = x0 y. 1 1 Choose x such that either x x x / x0 or x x0 x / x . 2 2 2 1 ^ 2 2 1 2 2 1 ^ 2 2 1 Assume the former (the latter case is similar). By the -axiom, x y 2 2 x1 y = x10 y.Sincex10 =0the -axiom gives x20 x10 such that 6 2 + x2 y = x20 y. As x2 / x10 we have x2 = x20 . By (a), x2 x1 tc (x0) + 2 6 2 ✓ and x0 x0 tc (x0 ). Thus x z, contradicting the minimality of x . 2 2 1 ✓ 0 2 2 1 For (d), an easy induction shows that u 0=u for all u.Item(d)isthen true immediately if any of x, y or z is 0. Otherwise it follows by induction on x,usingthe -axiom. ⇤ We write 2 := 1 1, 3 := 1 1 1, etc. Notice that, in T ,1 x = x . 0 { } We give an example of how we can now begin to build useful embeddings.

Example 4.2: There is a #-term tpair(x, y) such that T0 deﬁnes a 0(L0)- embedding from x, y into t (x, y). h i pair

Proof: We put t (x, y):=4 x 1 y. Consider the relations pair e := u, u : u tc+(y) {h i 2 } f := u, u 1 y : u tc+(x) {h i 2 } g := x , 2 x 1 y {h{ } i} h := x, y , 3 x 1 y . {h{ } i} Then e : y t (x, y) and f : x t (x, y), and these two embed- { } 4 pair { } 4 pair dings are compatible since they have disjoint ranges. So e f : x, y [ { } 4 t (x, y) (appealing to Lemma 3.10), hence e f h : x, y t (x, y). pair [ [ {{ }} 4 pair On the other hand f g : x t (x, y). These are compatible, so [ {{ }} 4 pair e f g h : x , x, y t (x, y), as required. All these embeddings [ [ [ {{ } { }} 4 pair can be expressed straightforwardly in T0 by 0(L0)-formulas. ⇤

4.2. Adding -bounded functions 2 + We give a small expansion T0 of T0. + Deﬁnition 4.3: Let L0 be the language obtained from L0 by adding a relation symbol R(~x ) for every 0(L0)-formula '(~x ), and a function symbol March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 74

74 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

f(~x ) for every (L )-formula (y,~x) such that T proves !y t(~x ) (y,~x) 0 0 0 9 2 for some L0-term t(~x ). + + The theory T0 has language L0 and is obtained from T0 by adding for + every relation symbol R(~x )inL0 as above the deﬁning axiom R(~x ) '(~x ), + $ and for every function symbol f(~x )inL0 as above the deﬁning axiom (f(~x ),~x).

+ + Proposition 4.4: T0 is a conservative extension of T0. Every 0(L0 )- + + formula is T0 -provably equivalent to a 0(L0)-formula. In particular, T0 + + proves 0(L0 )-Induction and 0(L0 )-Separation.

+ + We omit the proof. The language L0 and the theory T0 are introduced mainly for notational convenience. Interesting functions often do not have -bounded values. 2 + Lemma 4.5: Every function symbol introduced in L0 has a copy in Lcrsf for which Tcrsf proves the deﬁning axiom.

Proof: Suppose T proves !y t(~x ) (y,~x). Then using Lemma 3.18 we 0 9 2 can compute y in T as y t(~x ): (y,~x) . crsf { 2 } ⇤ S + Example 4.6: The language L0 contains the relation symbol IsPair(x) with defining axiom u, v tc(x)(x = u, v ), the function symbol 9 2 h i cond (x, y, u, v) with defining axiom as in Definition 3.15, and function 2 symbols ⇡ (x), ⇡ (x) and w’x such that T+ proves ⇡ ( x ,x )=x , 1 2 0 1 h 1 2i 1 ⇡ ( x ,x )=x and 2 h 1 2i 2 y if y is unique with x, y w w’x = h i2 0 otherwise. ⇢ We now formalize the graph isomorphism for # mentioned in Section 2. We introduce #”(u, y) below as an auxiliary function to formulate the defining axiom for x,y(u, v).

Lemma 4.7: There are function symbols #”(u, y), x,y(u, v), ⇡1,x,y(w) + + and ⇡2,x,y(w) in L0 such that T0 proves

#”(u, y)= u0#y : u0 u , { 2 } v #”(u, y) if u tc+(x) and v tc+(y) (u, v)= 2 2 x,y 0 otherwise. ⇢ + Moreover, T0 proves that March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 75

Cobham Recursive Set Functions and Weak Set Theories 75

+ + (a) x,y is injective on arguments u tc (x),v tc (y). + 2 2 (b) Every w tc (x#y) has a x,y-preimage (⇡1,x,y(w),⇡2,x,y(w)). 2 + + (c) For all u, u0 tc (x) and v, v0 tc (y), 2 2 (u0,v0) (u, v) (u0 = u v0 v) (u0 u v0 = y v = 0). x,y 2 x,y $ ^ 2 _ 2 ^ ^

Proof: The functions #”(u, y),x,y(u, v) have obvious defining axioms. Concerning bounding terms, from the #-axiom we get that #”(u, y) 2 tc+(u#y) and x#y = y #”(x, y). (4.1) By induction on x, using Lemma 4.1(b), we get that if u tc+(x)then 2 u#y tc+(x#y). Another use of Lemma 4.1(b) shows that if v tc+(y) 2 2 then v #”(u, y) tc+(u#y). Hence (u, v) tc+(x#y). 2 x,y 2 Observe that for u tc+(x),v tc+(y), 2 2 v =0 x,y(u, v)= x,y(u, v0):v0 v , 6 ! { 2 } (4.2) (u, 0) = (u0,y):u0 u . x,y { x,y 2 } The first line is from the definition. For the second, note z (u, 0) is 2 x,y equivalent to z #”(u, y), and hence to z = u0#y for some u0 u;but 2 2 u0#y = x,y(u0,y) by (4.1). For (a), let u, u,˜ . . . range over tc+(x) and v, v,˜ . . . range over tc+(y). We claim that x,y(u, v)=x,y(˜u, v˜)impliesu =˜u and v =˜v.By Lemma 4.1(c) it suces to show it implies u =˜u. Assume otherwise. By (L+)-Foundation choose u -minimal such that there existu, ˜ v, v˜ with 0 0 2 (u, v)= (˜u, v˜) and u =˜u; then chooseu ˜ -minimal such that there x,y x,y 6 2 are v, v˜ with this property, and so on for v, v˜. We distinguish two cases, as in (4.2). First suppose v = 0. Then there is v0 v such that (u, v0) 6 2 x,y 2 (˜u, v˜). Ifv ˜ = 0, then (u, v0)= (˜u, v00) for some v00 v˜, and x,y 6 x,y x,y 2 this contradicts the choice of v.If˜v = 0, then x,y(u, v0)=x,y(u0,y) for some u0 u˜, and this contradicts the choice ofu ˜. 2 Now suppose v = 0. Ifv ˜ = 0, then (u0,y):u0 u = { x,y 2 } (u00,y):u00 u˜ , so for each u0 u there is u00 u˜ such that { x,y 2 } 2 2 (u0,y)= (u00,y), so then u0 = u00 by choice of u;thusu u˜. x,y x,y ✓ Similarlyu ˜ u, contradicting our assumption u =˜u.If˜v = 0, then for ✓ 6 6 each u0 u there is v0 v˜ such that (u0,y)= (˜u, v0), so u0 =˜u 2 2 x,y x,y by choice of u.Thusu = 0 or u = u˜ ; the latter is impossible by choice { } of u,sou = 0; then x,y(˜u, v˜)=x,y(u, v)=x,y(0, 0) = 0, sov ˜ = 0, a contradiction. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 76

76 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

For (b) we will show surjectivity; ⇡1,x,y(w) and ⇡2,x,y(w) can then easily be constructed. So let w tc+(x#y). If w = x#y,putu := x, v := y. 2 Otherwise w tc(x#y)=tc(y #”(x, y)) by (4.1). By Lemma 4.1(c) we 2 have two cases. If w = v0 #”(x, y) for some v0 y,putu := x, v := v0.If + 2 + w tc(#”(x, y)), then w tc (x0#y) for some x0 x and, using 0(L0 )- 2 2 + + 2 + Induction on x,weﬁndu tc (x0) tc (x) and v tc (y)withw = 2 ✓ 2 x0,y(u, v). Since x,y(u, v) does not depend on x,wehavew = x0,y(u, v)= x,y(u, v). Claim (c) follows by (4.2). ⇤

4.3. Monotonicity lemma We can now formally derive non-uniform and weakly uniform versions of the Monotonicity Lemma 2.4, meaning “monotonicity of #-terms with respect to embeddings.” Note that Lemma 4.8 includes as a special case the transitivity of embeddings,

z x x y z y. (4.3) 4 ^ 4 ! 4

Lemma 4.8: (Monotonicity) For all #-terms t(x,w ~ ) the theory T0 proves

z t(x,w ~ ) x y z t(y,w ~ ). (4.4) 4 ^ 4 ! 4

Moreover, for all 0(L0)-formulas "0,"1 there is a 0(L0)-formula "2 such that T0 proves

" ( , ,x,z,w~ ):z t(x,w ~ ) " ( , ,x,y,w~ ):x y 0 · · 4 ^ 1 · · 4 " ( , ,x,y,z,w~ ):z t(y,w ~ ). ! 2 · · 4 Proof: We only verify the ﬁrst statement; the second follows by inspection + of the proof. We proceed by induction on t. We work in T0 ,whichis sucient by Proposition 5.5. If t(x,w ~ ) is 1 or a variable distinct from x, then there is nothing to show. If t(x,w ~ ) equals x then we have to show (4.3). So assume e : z 4 x and f : x 4 y.Then g := u, w tc(z) tc(y): v tc(x)( u, v e v, w f) {h i2 ⇥ 9 2 h i2 ^h i2 } exists by 0(L0)-Separation. We claim g : z 4 y. It is easy to see that u, w , u0,w g implies u = u0. Assume u0 u tc(x) and u, w g. h i h i2 2 2 h i2 Choose v such that u, v e and v, w f.Thenthereisv0 tc(v)such h i2 h i2 2 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 77

Cobham Recursive Set Functions and Weak Set Theories 77

that u0,v0 e. It now suces to show that, generally, for all v, v0,w we h i2 have

v0 tc(v) v, w f w0 tc(w) v0,w0 f. 2 ^h i2 !9 2 h i2 This is clear if v0 v.Otherwise,v0 tc(v00) for some v00 v. Then choose 2 2 2 w00 tc(w) such that v00,w00 f. Appealing to (L )-Induction, we can 2 h i2 0 0 ﬁnd w0 tc(w00) such that v0,w0 f.Thenw0 tc(w) by Lemma 4.1(a), 2 h i2 2 as claimed. As preparation for the induction step in our induction on t,weshow

x x0 y y0 x y x0 y0 x#y x0#y0. (4.5) 4 ^ 4 ! 4 ^ 4 Assume e : x 4 x0 and f : y 4 y0.By0(L0)-Separation the set

u, v tc(x y) tc(x0 y0): {h i2 ⇥ u0 tc(x) v0 tc(x0)(u = u0 y v = v0 y0 u0,v0 e) 9 2 9 2 ^ ^h i2 } exists. We leave it to the reader to check that its union with f witnesses x y x0 y0. For #, observe e := e x, x0 : x x0 and 4 + [{h i} { } 4 { } f := f y, y0 : y y0 . Let g be the set containing the pairs + [{h i} { } 4 { } x,y(u, v),x0,y0 (u0,v0) such that u, u0 e+ and v, v0 f+.Thisset h + i h i2 h i2 g exists by 0(L0 )-Separation. Using Lemma 4.7 it is straightforward to check that g : x#y 4 x0#y0. Now the induction step is easy. We are given embeddings z 4 t(x,w ~ ) and x y. Assume ﬁrst that t(x,w ~ )=t (x,w ~ ) t (x,w ~ ). By the identity 4 1 2 embedding t1(x,w ~ ) 4 t1(x,w ~ ), and applying the inductive hypothesis gives t1(x,w ~ ) 4 t1(y,w ~ ). Similarly t2(x,w ~ ) 4 t2(y,w ~ ). Applying (4.5) we get t (x,w ~ ) t (x,w ~ ) t (y,w ~ ) t (y,w ~ ), 1 2 4 1 2 that is, t(x,w ~ ) 4 t(y,w ~ ). This, together with (4.3), implies (4.4). The case of t1(x,w ~ )#t2(x,w ~ ) is analogous. ⇤

4.4. Some useful embeddings + We ﬁrst show that we can embed all L0 -terms into #-terms. Lemma 4.9: The theory T deﬁnes a (L )-embedding of x y into a 0 0 0 ⇥ #-term t (x, y). ⇥ + + Proof: By Proposition 4.4 it suces to prove this for T0 and a 0(L0 )- embedding. We set t (x, y)=(x#y) (x#y) x y x. ⇥ March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 78

78 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

The formula " (z,z0,x,y) implements the following informal procedure on ⇥ input z,z0,x,y. In the description of this procedure we understand that whenever a “check” is carried out then the computation halts, and the procedure rejects or accepts depending on whether the check failed or not. For example, line 2 is reached only if z/tc(x). 2 It is easy to check that the condition that z,z0,x,y is accepted is ex- pressible as a 0(L0)-formula.

