Note that second-order bounded quantiers range over sets whose elements are bounded by t, so by the absence of exponentiation, their nature is radical

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1 Notes on polynomially bounded arithmetic Domenico Zambella March 8, 1994 Abstract We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries The polynomially bounded hierarchy. : : : : : : : : : : : : : : : : : : : : : : : : : : The axioms of second-order bounded arithmetic. : : : : : : : : : : : : : : : : : : : Rudimentary functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Other fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Polynomial time computable functions. : : : : : : : : : : : : : : : : : : : : : : : : : Relations among fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Relations with Buss' bounded arithmetic. : : : : : : : : : : : : : : : : : : : : : : : 11 2 Witnessing theorems and conservativity results Closures : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : A model-theoretical version of Buss' witnessing theorem. : : : : : : : : : : : : : : : A model theoretical characterization of choice. : : : : : : : : : : : : : : : : : : : : Ultrapowers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 3 The collapse of BA versus the collapse of PH An interpolation theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Sucient conditions for the collapse of BA : : : : : : : : : : : : : : : : : : : : : : : Necessary conditions for the collapse of BA : : : : : : : : : : : : : : : : : : : : : : Krajcek, Pudlak and Takeuti's method : : : : : : : : : : : : : : : : : : : : : : : : 28 0 Introduction and motivation. In every model of I 0 numbers code nite sets. Sets coded by numbers are 0 -denable. In general, the converse is not true. Weak theories, which do not prove the totality of exponentiation, do not prove the existence of a code for every nite 0 -denable set. So, a natural way of strengthening I 0 is by adding to the language second-order variables X, Y, Z, etc. ranging over nite sets of numbers and introducing axioms of nite comprehension ensuring the existence of sets of the form fx <a : '(x)g for '(x) ranging over some class of second-order formulas. Interesting theories arise when we restrict the schema of nite comprehension to bounded formulas. These are formulas where all quantiers are of the form Qx <t or QX <t where t is a term (a polynomial). 1

2 Note that second-order bounded quantiers range over sets whose elements are bounded by t, so by the absence of exponentiation, their nature is radically dierent fram that of rst-order quantiers. We introduce the classes p i and p i counting alternations of (polynomially) bounded second-order quantiers. Restricting the strength of the schema of nite comprehension to formulas of a certain complexity one obtains the hierarchy of theories that we call p i -comp. The union of all these theories (i.e., nite comprehension for all bounded second-order formulas) is called second-order bounded arithmetic BA. We study the relative strength of various fragments of BA and in particular their provably total functions. Interestingly, all provably recursive functions of BA are of polynomial growth. Some theorems of partial conservativity are proved for some of these theories and the connection with complexity theory is briey discussed. In the last decades two subsystems of arithmetic, I 0 and S 2, have been studied especially for their connections with complexity theory (see e.g., [16] and [3] or [7]). In particular, Buss' S 2 is the most extensively studied. The theory S 2 coincides with (an extension by denition of) the also well-known I These theories are rst-order strengthening of I 0. In the case of I or S 2 the urge for the strengthenings is somehow technical; it arises from metamathematical and/or syntactical considerations. In fact, in order to have a reasonable formalization of computation and/or syntax one needs to be able to perform operations on strings such as the substitution of substrings. Such operations increases the code of the string superpolynomially and so, it is not provably total in I 0. Adding to I 0 an axiom (i.c., 1 ) asserting the totality of this function one obtains a stronger theory in which it is possible to formalize almost all basic notions of metamathematics. Buss introduced a hierarchy of theories S i 2 whose union is S 2. These theories are obtained by some weakening of the axiom of induction (while introducing suciently many new primitives to allow smooth bootstrapping). It is not surprising that BA coincides with Buss' S 2, modulo an appropriate translation. Namely, to each (rst-order) model M 0 of S 2 corresponds a (second-order) model M 00 of BA. The rst-order objects of M 00 are the logarithmic numbers of M 0 (i.e., numbers belonging to the domain of exponentiation). The sharp function guarantees that these numbers are closed under multiplication. As second-order objects we take those nite sets which have a code in M 0. In this way, p i -formulas get transformed into b i -formulas of Buss' language (see e.g., [3] or chapter 5 of [7]) in a very natural way, so, the constructed second-order model veries nite comprehension for all bounded formulas. Vice versa, from a model M 00 of BA one obtains a (rst-order) model M 0 of S 2 by the inverse procedure. As domain of M 0 we take the second-order objects of M 00. In M 00 we dene the primitives of S 2 as set operations. Intuitively, we think of a nite set X as the numbers P x 2X 2x and dene operations lead by this idea. We shall see that BA disposes over enough second-order recursion to formalize these operations and to prove the axioms of S 2 to hold in M 0. Note, parenthetically, that the cartesian product of two sets is mapped to a rst-order function with the growth rate of the sharp function. This procedure actually maps models of p i -comp into models of S i 2 and vice versa (for all i >0). A few details on this isomorphism (which was discovered in dierent ways by many authors) are contained in Section 1.7. The readers who are mainly interested in S i 2 are advised to read that section rst. In fact, afterwards they will be able to translate most of the results reported here into theorems about fragments of S 2. In particular, Theorem 2.2 is a strengthening of the main theorem of [3]. Our proof is model-theoretic and, to our knowledge, diers from those existing in the literature (but it is formally identical to an unpublished model-theoretic argument for the conservativity of I 1 over PRA by Albert Visser. In fact, formal similarities between I 1 and p i -comp are apparent when primitive recursive functions are replaced by polynomial time computable functions). Other conservativity results are obtainable with the same method. The author's personal motivation for using a second-order framework is that this approach allows economy of primitives, natural denitions and (again in the author's 2