Input: z,z0,x,y

1. if z tc(x) then check z0 = z 2 2. if z tc(y) then check z0 = z x 2 3. guess u x 2 4. if z = u then check z0 = u y x { } 5. guess v y 2 6. if z = u, v then check z0 = (u, v) x y x { } x,y 7. if z = u , u, v then check z0 = (u, v) (x#y) x y x {{ } { }} x,y 8. reject

It is clear that any z tc(x y) is mapped to at least one z0.Further, 2 ⇥ distinct z =˜z cannot be mapped to the same z0 =˜z0: any z0 satisfies the 6 check of at most one line and this line determines the pre-image z (Lemmas 4.7 and 4.1(c)). Assume z z˜ andz ˜ is mapped toz ˜0.Wehavetofindz0 tc(˜z0)such 2 2 that z is mapped to z0. This is easy if z tc(x) tc(y), so assume this is ⇢ [ not the case. Thenz ˜ cannot satisfy any “if” condition before line 7. Hence z˜ = u , u, v for some u x, v y andz ˜0 satisfies the check in line 7. {{ } { }} 2 2 As z z˜ we have z = u or z = u, v and for suitable guesses in lines 3 2 { } { } and 5, z satisfies the “if” condition of line 4 or 6. Then choose z0 satisfying the (first) corresponding check. ⇤

+ Lemma 4.10: For each L0 -term s(~x ) the theory T0 deﬁnes a 0(L0)- embedding of s(~x ) into a #-term s#(~x ).

Proof: This follows by an induction on s(~x ) using Lemma 4.8 once we + verify it for the base case that s(~x ) is a function symbol in L0 . + For any such s(~x ), there is an L0-term r(~x ) such that T0 proves s(~x ) r(~x ). By Lemma 4.1 tc(s(~x )) tc(r(~x )), so the identity embed- 2 ✓ ding (expressed by the formula u = v)embedss(~x )inr(~x ). By transitivity March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 79

Cobham Recursive Set Functions and Weak Set Theories 79

of 4 (Lemma 4.8), it thus suces to verify the lemma for L0-terms r(~x ). As for L0-terms, this follows by an induction on r(~x ) using Lemma 4.8 once we verify it for the base case that r(~x ) is a function symbol in L0. The only non-trivial case now is crossproduct , and this is handled by the previous ⇥ lemma. ⇤ For tuples ~u = u ,...,u let us abbreviate u z as ~u z.Weshow 1 k i i 2 2 that given a family of sets parametrized by tuples ~u z, where each set V 2 is uniformly embeddable in s, we can embed the whole family (if it exists as a set) in a #-term t(z,s). Note that the existence of V in the lemma is automatic in the presence of the Collection scheme.

+ Lemma 4.11: Let '(v,~u,w ~ ) and "(z,z0,v,~u,w~ ) be 0(L0 )-formulas and + s(w~ ) a #-term. There is a 0(L0 )-formula (z,z0,V,z,w~ ) and a #-term t(z,x) such that if ~u z 61v'(v,~u,w ~ ) 8 2 9 ~u z v '(v,~u,w ~ ) "( , ,v,~u,w~ ):v s(w~ ) (4.6) ^82 9 ^ · · 4 V = v : ~u z'(v,~u,w ~ ) ^ { 9 2 } then ( , ,V,z,w~ ):V t(z,s(w~ )), provably in T+. · · 4 0 Proof: For notational simplicity we suppress the side variables w~ .Weﬁrst consider the case in which ~u is a single variable u. Note that in the second line of the assumption (4.6) we may assume without loss of generality that we actually have "( , ,v,u): v s, since otherwise we could modify " · · { } 4 so that "(v, s, v, u) holds and replace the bound s with 1 s. Now put t(z,s):=z#s and deﬁne

"0(y, y˜0,V,u):= v V y0 tc(s)('(v, u) "(y, y0,v,u) y˜0 = (u, y0)). 9 2 9 2 ^ ^ z,s For each u z,if'(u, v) then the formula "0( , ,V,u) describes an embed- 2 · · ding of v into z#s which is a copy of the embedding "( , ,v,u), but with { } · · its range moved to lie entirely within the uth copy of s inside z#s.These embeddings have disjoint ranges for distinct u, so as in Lemma 3.10 their union (y, y˜0,V,z):= u z"0(y, y˜0,V,u) describes an embedding of V into 9 2 z#s,sincey tc(V )impliesy tc( v ) for some v V . 2 2 { } 2 When ~u is a tuple of k variables, we reduce to the first case by coding ~u as an ordered k-tuple in the usual way. So the quantifier ~u z becomes 8 2 u (z z) and we replace ' and " with formulas accessing the values 8 2 ⇥···⇥ of ~u from u using projection functions. The first case then gives an embedding of V into t(z z,s), and by Lemma 4.10 and the Monotonicity ⇥···⇥ Lemma 4.8 we get an embedding of V into some #-term t0(z,s). ⇤ March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 80

80 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

We finish this section by showing, in Lemma 4.13, that the non-uniform 4 embedding bounding the existential quantifier in a ⌃1 -formula (over any language) can be replaced with a weakly uniform embedding. This will be 4 useful when we want to show that structures satisfy ⌃1 -Induction. We first show a suitable embedding exists.

Lemma 4.12: There is a 0(L0)-formula "emb(u, v, e, x, y) and a #-term t (y) such that T proves e : x y " ( , ,e,x,y): e, x t (y). emb 0 4 ! emb · · h i 4 emb

Proof: Let " (u, v, x, y) 0(L0)describe(inT0) an embedding of ⇥ 2 x y into t (x, y). Then " (u, v, tc(x), tc(y)) describes an embedding of ⇥ ⇥ ⇥ tc(x) tc(y)intot (tc(x), tc(y)). The identity embedding embeds tc(x) ⇥ ⇥ into x. Combining these, Lemma 4.8 gives a 0(L0)-formula describing an embedding tc(x) tc(y) 4 t (x, y). But e : x 4 y implies e tc(x) tc(y), ⇥ ⇥ ✓ ⇥ so this formula also describes an embedding e 4 t (x, y). ⇥ Using Example 4.2 there is a 0(L0)-formula describing an embedding e, x t (e, x). By Lemma 4.8 and the previous paragraph, there is a h i 4 pair 0(L0)-formula describing an embedding e, x 4 tpair(t (x, y),x). Since h i ⇥ e : x 4 y it is easy to write a 0(L0)-formula with parameter e describing an embedding x 4 y, so using Lemma 4.8 again we can replace x by y, that is, construct a 0(L0)-embedding e, x 4 temb(y):=tpair(t (y, y),y). ⇤ h i ⇥

Lemma 4.13: Let L be a language extending L0. Provably in T0, every 4 ⌃1 (L)-formula ✓(~x ) is equivalent to a formula of the form v '(v,~x) "( , ,v,~x):v t(~x ) (4.7) 9 ^ · · 4

where ' is a 0(L)-formula, " is a 0(L0)-formula and t is a #-term.

Proof: Expanding the existential quantiﬁer implicit in the nonuniform embedding bound, there is a 0(L)-formula and a #-term s such that ✓(~x ) has the form

w e e : w s(~x ) (w,~x) . 9 9 4 ^

By Lemma 4.12, T0 proves

e : w s(~x ) " ( , ,e,w,s(~x )) : e, w t (s(~x )). 4 ! emb · · h i 4 emb Hence ✓(~x ) is equivalent to

e, w e : w s(~x ) (w,~x) " ( , ,e,w,s(~x )) : e, w t (s(~x )) . 9h i 4 ^ ^ emb · · h i 4 emb March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 81

Cobham Recursive Set Functions and Weak Set Theories 81

For clarity we have written this rather informally. Strictly speaking, e, w h i should be a single variable v, and e and w should be respectively ⇡1(v) and ⇡2(v); then apply Proposition 4.4. ⇤

5. Deﬁnability

u 4 This section develops KP1 with the goal of proving that it ⌃1 (L0)-deﬁnes all CRSF functions. The Deﬁnability Theorem 5.2 below states this in a syntactic manner, without reference to the universe of sets.

4 u u Deﬁnition 5.1: A⌃1 (L0)-expansion of KP1 is obtained from KP1 by adding a set of formulas of the following forms:

- '(f(~x ),~x)wheref(~x ) is a function symbol outside L0 and '(y,~x)is ⌃4(L ) such that KPu proves !y'(y,~x) 1 0 1 9 - R(~x ) '(~x )whereR(~x ) is a relation symbol outside L and '(~x )is $ 0 0(L0).

4 For example, it is not hard to give a ⌃1 (L0)-deﬁnition of a proper comprehension term for a formula ' (L ) as a new function symbol. 2 0 0 4 u Theorem 5.2: (Deﬁnability) There is a ⌃1 (L0)-expansion of KP1 which contains all function symbols in Lcrsf in its expanded language, and proves all axioms of Tcrsf .

4 u A⌃1 (L0)-expansion of KP1 is an expansion by definitions, and hence u is conservative over KP1 . Thus Theorem 5.2 immediately implies that every CRSF function is denoted by a symbol in the language, and hence is 4 u ⌃1 (L0)-definable in KP1 . Note that such an expansion does not include u the axiom schemes of KP1 for formulas in the expanded language. (Defini- u tion 3.14 describes the axiom schemes of KP1 .) The lack of these schemes is the main technical diculty in proving the Theorem 5.2. Some further comments can be found in Section 5.2, where this diculty is tackled. In Section 5.5 we describe a particular well-behaved expansion which proves all these axiom schemes in the expanded language. We will prove Theorem 5.2 indirectly. We first define an expansion u KP1 +Ldef which includes a function symbol for every function definable u 4 in KP1 by a particular kind of ⌃1 (L0)-formula. We will then show that these function symbols contain the basic functions from Lcrsf and satisfy the right closure properties. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 82

82 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

u Remark 5.3: For the results in this section about KP1 and its expansions, u we do not need the full strength of the collection scheme in KP1 .Every instance of 0(L0)-Collection we use is an instance of the apparently weaker scheme

- Uniformly Bounded 0(L0)-Replacement

u x 61v'(u, v, ~x) u x v'",t(u, v, ~x) 8 2 9 ^8 2 9 y (y = v : u x'(u, v, ~x) ) !9 { 9 2 } where '",t(u, v, ~x) abbreviates the formula

'(u, v, ~x) "( , ,v,u,~x):v t(u, ~x) ^ · · 4

and the scheme ranges over 0(L0)-formulas '," and #-terms t.

u 5.1. The deﬁnitional expansion KP1 +Ldef + It will be convenient to have the L0 relation and function symbols available, so the ﬁrst step in the expansion is a small one, allowing this.

u+ 4 u Deﬁnition 5.4: KP1 is the ⌃1 (L0)-expansion of KP1 which adds the deﬁning axioms for all symbols in L+ L . 0 \ 0 We will use the following proposition without comment. (Cf. Proposi- tion 4.4.)

u+ u Proposition 5.5: KP1 is a conservative extension of KP1 . Furthermore u+ u + KP1 proves the axiom schemes of KP1 with L0 replacing L0.

u+ 4 + We now expand KP1 with functions symbols f(~x )with⌃1 (L0 )- definitions of a special kind. The existentially quantified witness v in such a definition is not only bounded by a #-term t(~x ), but weakly uniformly + bounded. Moreover, the witness v is uniquely described by a 0(L0 )- formula '(v,~x). Intuitively, this formula says “v is a computation of the value of f on input ~x ”. The “output” function e is a very simple function that extracts the value e(v)=f(~x ) from the computation v.

Deﬁnition 5.6: A good deﬁnition is a tuple

('(v,~x),"(z,z0,v,~x),e(v),t(~x )) March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 83

Cobham Recursive Set Functions and Weak Set Theories 83

+ + where '," are 0(L0 )-formulas, e(v) is an L0 -term and t(~x )isa#-term u+ such that KP1 proves (Witness Existence) v'(v,~x) 9 (Witness Uniqueness) 61v'(v,~x) 9 (Witness Embedding) '(v,~x) "( , ,v,~x):v t(~x ). ! · · 4 u u+ Definition 5.7: The theory KP1 +Ldef is obtained from KP1 by adding for every such good definition a function symbol f(~x ) along with the defining axiom

v ('(v,~x) f(~x )=e(v)). (5.1) 9 ^ u We then speak of a good deﬁnition of f. The language Ldef of KP1 +Ldef + consists of L0 together with all such function symbols.

u u+ It is obvious that KP1 +Ldef is a conservative extension of KP1 . Again, u we stress that we do not adopt the axiom schemes of KP1 for the language u Ldef . For example, by deﬁnition, KP1 +Ldef has just 0(L0)-Separation, not 0(Ldef )-Separation.

u Theorem 5.8: KP1 +Ldef proves that Ldef satisﬁes the closure properties of Lcrsf from Deﬁnition 3.15. That is, (a) closure under composition, (b) closure under replacement and (c) closure under syntactic Cobham recursion.

Statement (a) is proved in Theorem 5.10, statement (b) in Theorem 5.15 (see Example 5.16) and statement (c) in Theorem 5.17. The Definability Theorem 5.2 follows easily from Theorem 5.8. We first 4 u observe that all function symbols in Ldef are ⌃1 (L0)-definable in KP1 , + since we can replace all L0 symbols in (5.1) by their 0(L0) definitions and u+ u appeal to the conservativity of KP1 over KP1 . We can then go through the function symbols in Lcrsf one-by-one and show that each one has a corresponding symbol in Ldef (see also Section 5.5 below). Notice that this 4 gives us more than just that every CRSF function f is ⌃1 (L0)-definable u in KP1 . In particular, we have that the witness v is unique, which we will use later in Corollary 6.17. Put di↵erently, the value f(~x )is0(L0)-definable from a set v which is 0(L0)-definable from the arguments ~x . A first step in the proof of Theorem 5.8 is to show that we can treat the language Ldef uniformly, in that every function symbol in it has a good definition. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 84

84 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

+ Lemma 5.9: For every f(~x ) in L0 there exists a good deﬁnition u+ ('(v,~x),"(u, v, ~x),e(v),t(~x )) such that KP1 proves (5.1).

Proof: For '(v,~x) choose v = f(~x ), for e(v) choose v, and for " and t use Lemma 4.10. ⇤

It is straightforward to show part (a) of Theorem 5.8, that Ldef is closed under composition. In this proof, and in the rest of the section, we will make frequent appeals to the Monotonocity Lemma 4.8 and will simply say “by monotonicity”.

Theorem 5.10: For all n-ary function symbols h(x1,...,xn) in Ldef and m-ary function symbols gi(y1,...,ym) for i =1,...,n in Ldef there is an u m-ary function symbol f(~y ) in Ldef such that KP1 +Ldef proves

f(~y )=h(g1(~y ),...,gn(~y )).

Proof: For notational simplicity, assume n = 1. Let h(x) and g(~y )befunc- tion symbols in Ldef with good deﬁnitions ('h,"h,eh,th) and ('g,"g,eg,tg). Set (v, v ,v ,~y):=(v = v ,v ' (v ,~y) ' (v ,e (v ))), g h h h gi^ g g ^ h h g g ' (v,~y):= v ,v tc(v) (v, v ,v ,~y), f 9 h g 2 g h ef (v):=eh(⇡1(v)).

We claim that there are "f ,tf such that ('f ,"f ,ef ,tf ) is a good deﬁnition, u+ i.e., such that KP1 proves (v, v ,v ,~y) " ( , ,v,~y): v ,v t (~y ). g h ! f · · h h gi 4 f u+ Argue in KP1 . Assume (v, vg,vh,~y). By Lemma 4.10 and monotonic- # # ity, we have eg(vg) 4 eg (tg(~y )) for some #-term eg . By monotonicity, # vh 4 th(eg(vg)) implies that vh 4 th(eg (tg(~y ))). Using the term tpair from # Example 4.2, tf (~y ):=tpair th(eg (tg(~y ))),tg(~y ) is as desired. It is easy to ﬁnd a formula " as desired. f ⇤ Before proving parts (b) and (c) of Theorem 5.8 we need a technical lemma. We return to the proof of part (b) in Section 5.3.

5.2. Elimination lemma u Recall that the axioms of KP1 +Ldef do not include the axiom schemes of u KP1 in the language Ldef but only in the language L0. However, in order to March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 85

Cobham Recursive Set Functions and Weak Set Theories 85

prove closure under syntactic Cobham recursion, Theorem 5.8 (c), we will need some version of these schemes. In the usual development of full Kripke Platek set theory KP (e.g., [5, Chapter I]), one shows that ⌃1-expansions prove each scheme for formulas mentioning new symbols from their bigger language L, for example 0(L)-Separation. This is done in two steps. First, one shows that occurrences of new ⌃1-deﬁned symbols can be eliminated in a way that trans- forms 0(L)-formulas into 1-formulas. Second, one proves 1-Separation in KP. An analogous procedure is employed in bounded arithmetic when 1 developing S2 (cf. [12]). u+ For our weak theory KP1 the situation is more subtle. The following lemma gives a version of the elimination step, just good enough for our pur- poses: it eliminates new function symbols by -bounding quantiﬁers, with 2 the help of an auxiliary parameter V .Intuitively,thisV is a set collecting enough computations of new functions to evaluate the given formula; it is uniquely determined by a simple formula and weakly uniformly bounded. The precise statement needs the following auxiliary notion.

+ Deﬁnition 5.11: We write 0 (Ldef ) for the class of 0(Ldef )-formulas all of whose -bounding terms are L+-terms. 2 0 Lemma 5.12: (Elimination) For every '(~x ) +(L ) there are (L+)- 2 0 def 0 0 formulas 'equ(~x , V ), 'aux(~x , V ), 'emb(z,z0,~x,V) and a #-term t'(~x ) such u that KP1 +Ldef proves

61V' (~x , V ) 9 aux V (' (~x , V ) ' ( , ,~x,V):V t (~x )) (5.2) 9 aux ^ emb · · 4 ' ' (~x , V ) ('(~x ) ' (~x , V )). aux ! $ equ

Proof: This is proved by induction on '(~x ). The base case for atomic '(~x ) is the most involved and is proved by induction on the number of occurrences of symbols in ' from L L+. If this number is 0, there is not def \ 0 much to be shown. Otherwise one can write

'(~x )= (~x , f (~s (~x ))),

where (~x , y ) has one fewer occurrence of symbols from L L+,the def \ 0 symbol f(~z ) is from L L+, and ~s (~x ) is a tuple of L+-terms. def \ 0 0 Let ('f ,"f ,ef ,tf ) be a good deﬁnition of f(~z ). By Lemma 4.10 and # # monotonicity, we have #-terms ef (v),~s (~x ) and 0(L0)-formulas "0,"1 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 86

86 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

u such that KP1 +Ldef proves ' (v,~s(~x )) f ! " ( , ,v,~x):v t (~s #(~x )) " ( , ,v,~x):e (v) e#(t (~s #(~x ))). 0 · · 4 f ^ 1 · · f 4 f f u By induction, there are equ, aux, emb,t such that KP1 +Ldef proves 61W (~x , e (v),W) 9 aux f W (~x , e (v),W) ( , ,~x,e (v),W):W t (~x , e (v)) 9 aux f ^ emb · · f 4 f (~x , e (v),W) ( (~x , e (v)) (~x , e (v),W)). aux f ! f $ equ f Deﬁne ' (~x , V ):= W, v tc(V )((~x , V , W , v )) where aux 9 2 (~x , V , W , v ):=(V = W, v (~x , e (v),W) ' (v,~s(~x ))). h i^ aux f ^ f + Monotonicity lets us construct from emb a0(L0 )-formula "2 such that u KP1 +Ldef proves (~x , V , W , v ) ! " ( , ,~x,V):W t (~x , e #(v)) " ( , ,⇡ (V ),~x):v t (~s #(~x )). 2 · · 4 f ^ 0 · · 2 4 f Using the term tpair from Example 4.2 we deﬁne

# # t'(~x ):=tpair(t (~x , e f (v)),tf (~s (~x ))) + u and get a 0(L0 )-formula 'emb such that KP1 +Ldef proves (~x , V , W , v ) ' ( , ,~x,V):V t (~x ). ! emb · · 4 ' Finally, we set

'equ(~x , V ):= equ(~x , e f (⇡2(V )),⇡1(V )). It is easy to verify (5.2). This completes the proof for the case that '(~x )is atomic. The induction step is easy if '(~x ) is a negation or a conjunction. We + consider the case that '(~x )= u s(~x ) (u, ~x) for some L0 -term s(~x ). By 8 2 u induction, there are equ, aux, emb,t such that KP1 +Ldef proves u s(~x ) 61W (u, ~x, W ) 8 2 9 aux u s(~x ) W (u, ~x, W ) ( , ,u,~x,W):W t (u, ~x) 8 2 9 aux ^ emb · · 4 u s(~x ) W (u, ~x, W ) ( (u, ~x) (u, ~x, W )) . 8 2 8 aux ! $ equ # + By monotonicity and Lemma 4.10 there is a #-term s (~x ) such that T0 deﬁnes a (L )-embedding of u into s#(~x )whenu s(~x ), and hence 0 0 2 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 87

Cobham Recursive Set Functions and Weak Set Theories 87

without loss of generality we can replace the bound t (u, ~x) above with # u t (s (~x ),~x). By 0(L0)-Collection, KP1 +Ldef proves that the set V = W : u s(~x ) (u, ~x, W ) (5.3) { 9 2 aux } exists, and by Lemma 4.11 it also 0(L0)-defines an embedding of V into some #-term t(~x ). For 'emb(z,z0,~x,V) we choose a formula describing this + embedding and we set t'(~x )=t(~x ). Define 'aux(~x , V )tobea0(L0 )- + formula expressing (5.3); this is 0(L0 ) because witnesses in aux are unique – recall the discussion of “collection terms” in Section 3.3. Define + 'equ(~x , V )tobethe0(L0 )-formula u s(~x ) W V ( (u, ~x, W ) (u, ~x, W )). 8 2 9 2 equ ^ aux u It is straightforward to verify (5.2) in KP1 +Ldef . ⇤ One can bootstrap the Elimination Lemma to yield a bigger auxiliary set V such that 'equ(~x , V ) is equivalent to '(~x ) simultaneously for all tuples ~x taken from a given set. As we shall use this stronger version too, we give details. Recall ~x z stands for x z. 2 i i 2 + + Lemma 5.13: For every '(~x,~y V) 0 (Ldef ) and every L0 -term s(~y ) there + s 2 s s are 0(L0 )-formulas 'equ(~x,~y , U ), 'aux(~y , U ), 'emb(z,z0,~y,U) and a #- s u term t'(~y ) such that KP1 +Ldef proves 61U's (~y , U ) 9 aux U ('s (~y , U ) 's ( , ,~y,U):U ts (~y )) 9 aux ^ emb · · 4 ' 's (~y , U ) ~x s(~y )('(~x,~y ) 's (~x,~y , U )). aux !8 2 $ equ

Proof: Using Lemma 5.12, choose 'equ, 'aux, 'emb and t' for which u KP1 +Ldef proves ~x s(~y ) 61V' (~x,~y , V ) 8 2 9 aux ~x s(~y ) V (' (~x,~y , V ) ' ( , ,~x,~y,V):V t (~x,~y )) 8 2 9 aux ^ emb · · 4 ' ~x s(~y ) ' (~x,~y , V ) ('(~x,~y ) ' (~x,~y , V )) . 8 2 aux ! $ equ From the first and second lines, exactly as in the universal quantification + s step in the proof of Lemma 5.12, we get a 0(L0 )-formula 'emb(z,z0,~y,U) u and a #-term t'(~y ) such that KP1 +Ldef proves the existence of a set U with U = V : ~x s(~y ) ' (~x,~y , V ) and 's ( , ,~y,U):U t (~y ). (5.4) { 9 2 aux } emb · · 4 ' s + For 'aux take a 0(L0 )-formula expressing the first conjunct of (5.4), and for 's take the (L+)-formula V U (' (~x,~y , V ) ' (~x,~y , V )). equ 0 0 9 2 aux ^ equ ⇤ March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 88

88 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

As a simple application of Lemma 5.12 we derive a separation scheme.

u + Corollary 5.14: The theory KP1 +Ldef proves 0 (Ldef )-Separation.

Proof: Let '(u, x,w ~ )bea+(L )-formula. We want to show that u 0 def { 2 x : '(u, x,w ~ ) exists. Choose 'x ,'x according to the previous lemma, } aux equ substituting x,w ~ for ~y ,substitutingu for ~x , and substituting x for s(~y ). Choose U such that 'x (x,w, ~ U). Then the set u x : '(u, x,w ~ ) equals aux { 2 } u x : 'x (u, x,w, ~ U) ,soexistsby(L+)-Separation. { 2 equ } 0 0 ⇤

5.3. Closure under replacement The following theorem is crucial. It provides a formalized version of Theo- u rem 2.6(a) showing, more generally, that KP1 +Ldef can handle comprehension terms coming from Replacement. Similar terms are basic computation steps in Sazonov’s term calculus [23] and in the logic of Blass et al. [10]. Recall that ~x u stands for x u. 2 i i 2 V + Theorem 5.15: Let ✓(u, ~y,~x) be a 0 (Ldef )-formula and g(~y,~x ) a function symbol in Ldef . Then there exists a function symbol f(u, ~y) in Ldef such u that KP1 +Ldef proves f(u, ~y)= g(~y,~x ):✓(u, ~y,~x) ~x u . { ^ 2 } Proof: For notational simplicity we assume ~y is the empty tuple. It is u sucient to prove the theorem for g such that KP1 +Ldef proves g(~x ) = 0. u 6 We ﬁrst show that KP1 +Ldef proves the existence of z := g(~x ):✓(u, ~x) ~x u { ^ 2 }

and furthermore describes an embedding of z into t1(u) for a suitable #- term t1. Let ('g,"g,eg,tg) be a good deﬁnition of g and choose ✓equ,✓aux,✓emb,t✓ u for ✓ according to the Elimination Lemma 5.12. Argue in KP1 +Ldef . For every ~x u there exists a unique w such that 2 y, v ,V tc(w) (w, y, v ,V,u,~x), 9 g 2 g where (w, y, v ,V,u,~x) expresses that w = y, v ,V where either g hh gi i (y = g(~x ) ✓(u, ~x)) or (y =0 ✓(u, ~x)), and the computations of g and ^ ^¬ ✓ are witnessed by vg and V . Formally, (w, y, vg,V,u,~x) is the following March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 89

Cobham Recursive Set Functions and Weak Set Theories 89

+ 0(L0 )-formula: w = y, v ,V ✓ (u, ~x, V ) ' (v ,~x) hh gi i^ aux ^ g g (y = e (v ) ✓ (u, ~x, V )) (y =0 ✓ (u, ~x, V )) . ^ g g ^ equ _ ^¬ equ As in the proof of Lemma 5.12, from ✓emb and "g we can construct a 0(L0)- formula " and #-term t (u, ~x) such that "( , ,w,u,~x):w t (u, ~x) for this 2 · · 4 2 w. By Collection the set W = w : ~x u y, v ,V tc(w) (w, y, v ,V,u,~x) , { 9 2 9 g 2 g } exists, and by Lemma 4.11 we have "0( , ,W,u):W t (u) for a suitable · · 4 3 0(L0)-formula "0 and #-term t3(u). The deﬁnition of W above can be + expressed by the 0(L0 )-formula (as discussed in Section 3.3) w W ~x u y, v ,V tc(w) ~x u w W y, v ,V tc(w) . 8 2 9 2 9 g 2 ^82 9 2 9 g 2 (5.5) We see that z exists by 0(L0)-Separation: z = y tc(W ): w W (y = ⇡ (⇡ (w)) y = 0) . (5.6) { 2 9 2 1 1 ^ 6 } Note "0( , ,W,u):z t (u)sincez is a subset of tc(W ). Recalling t · · 4 3 pair from Example 4.2, we construct a good deﬁnition (✓f ,"f ,ef ,tf ) of f(u): ✓ (v, u):= W, z tc(v)(v = W, z (5.5) and (5.6) hold) f 9 2 h i^ ef (v):=⇡2(v)

tf (u):=tpair(t3(u),t3(u)), and " such that " ( , ,v,u):v t (u) for the unique v with ✓ (v, u). f f · · 4 f f ⇤ u We can now show that, in KP1 +Ldef , weakly uniform embeddings (given by 0(L0)-formulas) and strongly uniform embeddings (given by function symbols) are closely related. For suppose we are given a 0(L0)- embedding "( , ,~x):s(~x ) t(~x ). Then a function ⌧ satisfying ⌧(z,~x)= · · 4 z0 tc(t(~x )) : "(z,z0,~x) is in L by Theorem 5.15, and we have ⌧( ,~x): { 2 } def · s(~x ) t(~x ). On the other hand, suppose ⌧ L and ⌧( ,~x):s(~x ) t(~x ). 4 2 def · 4 If we deﬁne "(z,z0,~x) as z0 ⌧(z,~x), then "( , ,~x):s(~x ) 4 t(~x ). The + 2 · · embedding " is 0 (Ldef ) rather than 0(L0), but using the Elimination Lemma 5.12 we can ﬁnd an equivalent 0(L0)-embedding, at the cost of involving a unique, bounded parameter V . We will use constructions like this in the next subsection, where we need to show, using induction with bounds given by weakly uniform embeddings, that Ldef is closed under Cobham recursion where the bound is given by a strongly uniform embedding. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 90

90 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

Example 5.16: Let f(x,w ~ ) be a function symbol in Ldef .ThenLdef con- u tains a function symbol f”(x,w ~ ) such that KP1 +Ldef proves f”(x,w ~ )= f(u,w ~ ):u x . { 2 } Furthermore, Ldef contains the function symbols x y and x y and u \ \ KP1 +Ldef proves the usual deﬁning axioms for them.

5.4. Closure under syntactic Cobham recursion

We are ready to verify statement (c) of Theorem 5.8, that Ldef is closed under syntactic Cobham recursion.

Theorem 5.17: For all function symbols g(x, z,w ~ ) and ⌧(u, v, x,w ~ ) in Ldef and all #-terms t(x,w ~ ) there is a function symbol f(x,w ~ ) in Ldef such that u KP1 +Ldef proves g(x, f”(x,w ~ ),w~ )if⌧ is an embedding into t at x,w ~ f(x,w ~ )= (5.7) 0 otherwise, ⇢ where “⌧ is an embedding into t at x,w ~ ”standsforthe0(Ldef )-formula ⌧( ,g(x, f”(x,w ~ ),w~ ),x,w~ ):g(x, f”(x,w ~ ),w~ ) t(x,w ~ ). · 4 Proof: Let g, ⌧, t be as stated. For notational simplicity we assume w~ is the empty tuple. We are looking for a good definition ('f ,"f ,ef ,tf ) of the func- u tion f(x), that is, for a good definition for which KP1 +Ldef proves (5.7) for the associated function symbol f(x)inLdef . We intend to let 'f (v, x) say that v encodes the course of values of f, namely the set of all pairs u, f(u) ,u tc+(x). More precisely, we will +h i 2 express this by writing a 0 (Ldef )-formula (w, x) which asserts that the values in a sequence w are recursively computed by g, and then applying + the Elimination Lemma 5.12 to get the required 0(L0 )-formula 'f (v, x). Hence the witness v will consist of w plus some parameters needed for the elimination of Ldef -symbols. By Theorem 5.15 there is a binary function symbol w”y in Ldef such that KPu +L proves w”y = w’z : z y . We define an auxiliary formula 1 def { 2 } ⇠(w, y):=⌧( ,g(y, w”y),y):g(y, w”y) t(y). · 4 We then let (w, x) express that w is a function with domain tc+(x)such that y tc+(x) (⇠(w, y) w’y = g(y, w”y)) ( ⇠(w, y) w’y = 0) . 8 2 ^ _ ¬ ^ March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 91

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+ Claim 1. There is a 0 (Ldef )-formula and a #-term s such that u KP1 +Ldef proves (w, x) ( , ,w,x):w s(x). ! · · 4 u Proof of Claim 1. Argue in KP1 +Ldef . Suppose that (w, x) holds. Then + for all y tc (x)wehave⌧( ,w’y, y):w’y 4 t(y). Let 0(z,z0,w,y)be 2 · + the formula z0 ⌧(z,w’y, y). Then we have a weakly uniform 0 (Ldef )- 2 + embedding 0( , ,w,y):w’y 4 t(y) for all y tc (x), and by the proof · · 2 + of the Monotonicity Lemma 4.8, adapted for 0 (Ldef )-formulas, we can + construct a 0 (Ldef )-embedding 1( , ,w,y,x):w’y 4 t(x). · · + Using t from Example 4.2 we can ﬁnd (L ) and a #-term t0 pair 2 2 0 def such that

( , ,w,y,x): y, w’y t0(x) 2 · · h i 4 for y tc+(x). Writing '(v, y, w) for the formula v = y, w’y ,wehave 2 h i !v'(v, y, w) and an embedding of v for each y, so can apply Lemma 4.11, 9 + adapted for 0 (Ldef )-embeddings, to combine these into a single embed- + ding of w into t00(tc (x),t0(x)) for a #-term t00. As usual by Lemma 4.10 and monotonicity we can replace this bound with a #-term s(x). a We can begin to construct a good deﬁnition of f.Since is a + + 0 (Ldef )-formula there exist 0(L0 )-formulas equ(w, x, V ), aux(w, x, V ), emb(z,z0,w,x,V) and a #-term t (w, x) satisfying the Elimina- tion Lemma 5.12 for . Choose and s to satisfy Claim 1. By + + Lemma 5.13, since is a 0 (Ldef )-formula there exist 0(L0 )-formulas r r r r equ(z,z0,w,x,U), aux(w, x, U), emb(z,z0,w,x,U) and a #-term t(w, x) u such that KP1 +Ldef proves 61Ur (w, x, U) 9 aux U (r (w, x, U) r ( , ,w,x,U):U tr(w, x)) 9 aux ^ emb · · 4 r r (w, x, U) z,z0 r(w, x)((z,z0,w,x) (z,z0,w,x,U)), aux !8 2 $ equ where r(w, x)isthetermtc( w, s(x) ). From the third line it follows that { } r (w, x, U) (( , ,w,x):w s(x) r ( , ,w,x,U):w s(x)). aux ! · · 4 $ equ · · 4 Now deﬁne ' (v, x):= w, V, U tc(v) f 9 2 v = w, V,U (w, x, V ) (w, x, V ) r (w, x, U) h h ii ^ aux ^ equ ^ aux ef (v):=⇡1(v)’top(⇡1(v)), March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 92

92 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

+ where top(w) is an L0 -term that recovers x from w, as the unique member x0 of the domain of w such that tc(x0) contains all other members of the domain of w. We obtain "f and tf from the following claim. Recall that the properties u+ of a good deﬁnition of an Ldef symbol must be provable in KP1 . + u+ Claim 2. There is a 0(L0 )-formula "f and a #-term tf such that KP1 proves ' (v, x) " ( , ,v,x):v t (x). f ! f · · 4 f u Proof of Claim 2. By conservativity we can argue in KP1 +Ldef . Assume + r 'f (v, x) and write v = w, V,U .Wehave0(L0 )-formulas emb, emb r h h ii and equ such that - ( , ,w,x,V):V t (w, x), emb · · 4 - r ( , ,w,x,U):U tr(w, x), emb · · 4 - r ( , ,w,x,U):w s(x), equ · · 4 r where t (w, x), t(w, x) and s(x) are #-terms. Deﬁne

r tf (x):=tpair s(x),tpair t (s(x),x),t(s(x),x)

⇣ + ⌘ and use monotonicity to get a 0(L0 )-formula "f (z,z0,v,x) describing an embedding of v into t (x). f a We must show that ('f ,"f ,ef ,tf ) is a good deﬁnition. Claim 2 gives (Witness Embedding) and the next two claims show (Witness Uniqueness) and (Witness Existence).

Claim 3. The tuple ('f ,"f ,ef ,tf ) satisﬁes (Witness Uniqueness). u Proof of Claim 3. It suces to prove in KP1 +Ldef that (w, x) (˜w, x) w =˜w ^ ! r since uniqueness of V and U is then guaranteed by aux and aux.Sosup- pose (w, x), (˜w, x) and w =˜w.Thentheset y tc+(x):w’y =˜w’y 6 { 2 6 } is nonempty. By set foundation it contains an -minimal element y .Then 2 0 w”y =˜w”y since w, w˜ both have domain tc+(x) y . It follows that 0 0 ◆ 0 g(y ,w”y )=g(y , w˜”y ) and ⇠(w, y ) ⇠(˜w, y ). Since (w, x) and 0 0 0 0 0 $ 0 (˜w, x), we get w’y =˜w’y , a contradiction. 0 0 a The proof of the next claim is the only place where we use the full u strength of induction available in KP1 . March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 93

Cobham Recursive Set Functions and Weak Set Theories 93

Claim 4. The tuple ('f ,"f ,ef ,tf ) satisﬁes (Witness Existence).

u Proof of Claim 4. Again we will work in KP1 +Ldef and appeal to conser- 4 vativity. We will use uniformly bounded unique ⌃1 (L0)-Induction to prove v' (v, x). We already know by (Witness Uniqueness) that 61v' (v, x). 9 f 9 f Furthermore by Claim 2, the witness v is automatically uniformly bounded by the embedding "f .Itthussucestoshow

y x v' (v, y) v' (v, x). 8 2 9 f !9 f + Suppose the antecedent holds. By 0(L0)-Collection and 0(L0 )- Separation the set

W := ⇡ (v): y x' (v, y) { 1 9 2 f }

exists. For each y x this contains exactly one wy such that (wy,y), that 2 + is, such that wy is a function with domain tc (y) which recursively applies g. By the same argument as in the proof of Claim 3, any two such functions agree on arguments where they are both deﬁned. Hence, w := W is a function with domain tc(x), and we put w0 := w x, y) where y = [{h i} S g(x, w”x)if⇠(x, w), and y = 0 otherwise. Then (w0,x) holds. Furthermore u KP1 +Ldef proves that there exist U and V such that aux(w0,x,V) and r (w0,x,U). This yields ' (v, x) for v = w0, V,U . aux f h h ii a

We have shown ('f ,"f ,ef ,tf ) is a good deﬁnition. Let f be the symbol in Ldef associated to this deﬁnition. To conclude the proof we verify the u conclusion of the theorem, that is, that KP1 +Ldef proves

g(x, f”(x)) if ⌧( ,g(x, f”(x)),x):g(x, f”(x)) t(x) f(x)= · 4 0 otherwise. ⇢ Argue in KPu +L .Thewitnessv for f(x) has the form w, V,U 1 def h h ii such that (w, x) and f(x)=ef (v)=w’x. From (w, x) we get

g(x, w”x)if⌧( ,g(x, w”x),x):g(x, w”x) t(x) w’x = · 4 0 otherwise. ⇢ It now suces to verify f”(x)=w”x. This follows from f(y)=w’y for every y tc+(x) which is seen similarly as in the proofs of Claims 3 and 4. 2 ⇤ This completes the proof of Theorem 5.8 and thus of Theorem 5.2. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 94

94 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

5.5. The expanded theories u We ﬁx a fragment of KP1 +Ldef whose language is exactly Lcrsf .Weshow it proves the schemes in the language Lcrsf .

u Definition 5.18: KP1 (Lcrsf ) is the theory in the language Lcrsf which u consists of KP1 together with, for each function symbol f in Lcrsf L0,a u u\ defining axiom for f from KP1 +Ldef chosen in such a way that KP1 (Lcrsf ) proves the defining axiom for f from Tcrsf .

u u Since KP1 (Lcrsf ) is an expansion by deﬁnitions of KP1 ,wehave:

u u Proposition 5.19: KP1 (Lcrsf ) is conservative over KP1 and proves Tcrsf .

u We will later show that KP1 (Lcrsf )is⇧2(Lcrsf )-conservative over Tcrsf u (Theorem 6.9). Observe that, by Lemma 3.18, KP1 (Lcrsf )proves0(Lcrsf )- Separation.

Lemma 5.20: Let the theory T consist of Tcrsf together with the axiom u schemes of KP1 expanded to the language Lcrsf ,thatis,the0(Lcrsf )- 4 Collection and uniformly bounded unique ⌃1 (Lcrsf )-Induction schemes, where the uniform embeddings may be given by 0(Lcrsf )-formulas. Then u KP1 (Lcrsf ) is equivalent to T .

u Proof: Consider a model M of T ;wemustshowM ✏ KP1 (Lcrsf ). The u reduct of M to L0 is a model of KP1 , and Definition 5.18 is an extension by ˜ u ˜ definitions. Thus M can be expanded to a model M of KP1 (Lcrsf ), where L˜crsf is a disjoint copy of Lcrsf containing a function symbol f˜ for every function symbol f in Lcrsf .TheL˜crsf functions satisfy the defining axioms from Tcrsf ; we claim that this implies that the L˜crsf functions are identical to the original functions from L L in M, and thus M KPu(L ). crsf \ 0 ✏ 1 crsf The claim is immediate in the case of functions defined by composition or replacement, but for recursion we need to appeal to induction in M. Suppose f = fg,⌧,s is defined by syntactic Cobham recursion, where g =˜g and ⌧ =˜⌧ in M˜ and s is a #-term. We have in M˜ , where for clarity we suppress the side variables w~ , both

g(x, f˜”(x)) if ⌧ is an embedding into s at x x f˜(x)= 8 0 otherwise ⇢ and the same formula with f and f” in place of f˜ and f˜”. The function f˜ + has a good deﬁnition in the sense of Deﬁnition 5.6, so there are 0(L0 )- March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 95

Cobham Recursive Set Functions and Weak Set Theories 95

formulas ', ", an L+-term e and a #-term t such that for all x M˜ , 0 2 M˜ = !v'(x, v) | 9 v ('(x, v) "( , ,x,v):v t(x)) ^8 ! · · 4 f˜(x)=f(x) v ('(x, v) e(v)=f(x)) . ^ $9 ^ ˜ 4 Hence f(x)=f(x) can be expressed as a uniformly bounded ⌃1 (Lcrsf )- formula, for which witnesses are unique. Therefore we can prove it holds for all x by induction in M, as the induction step follows immediately from the recursive equations for f˜ and f. For the other direction, suppose M =KPu(L ). We must show that | 1 crsf M satisﬁes the induction and collection schemes of T . Suppose '(x, y) and "(z,z0,x,y) are 0(Lcrsf )-formulas and t(x)isa 1 #-term, all with parameters from M, and that M = x  y'(x, y). Let ",t | 8 9 ' (x, y) abbreviate the formula '(x, y) "( , ,x,y):y 4 t(x). We will ﬁnd + ^ · · ˜ 0(L0 )-formulas' ˜(x, w)," ˜0(z,z0,x,w) and a #-term t(x), with the same, unwritten, parameters, such that

1 ",t "˜,t˜ M = x  w '˜(x, w) x y' (x, y) w '˜ (x, w) (5.8) | 8 9 ^8 9 $9 4 from which it follows that M satisfies uniformly bounded unique ⌃1 (Lcrsf )- Induction. Let (u, x, y)expressu = z,z0 tc(y) tc(t(x)) : "(z,z0,x,y) , " {h i2 ⇥ } which implies in M that u : y t(x) "( , ,x,y):y t(x). By 4 $ · · 4 Lemma 3.18 there is f L such that 2 crsf M =('",t(x, y) (u, x, y)) f(x, y, u) =0. | ^ " $ 6 The function f has a good definition in the sense of Definition 5.6. Therefore + + there are 0(L0 )-formulas , , an L0 -term e and a #-term s such that for all x, y M, 2 M = !v (x, y, u, v) | 9 v ( (x, y, u, v) ( , ,x,y,u,v):v s(x, y, u)) ^8 ! · · 4 ('",t(x, y) (u, x, y)) v ( (x, y, u, v) e(v) = 0) . ^ ^ " $9 ^ 6 We can now define '˜(x, w):= y, u, v tc(w) w = u, y ,v (x, y, u, v) e(v) =0 . 9 2 hh i i^ ^ 6 Then' ˜ satisfies the uniqueness condition; furthermore, the right-to-left im- plication in (5.8) will hold for any choice of" ˜ and t˜. For the other direction, 0(Lcrsf )-Separation yields u satisfying "(u, x, y). To construct" ˜ and t˜for March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 96

96 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

the embedding, Lemma 4.12 gives " (L ) and a #-term t such emb 2 0 0 emb that M = u : y t(x) " ( , ,u,y,t(x)) : u, y t (t(x)). Thus, as | 4 ! emb · · h i 4 emb in the proof of the Elimination Lemma 5.12, using monotonicity and the + ˜ term tpair we can ﬁnd a 0(L0 )-formula" ˜ and a #-term t such that for all x, y, v M, 2 M = u : y t(x) ( , ,x,y,v):v s(x, y, u) | 4 ^ · · 4 "˜( , ,x, u, y ,v ): u, y ,v t˜(x). ! · · hh i i hh i i 4 Then' ˜,˜" and t˜ satisfy (5.8). For collection, suppose M = x u y'(x, y) for ' (L ). Then | 8 2 9 2 0 crsf '(x, y) is equivalent to f(x, y) = 0 for some f L . The good deﬁnition 6 2 crsf of f gives (L+) such that '(x, y) v (x, y, v)inM for all x, y. 2 0 0 $9 By (L+)-Collection there is W M such that 0 0 2 M = x u y, v W (x, y, v). | 8 2 9 2 Thus M = x u y W'(x, y) as required. | 8 2 9 2 ⇤

4 4 u Lemma 5.21: Let KP1 (Lcrsf ) be the theory KP1 +KP1 (Lcrsf ). Then 4 4 KP1 (Lcrsf ) is conservative over KP1 , and is equivalent to the theory 4 consisting of Tcrsf plus the 0(Lcrsf )-Collection and ⌃1 (Lcrsf )-Induction schemes.

Proof: By Lemma 5.20 it is sucient to show that any model M of 4 4 KP1 (Lcrsf ) satisﬁes the ⌃1 (Lcrsf )-Induction scheme. By Lemma 4.13 it 4 is enough to show that uniformly bounded ⌃1 (Lcrsf )-Induction holds, and this follows by the same argument as in the proof of Lemma 5.20, ignoring the conditions about the witnesses y and w being unique. ⇤

6. Witnessing

4 4 Theorem 5.2 established that every CRSF function is ⌃1 -deﬁnable in KP1 , u 4 and in fact already in KP1 . We would like to show that every function ⌃1 - 4 deﬁnable in KP1 is in CRSF. By analogy with bounded arithmetic, one could aim to prove that whenever KP4 y'(y,~x)with' (L ), 1 `9 2 0 0 then T '(f(~x ),~x) or at least ZFC '(f(~x ),~x) for some “witnessing” crsf ` ` function f(~x ) in CRSF. As mentioned in the introduction, this fails: a witnessing function C(x) for (x =0 y (y x)) would satisfy 6 !9 2 (x =0 C(x) x) 6 ! 2 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 97

Cobham Recursive Set Functions and Weak Set Theories 97

and not even ZFC can define such a C as a CRSF function.d This section shows two ways around this obstacle. The first is to weaken the conclusion of the witnessing theorem from '(f(~x ),~x)to y f(~x ) '(y,~x). We prove such a witnessing theorem for u 9 2 KP1 (Theorem 6.10), and this has as a corollary the following definability theoretic characterization of CRSF. We do not know whether Theorem 6.10 4 u or Corollary 6.1 hold for KP1 instead of KP1 .

Corollary 6.1: A function is in CRSF if and only if it is ⌃1(L0)-definable u in KP1 . The second is to simply add a global choice function C to CRSF as one of the initial functions, resulting in CRSFC (Remark 2.10). In this way 4 we are able to prove full witnessing, even for the stronger theory KPC1 obtained by adding the axiom of global choice (Theorem 6.10). Again this has as a corollary the following definability theoretic characterization of CRSFC where we write LC := L C . 0 0 [{ } C C Corollary 6.2: A function is in CRSF if and only if it is ⌃1(L0 )- 4 definable in KPC1 .

4 We do not know whether some form of witnessing holds for KP1 without choice. In particular, the following question is open: if KP4 !y'(y, x), 1 `9 for ' a0(L0)-formula, does this imply that there is a CRSF function f such that (provably in KP4) x'(f(x),x) holds? 1 8 It would also be interesting to prove a result of this type that needs only an appropriate form of local choice, rather than global choice. For example: if KP4 !y'(y, x), for ' a(L )-formula, does this imply that there is 1 `9 0 0 a CRSF function f(x, r) such that (provably in KP4) x'(f(x, r),x) holds 1 8 whenever r is a well-ordering of tc(x)?

6.1. Witnessing Tcrsf and Herbrand saturation We use a method introduced by Avigad in [4] as a general tool for model- theoretic proofs of witnessing theorems, in particular subsuming Zambella’s witnessing proof for bounded arithmetic [24]. A structure is Herbrand saturated if it satisﬁes every sentence, with parameters, which is consistent 98 with its universal diagram. To get a witnessing theorem for a theory T , one

dThis is wellknown: otherwise ZFC would deﬁne a global well-order and thus prove V=HOD;butV=HODisrelativelyconsistent(seee.g.,[19,p.222]). 6 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 98

98 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

uses Herbrand saturation to show that T is -conservative over a suitable 89 universal theory S.SinceS is universal, a form of witnessing for S follows directly from Herbrand’s theorem; conservativity means that this carries over to T . u We want to use this approach where T is KP1 and S is Tcrsf . We cannot do this directly since Tcrsf is not universal but, as Tcrsf is ⇧1, it turns out that something similar works. Below we prove a version of Herbrand’s theorem for Tcrsf ,inwhichawitnesstoa⌃1(Lcrsf ) sentence is not necessarily equal to a term, but is always contained in some term.

Theorem 6.3: Suppose T y'(y,~x) where ' is (L ). Then there crsf `9 0 crsf is an L function symbol f such that T y f(~x ) '(y,~x). crsf crsf `9 2

Proof: Take a new tuple ~c of constants and let P (~c ) be the theory

T + y t(~c ) '(y,~c):t(~x ) an L -term . crsf {8 2 ¬ crsf }

It suces to show that P (~c ) is inconsistent. Then Tcrsf proves y t (~x ) '(y,~x) for ﬁnitely many terms t (~x ),...,t (~x ); we can choose i 9 2 i 1 k f(~x ) so that T proves f(~x )=t (~x ) t (~x ) using closure under W crsf 1 [···[ k composition. For the sake of a contradiction assume P (~c ) has a model M.Deﬁne

N := a M : M = a t(~c ) for some L -term t(~x ) . { 2 | 2 crsf } Note N contains each component cM of ~c M via the term c .Weﬁrstshow i { i} that N is a substructure of M. To see this, suppose g is an r-ary function symbol in L and ~a N r.Wemustshowg(~a ) N. For each component crsf 2 2 ai of ~a there is a term ti(~x ) such that M = ai ti(~c ). Choose a function | 2 r r symbol G(z)inLcrsf such that Tcrsf proves G(z)=g(⇡1(z),...,⇡r (z)), r where ⇡i is the standard projection function for ordered r-tuples (which is in L+). Then in M we have g(~a ) G”(t (~c ) t (~c )). 0 2 1 ⇥···⇥ r Next we show that N is a 0(Lcrsf )-elementary substructure of M, that is, for every (L )-formula ✓ and ~a N,wehaveN = ✓(~a ) M = 0 crsf 2 | , | ✓(~a ). This is proved by induction on ✓, and the only non-trivial case is where ✓(~a ) has the form u t(~a ) (u,~a) for some term t, and we have 9 2 M = b t(~a ) (b,~a) for some b in M. As N is a substructure, t(~a ) N | 2 ^ 2 and hence M = t(~a ) s(~c ) for some term s(~x ). Thus M = b s(~c ), so | 2 | 2 b N. By the induction hypothesis N = (b,~a) which gives N = ✓(~a ) as 2 | |S required. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 99

Cobham Recursive Set Functions and Weak Set Theories 99

Thus N =T since T is ⇧ (L ). Further, N = y ( '(y,~c)), | crsf crsf 1 crsf | 8 ¬ since in M there is no witness for '(y,~c) inside any term in ~c . This contradicts the assumption of the theorem. ⇤

Corollary 6.4: If T !y'(y,~x), where ' is (L ), then there is crsf `9 0 crsf an L function symbol g such that T '(g(~x ),~x). crsf crsf ` Proof: Appealing to Lemma 3.18, take g(~x ) computing y f(~x ): { 2 '(y,~x) where f is given by Theorem 6.3. } S ⇤ We give our version of Herbrand saturation. Let L L be a countable ◆ 0 language.

Deﬁnition 6.5: AstructureM is 0(L)-Herbrand saturated if it satisﬁes every ⌃2(L)-sentence with parameters from M which is consistent with the ⇧1(L)-diagram of M.

The next two lemmas do not use any special properties of the class 0(L), beyond that it is closed under subformulas, negations and substitu- tion.

Lemma 6.6: Every consistent ⇧1(L) theory T has a 0(L)-Herbrand saturated model.

Proof: Let L+ be L together with names for countably many new con- + stants. Enumerate all 0(L )-formulas as '1,'2,.... Let T1 = T and de- ﬁne a sequence of theories T T ... as follows: if T + ~x ~y' (~x,~y )is 1 ✓ 2 ✓ i 9 8 i consistent, let T = T + ~y' (~c,~y )where~c is a tuple of constant symbols i+1 i 8 i that do not appear in Ti or 'i.OtherwiseletTi+1 = Ti. Let T ⇤ = i Ti. By construction, T is consistent and ⇧ (L+). ⇤ 1 S Let M be a model of T ⇤ and let N be the substructure of M consisting of + elements named by L -terms. We claim that N = T ⇤. It is enough to show | that for every (L+)-formula ' and every tuple ~a from N,wehaveN = 0 | ✓(~a ) M = ✓(~a ). We prove this by induction on ✓. For the only interesting , | case, suppose M = x (~a , x ) where the inductive hypothesis holds for . | 9 Since the components of ~a are named by terms, (~a , x ) is equivalent in M to some formula ' (x) from our enumeration. But M = x' (x)implies i | 9 i that x' (x) is consistent with T and hence that ' (c)isinT for some 9 i i i i+1 constant c.ThusM = ' (c) and therefore M = (~a , c ), so N = (~a , c )by | i | | the inductive hypothesis. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 100

100 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

Finally, N is 0(L)-Herbrand saturated. For suppose that is 0(L) and ~x ~y (~x,~y,~a ) is consistent with the ⇧ (L)-diagram of N, and hence 9 8 1 with T ⇤. Then as above (~x,~y,~a ) is equivalent to 'i(~x,~y ) for some i, and since ~x ~y' (~x,~y ) is consistent with T it is witnessed in T by a tuple 9 8 i i i+1 of constants and hence is true in N. ⇤

Lemma 6.7: If S, T are theories such that S is ⇧1(L) and every 0(L)- Herbrand saturated model of S is a model of T , then T is ⇧2(L)-conservative over S.

Proof: Suppose T proves ~x ~y' (~x,~y )butS does not, where ' is (L). 8 9 0 Then, letting ~c be a tuple of new constants, the theory S + ~y ( '(~c,~y )) 8 ¬ has a 0(L)-Herbrand saturated model by Lemma 6.6. This contradicts the assumptions about S and T . ⇤

We now describe the most useful property of 0(Lcrsf )-Herbrand saturated models.

Lemma 6.8: Suppose that M =T is (L )-Herbrand saturated and | crsf 0 crsf that '(y,~x,~a) is a (L )-formula with parameters ~a M such that 0 crsf 2 M = ~x y'(y,~x,~a). Then there exist a function f L and parameters | 8 9 2 crsf m~ M such that M = ~x y f(~x,m~ ) '(y,~x,~a). 2 | 8 9 2

Proof: Let T ⇤ be the ⇧ (L )-diagram of M.ThenT ⇤ ~x y'(y,~x,~a) 1 crsf `8 9 since otherwise M = ~x y ( '(y,~x,~a)) by Herbrand saturation. The rest | 9 8 ¬ of the argument is standard. By compactness, there are ~b M and ✓ in 2 (L ) such that M = ~z✓ (~a , ~b, ~z) and 0 crsf | 8 T + ~z✓ (~a , ~b, ~z) ~x y'(y,~x,~a) crsf 8 `8 9 where we treat ~a , ~b as constant symbols. Hence, replacing ~a , ~b with variables ~u,~v , T ~z ✓(~u,~v,~z ) y'(y,~x,~u). (6.1) crsf `9 ¬ _9 Using pairing and projection functions to code tuples of sets as single sets, we can apply Theorem 6.3 to formulas with more than one unbounded existential quantiﬁer. In particular from (6.1) we get an Lcrsf function symbol f with T ~z ✓(~u,~v,~z ) y f(~x,~u,~v ) '(y,~x,~u). crsf `9 ¬ _9 2 Since M = ~z✓ (~a , ~b, ~x) it follows that M = ~x y f(~x,~a , ~b) '(y,~x,~a). | 8 | 8 9 2 ⇤ March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 101

Cobham Recursive Set Functions and Weak Set Theories 101

u 6.2. Witnessing KP1 u We prove witnessing for KP1 (Lcrsf ) as a consequence of witnessing for Tcrsf , together with the following conservativity result.

u Theorem 6.9: The theory KP1 (Lcrsf ) is ⇧2(Lcrsf )-conservative over Tcrsf .

Proof: Let M be an arbitrary 0(Lcrsf )-Herbrand saturated model of u Tcrsf . By Lemma 6.7 it is enough to show that M is a model of KP1 (Lcrsf ). By Lemma 5.20 it is enough to show that M satisﬁes 0(Lcrsf )-Collection 4 and uniformly bounded unique ⌃1 (Lcrsf )-Induction. For collection, suppose that for some a M we have 2 M = u a v'(u, v), | 8 2 9 where ' is 0(Lcrsf ) with parameters. We rewrite this as M = u v (u a '(u, v)). | 8 9 2 ! By Lemma 6.8, for some L function symbol f and tuple ~b M, crsf 2 M = u v f(u,~b)(u a '(u, v)). | 8 9 2 2 ! Hence if we let c = f”(a,~b) we have, as required for collection, S M = u a v c'(u, v). | 8 2 9 2 For induction, let '(u, v),"(z,z0,v,u) (L ) and t(u)bea#- 2 0 crsf term, all possibly with parameters, and let '",t(u, v) abbreviate '(u, v) ^ "( , ,v,u):v t(u). Working in M, suppose · · 4 1 ",t ",t u 6 v'(u, v) x u x v' (u, v) v0 ' (x, v0) . 8 9 ^8 8 2 9 !9 Then in particular ",t ",t x, w u x v w' (u, v) v0 ' (x, v0) . 8 8 2 9 2 !9 By Lemma 6.8 there is an L -function symbol h and a tuple ~a M such crsf 2 that we can bound the witness v0 as a member of h(x, w,~a). Then, since witnesses v to ' are unique, if we let g(x, w,~a) compute

v0 h(x, w,~a):'(x, v0) { 2 } we have S x, w u x v w'",t(u, v) '",t(x, g(x, w,~a)) . (6.2) 8 8 2 9 2 ! We must be careful here with our parameters. We may assume without loss of generality that the so-far unwritten parameters in ', " and t are contained March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 102

102 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

in the tuple ~a , and further that '(u, v) is really '(u, v,~a), "(z,z0,v,u)is "(z,z0,v,u,~a) and t(u)ist(u,~a). We now use syntactic Cobham recursion to iterate g.Tousethere- cursion available in Tcrsf we need to turn the weakly uniform embedding given by " into a strongly uniform embedding. So let ⌧ be an Lcrsf -function symbol for which Tcrsf proves

⌧(z,v,u,w ~ )= z0 t(u,w ~ ):"(z,z0,v,u,w~ ) . { 2 } Let f be the Lcrsf function symbol fg,⌧,t with defining axiom g(u, f”(u,w ~ ),w~ )if⌧ is an embedding into t at u,w ~ f(u,w ~ )= 0 otherwise, ⇢ where “⌧ is an embedding into t at u,w ~ ” stands for the 0(Lcrsf )-formula ⌧( ,g(u, f”(u,w ~ ),w~ ),u,w~ ):g(u, f”(u,w ~ ),w~ ) t(u,w ~ ). · 4 It suces now to show that x'",t(x, f(x,~a),~a). We will use 8 0(Lcrsf )-Induction, which is available by Lemma 3.18. Suppose u x'",t(u, f(u,~a),~a). Let w = f”(x,~a) and let v = g(x, w,~a). By (6.2) 8 2 we have '",t(x, v,~a), and in particular "( , ,v,x,~a):v t(x,~a). Hence · · 4 also ⌧( ,v,x,~a):v t(x,~a), that is, ⌧ is an embedding into t at x,~a. · 4 From the defining axiom for f we conclude that f(x,~a)=v, and thus ",t ' (x, f(x,~a),~a). This completes the proof. ⇤ u u From this we get witnessing for KP1 (Lcrsf ), and a fortiori for KP1 : Theorem 6.10: Suppose KPu(L ) y'(y,~x) where ' is (L ). 1 crsf `9 0 crsf Then there is an L -function symbol f such that T y f(~x ) '(y,~x). crsf crsf `9 2 Proof: By Theorem 6.9, T y'(y,~x). Then apply Theorem 6.3. crsf `9 ⇤ Corollary 6.11: If KPu(L ) !y'(y,~x) where ' is ⌃ (L ), then 1 crsf `9 1 crsf there is an L -function symbol g such that T '(g(~x ),~x). crsf crsf ` Proof: Suppose KPu(L ) !y v✓(y, v,~x)where✓ is (L ). Using 1 crsf `9 9 0 crsf Theorem 6.10 it is not hard to show that T y, v f(~x ) ✓(y, v,~x) crsf `9 2 for some L -function symbol f, and from Theorem 6.9 we get T crsf crsf ` 61y✓(y, v,~x). Thus we can define the witnessing function as g(~x ):= y 9 { 2 f(~x ): v f(~x ) ✓(y, v,~x) . 9 2 } S ⇤ Together with the Definability Theorem 5.2, the above implies Corol- lary 6.1. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 103

Cobham Recursive Set Functions and Weak Set Theories 103

6.3. Witnessing with global choice

We add to our basic language L0 and theory T0 a symbol C for a global choice function, with deﬁning axiom

(GC) : C(0) = 0 (x =0 C(x) x). ^ 6 ! 2 C C We denote the augmented language and theory by L0 and T0 .Wewrite C C C C Lcrsf and Tcrsf for Lcrsf and Tcrsf deﬁned using L0 and T0 in place of L0 C C and T0. The symbols in Lcrsf correspond to the functions in CRSF ,deﬁned like CRSF but with the global choice function C(x) as an additional initial function (cf. Remark 2.10). 4 4 We write KPC1 for the corresponding version of KP1 , that is, the C C 4 C theory consisting of T0 and 0(L0 )-Collection and ⌃1 (L0 )-Induction u C C schemes. Similarly KPC1 consists of T0 and the 0(L0 )-Collection and 4 C uniformly bounded unique ⌃1 (L0 )-Induction schemes (where we allow em- C beddings to be 0(L0 )). Earlier results about the theories without choice carry over to the theories with choice as expected; there is one extra case in Lemma 4.10, taken care of by noting that the identity embedding embeds 4 C 4 C 4 C(x) 4 x. In particular, KPC1 (Lcrsf )isa⌃1 (L0 )-expansion of KPC1 C C to the language Lcrsf which is equivalent to the theory consisting of Tcrsf 4 C and the schemes of KPC1 over the language Lcrsf (cf. Lemma 5.21). Thus, Corollary 6.2 follows from Theorem 6.13 below, a witnessing theorem for 4 C C KPC1 (Lcrsf ). We prove it by showing conservativity over Tcrsf :

4 C C Theorem 6.12: The theory KPC1 (Lcrsf ) is ⇧2(Lcrsf )-conservative C over Tcrsf .

C Proof: Let M be an arbitrary 0(Lcrsf )-Herbrand saturated model of C Tcrsf . As before, by Lemma 6.7 it is enough to show that M is a model 4 C of KPC1 (Lcrsf ). By exactly the same argument as in the proof of Theo- C rem 6.9, we get that M is a model of 0(Lcrsf )-Collection. 4 C It remains to show that ⌃1 (Lcrsf )-Induction holds in M.By 4 C Lemma 4.13, it is enough to show that uniformly bounded ⌃1 (Lcrsf )- Induction holds. That is, exactly the induction shown for the formula v'",t(u, v) in the proof of Theorem 6.9, except without the uniqueness 9 assumption that M = u 61v'(u, v). Working through that proof, we see | 8 9 that uniqueness is used only in one place, to construct an Lcrsf -function symbol g satisfying

x, w u x v w'",t(u, v) '",t(x, g(x, w,~a)) 8 8 2 9 2 ! March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 104

104 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

from an Lcrsf -function symbol h satisfying

",t ",t x, w u x v w' (u, v) v0 h(x, w,~a) ' (x, v0)) . 8 8 2 9 2 !9 2 C In Lcrsf this can be done without the assumption, by setting

",t g(x, w,~a)=C( v0 h(x, w,~a):' (x, v0) ). { 2 } The rest of the proof goes through as before. ⇤

Theorem 6.13: Suppose KPC4(LC ) y'(y,~x) where ' is (LC ). 1 crsf `9 0 crsf Then there is an LC -function symbol f such that TC '(f(~x ),~x). crsf crsf ` Proof: By Theorem 6.12, TC y'(y,~x). Using Theorem 6.3 for TC crsf `9 crsf there is g in LC such that TC y g(~x )'(y,~x). Using Lemma 3.18 for crsf crsf `9 2 TC we ﬁnd h such that TC h(~x )= y g(~x ):'(y,~x) . Then choose crsf crsf ` { 2 } f such that TC f(~x )=C(h(~x )). crsf ` ⇤ For the theory without choice, we get a weak result in the style of Parikh’s theorem [21].

Corollary 6.14: Suppose KP4(L ) y'(y,~x) where ' is (L ). 1 crsf `9 0 crsf Then in the universe of sets we can bound the complexity of the witness y in the following sense: there is a #-term t such that ~x y t(~x ) '(y,~x) 8 9 4 holds.

C Proof: It is easy to show that for every Lcrsf -function symbol f(~x )there C is an Lcrsf -function symbol ⌧(z,~x) and a #-term t(~x ) such that ⌧( ,~x): C C · f(~x ) 4 t(~x ), provably in Tcrsf . For the initial symbols from L0 this is by Lemma 4.10 (extended to cover C). For function symbols obtained by composition we use monotonicity, for replacement we use Lemma 4.11, and for syntactic Cobham recursion we are explicitly given such a bound. Now suppose the assumption of the corollary holds. Then also KPC4(LC ) y'(y,~x), hence TC '(f(~x ),~x) for some LC -function 1 crsf `9 crsf ` crsf symbol f by Theorem 6.13. It follows that TC y t(~x ) '(y,~x), by the crsf `9 4 previous paragraph and using 0(Lcrsf )-Separation to get a nonuniform embedding. In ZFC, global choice can be forced without adding new sets (see for example [17]) so we can expand the universe V of sets to a model (V,C) of ZF + (GC) and in particular of TC .Then ~x y t(~x ) '(y,~x) holds in crsf 8 9 4 (V,C), and thus also in V, since it does not mention the symbol C. ⇤ March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 105

Cobham Recursive Set Functions and Weak Set Theories 105

6.4. Uniform Cobham recursion We can use our definability and witnessing theorems to partially answer a question that arose from [7]. Namely, the embedding giving the bound on a Cobham recursion is given by a CRSF function. If we only allow sim- pler embeddings, given by 0(L0)-formulas, does the class CRSF change? We show that it does not. This is a partial answer because we only consider what happens if we make this change in our definition of CRSF from Proposition 2.9, which is slightly di↵erent from the original definition in [7].

Deﬁnition 6.15: In the universe of sets, the CRSFu functions are those obtained from the projections, zero, pair, union, conditional, transitive closure, cartesian product, set composition and set smash functions by composition, replacement and “weakly uniform syntactic Cobham recursion”. This is the following recursion scheme: suppose g(x, z,w ~ )isaCRSFu function, "(z,z0,y,x,w~ )isa0(L0)-formula and t(x,w ~ )isa#-term.ThenCRSFu contains the function symbol f = fg,",t deﬁned by g(x, f”(x,w ~ ),w~ )if" is an embedding into t at x,w ~ f(x,w ~ )= 0 otherwise ⇢ where the condition “" is an embedding into t at x,w ~ ” stands for

"( , ,g(x, f”(x,w ~ ),w~ ),x,w~ ):g(x, f”(x,w ~ ),w~ ) t(x,w ~ ). · · 4

The language Lcrsfu and theory Tcrsfu are deﬁned by changing the syntactic Cobham recursion case in the deﬁnitions of Lcrsf and Tcrsf to match the description above.

u Theorem 6.16: The theory KP1 is ⇧2(L0)-conservative over Tcrsfu.

Proof: It is straightforward to show that Tcrsfu proves T0, just as Tcrsf does. Similarly the results about Herbrand saturation go through for Tcrsfu. By Lemma 6.7 it is enough to show that any 0(Lcrsfu)-Herbrand saturated model of Tcrsfu is a model of 0(L0)-Collection and uniformly bounded 4 unique ⌃1 (L0)-Induction. For this we can simply repeat the proof of The- orem 6.9 with Tcrsfu in place of Tcrsf , observing that the proof becomes more direct, since in the application of syntactic Cobham recursion we can use the embedding " directly without needing to construct the function symbol ⌧. ⇤

Corollary 6.17: In the universe of sets, CRSFu =CRSF. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 106

106 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

Proof: It is clear that every CRSFu function is CRSF. For the other direction, suppose f(~x ) is CRSF. Then there is a good deﬁnition of f in the sense of Deﬁnition 5.6, and in particular there is a 0(L0)-formula '(v,~x) and an L+-term e such that KPu proves !v'(v,~x) and such that 0 1 9 '(v,~x) f(~x )=e(v) holds in the universe for all sets v,~x.Thereisasimi- ! lar version of Herbrand’s theorem for Tcrsfu as there is for Tcrsf . Combining this with Theorem 6.16 we get that there is an Lcrsfu function symbol g such that Tcrsfu proves '(g(~x ),~x). Hence in the universe f(~x )=e(g(~x )), which is a CRSFu function. ⇤

7. Partial Conservativity of Global Choice Recall from Section 6.3 the versions of our theories with global choice (GC).

C Proposition 7.1: T0 is not ⇧2(L0)-conservative over ZF.

C Proof: The theory T0 proves (AC) in the form: for every set x of disjoint, nonempty sets, there is a set z containing exactly one element from every member of x.Indeed,z := v tc(x): u xC(u)=v can be obtained C { 2 9 2 } by 0(L0 )-Separation. ⇤

4 C 4 In particular, the extension KPC1 (Lcrsf ) of KP1 (Lcrsf ) is not ⇧2(L0)- 4 C conservative. Informally, we ask how much stronger KPC1 (Lcrsf ) is com- 4 pared to KP1 (Lcrsf ). More formally, we aim to encapsulate the di↵erence in some local choice principles, namely a strong form of (AC) plus a form of dependent choice.e

7.1. Dependent Choice + The class of ordinals is denoted by Ord(x)inL0 with deﬁning axiom y x x (tc(y)=y). It is routine to verify in T+ some elementary proper- 8 2 [{ } 0 ties of ordinals, e.g., elements of ordinals are ordinals and, given two distinct ordinals, one is an element of the other. We let ↵,,... range over ordinals. By this we mean that ↵... and ↵...stand for ↵ (Ord(↵) ...) and 8 9 8 ! ↵ (Ord(↵) ...)respectively. 9 ^

eThis aims at technical simplicity of the argument rather than the strongest possible result. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 107

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The scheme 0(Lcrsf )-Dependent Choice gives for every 0(Lcrsf )- formula '(x, y, ~x) x y'(x, y, ~x) ↵ z (Fct(z) dom(z)=↵ 8 9 !8 9 ^ ↵'(z ,z’,~x)). (7.1) ^8 2 ⇠ + We assume L0 Lcrsf (cf. Lemma 4.5), Fct(y) is a unary relation symbol in + ✓ L0 expressing that y is a function, and dom(x), im(x), x⇠y are function sym- 4 bols in Lcrsf such that KP1 (Lcrsf )provesdom(x)=⇡1”(x), im(x)=⇡2”(x) and x y = z x : ⇡ (z) y . ⇠ { 2 1 2 } We further consider the following strong version of (AC) that we refer to as the well-ordering principle (WO): x ↵ y(“y is a bijection from ↵ onto x” , ↵ (y’ y’ )). 8 9 9 ^8 2 2 ! 2 The goal of this section is to prove:

4 C 4 Theorem 7.2: The theory KPC1 (Lcrsf ) is conservative over KP1 (Lcrsf ) plus 0(Lcrsf )-Dependent Choice plus (WO).

Note this just says that every Lcrsf -formula proved by the former theory is also proved by the latter. But the former theory is not an extension of the latter:

4 C Proposition 7.3: KPC1 (Lcrsf ) does not prove (WO).

4 C Proof: By Theorem 6.13, if KPC1 (Lcrsf ) proves the existence of an ordinal ↵ bijective to a given x,thenitprovesf(x) is such an ↵ for some function C symbol f(x)inLcrsf . Fix a universe of sets with a global choice function C, C and view it as a structure interpreting Lcrsf .There,f(x) denotes a function in CRSFC . By Theorem 2.5 (for CRSFC instead CRSF, recall Remark 2.10), there is a #-term t(x) such that ↵ = f(x) 4 t(x) holds for all x.Thenthere is a polynomial p such that the von Neumann rank rk(f(x)) = ↵ x is at | | most p(rk(x)) (cf. [7, Lemma 2, 4, Proposition 10]). This is false for many x. ⇤

7.2. The forcing

4 Let M be a countable model of KP1 (Lcrsf ) and 0(Lcrsf )-Dependent Choice and (WO). We intend to produce a generic extension of M modelling (GC). Note we do not assume that M is standard, in particular, M possibly does not interpret by . While the forcing frame is the class forcing commonly 2 2 used to force global choice, we use a technically simpliﬁed forcing relation March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 108

108 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

avoiding the use of names. This is similar to [17]. The argument that the 4 forcing preserves KP1 (Lcrsf ) needs some care since this theory and hence M is very weak. The forcing frame (P, P) is deﬁned as follows: P M contains p M 6 ✓ 2 if and only if p is a choice function in the sense of M, that is, M satisﬁes

(Fct(p) 0, 0 p x, y ( x, y p x =0 y x)). ^h i2 ^8 h i2 ^ 6 ! 2 Further, p P q means M = q p.Then(P, P) is a partial order. In 6 | ✓ 6 the following we let p, q, r, . . . range over conditions, i.e., elements of P.A subset X of P is dense below p if for all q 6P p there is r 6P q such that r X.Beingdense means being dense below 1P := 0, 0 (calculated 2 {h i} in M). A subset X of P is a filter if p M q X whenever p, q X, and [ 2 2 q X whenever p P q and p X.Beinggeneric means being a filter that 2 6 2 intersects all dense subsets of P that are definable (with parameters) in M. The forcing language is L R for a new binary relation symbol R. crsf [{ } The forcing relation relates conditions p to sentences of the forcing language with parameters from M. It is defined as follows. For an atomic sentence ' that does not mention R we let p ' if and only if M = '. For | an atomic sentence of the form Rts with closed terms t, s we let

p Rts M =(t = s t, s p). () | { }_h i2 We extend this deﬁnition via the recurrence:

p (' ) p ' and p , ^ () p ' for all q P p : q ', ¬ () 6 6 p x'(x) for all a M : p '(a). 8 () 2 This deﬁnes p ' for all sentences ' of the forcing language with parameters from M which are written using the logical symbols , , .We ^ ¬ 8 freely use the symbols , , understanding these as classical abbrevia- _ ! 9 tions. Namely, (' ), (' ), x(x) stand for ( ' ), (' ), _ ! 9 ¬ ¬ ^¬ ¬ ^¬ x ( (x)) respectively. Lemma 7.4 (f) below shows that p ' does not ¬8 ¬ depend of the choice of these abbreviations.

Lemma 7.4: Let ' be a sentence of the forcing language with parameters from M.

(a) (Conservativity) If R does not occur in ', then p ' if and only if M = '. | (b) (Extension) If p 6P q and q ', then p '. March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 109

Cobham Recursive Set Functions and Weak Set Theories 109

(c) (Stability) p ' if and only if p ',thatis,ifandonlyif q ¬¬ { | q ' is dense below p. } (d) (Truth Lemma) For every generic G there is R M 2 such that for G ✓ every ' we have that (M,R ) = ' if and only if some p G forces '. G | 2 (e) (Forcing Completeness) p ' if and only if (M,R ) = ' for every G | generic filter G containing p. (f) ' p ' is closed under logical consequence. { | } Proof: (a) and (b) obviously hold for atomic '; for general ' the claim follows by a straightforward induction (see, e.g., [3, Lemma 2.6]). Similarly, it suces to show (c) for atomic '. Assume ' = Rts for closed terms t, s. The second equivalence is trivial. The forward direction follows from (b): if p ',then q q ' q q P p is dense below p. Conversely, it { | }◆{ | 6 } is enough to find given p with p Rts some q P p forcing Rts.Wehave 6 6 ¬ M = t = s and M = t, s / p.IfM = t = 0, then M = s = 0 and no | 6 { } | h i 2 | | 6 condition forces Rts,soq := p Rts.IfM = t = 0, then there is a M ¬ | 6 2 such that M =(s = a a t) and q := p t, a calculated in M is a | 6 ^ 2 [{h i} condition. Then no r 6P q forces Rts. The remaining claims can be proved by standard means. We give precise references from [3]. A generic G is generic in the sense of [3, Definition 2.9], and M[G] is defined for every such G (cf. [3, Definition 2.16]). [3, Proposition 2.26] states that up to isomorphism each such model M[G] has the form (M,RG) as in (d). Then (d), (e), (f) are [3, Theorem 2.19, Corollary 2.20 (2), Corollary 2.20 (3)]. ⇤ It is easy to see that for each '(~x ) of the forcing language the set (p,~a): { p '(~a ) is definable in M. There is, however, no good control of the logical } complexity of the defining formula. Therefore we use the following auxiliary strong forcing relation between conditions and sentences of the forcing language with parameters from M. It is defined via the same recurrence as except for the negation clause. Namely, p ' is defined as p ' for ¬ ¬ atomic ' and otherwise via the recursion:

p ( ) p or p , ¬ ^ () ¬ ¬ p p , ¬¬ () p x (x) there is a M : p (a). ¬8 () 2 ¬ Remark 7.5: One can check that p x'(x) if and only if there is a M 9 2 such that p '(a), and p (' ) if and only if p ' or p . Here we _ understand , by the particular abbreviations mentioned earlier. In this 9 _ March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 110

110 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

sense commutes with quantiﬁers and connectives , .Thepricetopay ^ _ for these nice properties is that does not behave like a notion of forcing. For example, let a, b M,a = b, and calculate c := a, b in M;then 2 6 { } 1P Rca and 1P Rca,so1P (Rca Rca), and hence Lemma 7.4 (f) 6 6 ¬ 6 _¬ fails for . A formula is in negation normal form (NNF) if negations appear only in front of atomic subformulas.

Lemma 7.6: Let ' be a sentence of the forcing language with parameters from M.

(a) If p 6P q and q ', then p '. (b) If p ', then p '. (c) Let L L and (~x ) be a (L+ L R )-formula with param- ✓ crsf 0 0 [ [{ } eters from M. Then there exists a (L L)-formula ⇠(u, ~x) with 0 0 [ parameters from M such that ⇠(u, ~x) deﬁnes (p,~a):p (~a ) in M. { } (d) If ' is a ⌃ (L R )-sentence in NNF with parameters from M and 1 crsf [{ } p ', then there is q 6P p such that q '.

Proof: (a) and (b) are straightforward. (c) is proved by induction on . We only verify the case when (~x ) equals Rts for terms t = t(~x ),s= s(~x ). ¬ Then deﬁne ⇠(u, ~x) as

u P (t =0 s = 0) (t =0 s/t) x t (x = s t, x u) . 2 ^ ^ 6 _ 6 ^ 2 _9 2 6 ^h i2 Here, u P abbreviates a suitable 0(L0)-formula deﬁning P M inM. 2 ✓ We have to show that for all p P and ~a from M: 2 M = ⇠(p,~a) for all q P p : q Rt(~a )s(~a ). | () 6 6 The direction from left to right is easy to see. Conversely, assume no condition q P p forces Rt(~a )s(~a ) and note M = p P. Arguing in M,then 6 | 2 t(~a ) = s(~a ) and t(~a ),s(~a ) / p, in particular t(~a ),s(~a ) are not both 0. 6 { } h i 2 If t(~a ) = 0, then s(~a ) = 0 and ⇠(p,~a) is true. So suppose t(~a ) = 0. Then 6 6 q := p t(~a ),s(~a ) / P. Hence s(~a ) / t(~a ) or there is a t(~a )with [{h i} 2 2 2 a = s(~a ) and t(~a ),a p. Both cases imply ⇠(p,~a). 6 h i2 (d). Let '(~x ) be a formula of the forcing language with parameters from M. Call '(~x ) good if for all ~a from M and p P:ifp '(~a ), then there is 2 q 6P p with q '(~a ). Atomic and negated atomic formulas are good, as we can take q := p. Good formulas are closed under conjunctions and disjunctions, and

y (y,~x) is good whenever (y,~x) is good: if p y (y,~a), then b M q 9 9 2 { | S March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 111

Cobham Recursive Set Functions and Weak Set Theories 111

q (b,~a) is dense below p, so there are q P p and b M such that } 6 2 q (b,~a); as (y,~x) is good, there is r 6P q such that r (b,~a) and hence r y (y,~a). 9 Finally, we show that for a good (L R )-formula (y,~x), also 0 crsf [{ } y t(~x ) (y,~x) is good, where t is a term. If p y t(~a ) (y,~a), then 8 2 8 2 by Conservativity p (b,~a) for all b with M = b t(~a ). As (y,~x)is | 2 good, we find for every q 6P p and every such b some qb 6P q such that q (b,~a). By (WO) we find s M which is, in the sense of M, a bijection b 2 from an ordinal ↵ onto t(~a ). It suces to find ⇡ M such that ⇡ is, in the 2 sense of M, a function with domain ↵ and such that for all M M ↵: 2 2 ⇡’ 6P ⇡’ 6P p and ⇡’ (s’,~a). (7.2) More precisely, the first three ’ should read ’M . This suces indeed: by (a), then q := im(⇡), calculated in M, is a condition extending p such that q (s’,~a) for all M ↵.Thusq (b,~a) for all b with M = b t(~a ), S 2 | 2 and hence q y t(~a ) (y,~a). 8 2 To find such ⇡ we apply 0(Lcrsf )-Dependent Choice in M with the following 0(Lcrsf )-formula '(x, y) with parameters from M:

dom(x) ↵ p im(x) P 2 ^ [ 2 ⇠ p im(x) Sy y P (y, s’dom(x),~a) . ! [ ✓ ^ 2 ^ where ⇠ is as in (c).S Since is (L R ), we have ⇠ (by (c)) and 0 crsf [{ } hence ' in (L ). We show that M models x y'(x, y). Argue in M: 0 crsf 8 9 given c M with := dom(c) ↵ and q := p im(c) P a witness for 2 2 [ 2 y is given by q for b := s’. b S For ↵ as above, choose ⇡ witnessing z in (7.1). We claim ⇡ satisﬁes (7.2). It suces to show M = ↵ (p im(⇡⇠) P), or equivalently, M = | 8 2 [ M 2 | ,0 (p ⇡’ ⇡’0 P) for all ↵. This follows by 0(L0)- 8 2 [ [ 2 S 2 Induction on and elementary properties of ordinals. ⇤

7.3. Proof of Theorem 7.2

4 It suces to show that every countable model M of KP1 (Lcrsf )plus 0(Lcrsf )-Dependent Choice plus (WO) has an expansion to a model 4 C 4 C 4 C 4 of KPC1 (Lcrsf ). Recall, KPC1 (Lcrsf )isa⌃1 (L0 )-expansion of KPC1 (cf. Lemma 5.21). It thus suces to ﬁnd an expansion of M to a model 4 of KPC1 . Using the notation of the Truth Lemma 7.4(d), for every generic G we have that RG is the graph of a function C satisfying the March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 112

112 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

C axiom of global choice (GC). A 0(L0 )-formula in the corresponding expansion is equivalent to a (L R )-formula. It thus suces to show 0 0 [{ } that (M,R ) satisﬁes ⌃4(L R )-Induction, (L R )-Separation G 1 0 [{ } 0 0 [{ } and (L R )-Collection. 0 0 [{ } We start with Induction. So, given a (L R )-formula (y, z)with 0 0 [{ } parameters from M,a#-termt(y) with parameters from M and b M we 2 have to show that

(M,R ) = x y x z t(y) (y, z) z t(x) (x, z) G | 8 8 2 9 4 !9 4 z t(b) (b, z). !9 4 Recall IsPair(x) from Examples 4.6. We deﬁne

0(y, z):=(IsPair(z) ⇡ (z):⇡ (z) t(y) (y, ⇡ (z))). ^ 1 2 4 ^ 2 + + We can assume that 0 is in NNF. Recall L L ,soM interprets L . 0 ✓ crsf 0 We assume

(M,R ) = x ( y x z 0(y, z) z 0(x, z)) (7.3) G | 8 8 2 9 !9

and aim to show (M,R ) = z 0(b, z). By the Truth Lemma there exists G | 9 p G such that p forces (7.3). It suces to show that p forces z 0(b, z). By 2 9 Stability it suces to find, given p0 P p, some q P p0 forcing z 0(b, z). 6 6 9 By (WO) we find s M such that, in the sense of M, s is a bijection 2 from some ordinal ↵ onto tc+(b) that respects ,i.e.,M =(s’ s’ 2 | 2 2 tc+(b) ). So by Lemma 7.6(b), it suces to find for every M ↵ ! 2 2 a pair q ,a (in the sense of M) such that q P p0 and h i 6

q 0(s’,a). (7.4)

We intend to apply 0(Lcrsf )-Dependent Choice with the following formula '(x, y):

'(x, y):=(' (x) ' (x, y)), 0 ! 1 ' (x):=Fct(x) dom(x) ↵ 0 ^ 2 ,0 dom(x)(p0 ⇡1(x’) ⇡1(x’0) P) ^8 2 [ [ 2 ⇠ dom(x) IsPair(x’) 0(⇡ (x’),s’,⇡ (x’)) , ^8 2 ^ 1 2 ' (x, y):=IsPair(y) dom(x)(p0 ⇡ (x’) ⇡ (y)) 1 ^8 2 [ 1 ✓ 1 ⇠ 0(⇡ (y),s’dom(x),⇡ (y)), ^ 1 2 ⇠ where 0 is deﬁned as in Lemma 7.6(c). March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 113

Cobham Recursive Set Functions and Weak Set Theories 113

We have ' (L ) by Lemma 7.6(c). We show M = x y'(x, y). 2 0 crsf | 8 9 Let c M and assume M = ' (c). We have to show M = y' (c, y). 2 | 0 | 9 1 Compute

:= dom(c)

q := p0 ⇡1(c’) [ 2 S 4 in M (this can be done: for f(x) such that KP1 (Lcrsf )provesf(x)= M ⇡1(⇡2(x)), we have q = p0 f”(c)inM). Then q P extends (⇡1(c’)) M [ 2 M for all . By Lemma 7.6(a), q 0(s’,⇡2(c’)) for all , 2 S M 2 and hence q z 0(s’,z) for all .Thisimpliesq z 0(d, z) 9 2 9 for all d M with M = d s’. By Lemma 7.6 (b), we see that q 2 | 2 forces y s’ z 0(y, z). But q P p0 P p, so by Extension q forces (7.3). 8 2 9 6 6 Plugging s’ for x in (7.3) and recalling Lemma 7.4 (f) we see that q z 0(s’,z). This is a ⌃ (L R )-sentence in NNF with parameters 9 1 crsf [{ } from M, so Lemma 7.6 (d) gives q P q and a M such that q 6 2 0(s’,a ). Then M = ' (c, q ,a ) and thus M = y' (c, y). | 1 h i | 9 1 By Dependent Choice there is ⇡ M, in the sense of M a function with 2 domain ↵, such that M = '(⇡ ,⇡’) for all M ↵. | ⇠ 2 To show (7.4) it suces to show M = ' (⇡ ) for all M ↵, or | 0 ⇠ 2 equivalently

,0 (p0 ⇡1(⇡’) ⇡1(⇡’0) P) 8 2 [ [ 2 ⇠ (IsPair(⇡’) 0(⇡ (⇡’),s’,⇡ (⇡’))) ^82 ^ 1 2 holds in M for all M ↵. By Lemma 7.6(c), this can be written () for + 2 + a0(L0 )-formula (x) with parameters from M.Since0(L0 )-Induction holds in M,itsucestoverifyM = () assuming M = (). This | | 8 2 is easy. Thus (M,R ) satisfies (L R )-Induction. G 0 0 [{ } We show that (M,R ) satisfies (L R )-Collection. Let (y, z)be G 0 0 [{ } a(L R )-formula with parameters from M and a M such that 0 0 [{ } 2 (M,RG) satisfies

' := y a z (y, z). 8 2 9 By the Truth Lemma, ' is forced by some p G. Arguing as for (7.4) 2 we can ﬁnd q P p such that for all b M a there is c M such that 6 2 2 q (b, c) (observe that the proof of (7.4) gave a descending chain of q’s) – equivalently: q '. Note q ' if and only if M = y a z ⇠(q, y, z), | 8 2 9 March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 114

114 A. Beckmann, S. Buss, S.-D. Friedman, M. M¨uller & N. Thapen

where ⇠ is (L ), chosen according Lemma 7.6(c). Then q : q ' is 0 0 { } M-deﬁnable and dense below p,soweﬁndsuchq in G. Applying 0(L0)- Collection in M we get

M = V y a z V ⇠(q, y, z). | 9 8 2 9 2 But for all b, c M we have 2 (M,R ) =( ⇠(q, b, c) (b, c)), G | ! by Lemma 7.6(b) and the Truth Lemma. Thus (M,R ) satisfies (L G 0 0 [ R )-Collection. { } We show that (M,R ) satisfies (L R )-Separation. Let a M G 0 0 [{ } 2 and '(x)bea(L R )-formula with parameters from M. We can 0 0 [{ } assume '(x) is in NNF. Let '(x) be logically equivalent to '(x) and in ¬ NNF. By the Truth Lemma it suces to show 1P z (z = x a : '(x) ). 9 { 2 } By Stability it suces to show that for every p P there is q P p such 2 6 that q z (z = x a : '(x) ). 9 { 2 } Let p P be given. We claim that it suces to find q P p that strongly 2 6 decides '(b) for every b M a in the sense that q '(b) or q '(b). 2 Indeed, such a q forces z (z = x a : '(x) ). By Forcing Completeness 9 { 2 } we have to show (M,R ) = z (z = x a : '(x) ) for every generic G0 G0 | 9 { 2 } containing q.Butz is witnessed by x a : '⇠(q, x) , a set obtainable in { 2 } M by 0(L0)-Separation (Lemma 7.6(c)). To see this, we verify for every b M a: 2

(M,R ) =('(b) '⇠(q, b)). G0 | $ The direction from right to left follows from Lemma 7.6(b) and the Truth Lemma. Conversely, assuming (M,R ) = '(b) the Truth Lemma gives G0 | r G0 forcing '(b); then r q G0 since G0 is a ﬁlter, so r q forces '(b)by 2 [ 2 [ Extension, so cannot force '(b) by Lemma 7.4(f), so q '(b) by Extension, 6 so q '(b) by Lemma 7.6(b), so q '(b) and (M,R ) = '⇠(q, b)sinceq 6 G0 | strongly decides '(b). Thus, given a condition p, we are looking for q 6P p that strongly decides '(b) for every b M a. By (WO) choose s M such that, in the 2 2 sense of M, s is a bijection from ↵ onto a. A condition q as desired is obtained in M as the union of a descending sequence (q) ↵ with q0 6P p 2 such that each q strongly decides '(q,s’). To get such a sequence in M March 16, 2017 Lecture Note Series, IMS, NUS — Review Vol. 9in x 6in 10532-05 page 115

Cobham Recursive Set Functions and Weak Set Theories 115

we apply 0(Lcrsf )-Dependent Choice on the following formula (x, y):

(Fct(x) dom(x) ↵ ,0 dom(x) x’ x’0 P ^ 2 ^8 2 [ 2 dom(x)(p x’ y) ('⇠(y, s’dom(x)) '(y, s’dom(x)))). !8 2 [ ✓ ^ _ (7.5) e A function ⇡ with domain ↵ (in the sense of M) such that M = (⇡ ,⇡’) | ⇠ for all M ↵, is a sequence as desired. We are left to show 2 M = x y (x, y). | 8 9 Let c M satisfy the antecedent of (7.5), and compute := dom(c) 0 2 0 1 0 and q := im(c)inM.Thenq P.Thereexistsq 6P q such that 1 1 2 q '(s’) or q '(s’). Indeed, by Stability, if q0 '(s’), then there 1 0S 1 6 1 is q 6P q such that r '(s’) for all r 6P q ,i.e.,q '(s’) and 1 6 2¬ 1 hence q '(s’) by Lemma 7.4(f). Lemma 7.6(d) gives q 6P q such that q2 '(s’) or q2 '(s’)respectively.ThenM = (c, q2). |

Acknowledgements We thank the anonymous referee for a careful reading and detailed comments. Part of this work was done when the ﬁrst four authors attended the workshop Sets and Computations at the Institute for Mathematical Sciences, National University of Singapore.

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