3 opinion) a clear heuristic. In the hierarchy of fragments of BA very few inclusions are known to be strict. In general the problem of proving inclusions to be strict seems to be a very dicult one. A more realistic goal is to characterize the collapse of theories in terms of the provable collapse of some complexity classes. A corollary of Theorem 2.2 is that, if P-def (i.e., the 8 p fragment of 1 p( S )) proves p = 2 p, 2 then all of BA collapses to P-def. So, a very satisfactory result would be to prove the converse. One of the best known results in this direction is the celebrated KPT theorem (see Theorem 3.4): in [11] Krajcek, Pudlak and Takeuti, proved that if P-def proves p 1-comp, then in the standard model the polynomial time hierarchy collapses to the second level. Unfortunately, it is still unclear whether their proof is formalizable in BA, so, their result cannot be used to answer questions like: if P-def proves p 1-comp does BA collapses The main achievement of this paper is the following theorem. It gives a satisfactory characterization of the collapse of BA in terms of the provable collapse of PH. (On the right hand side we include the translation into Buss' language. For uniformity, we set T2 0 := PV 1. For the denition of BB b i+1 see [7].) Theorem. The following are equivalent (i) P i -def ` p i+1 -comp (ii) P i -def ` p i+1 p i+1 =poly T i 2 ` Si+1 2 T i 2 ` b i+1 b i+1 =poly (iii) P i -def ` BA T i 2 ` S 2 (iv) P i -def + p i+1 -choice ` p i+1 -comp T i 2 + BB b i+1 ` Si+1 2 The implication from (i) to (ii) is Theorem 3.3. The implication from (i) to (iii) can be reconstructed from the proof of Theorem 3.2. (To read these two proofs the reader need only to rush through Section 1.) From Theorem 3.2 it actually follows that (ii) implies (iii) while in Corollary 2.3 is proved that (iv) implies (i). Acknowledgments. The numerous discussions with Rineke Verbrugge, Harry Buhrman and Volodya Shavrukov have been pleasant and stimulating. When the rst draft of this manuscript was ready I had interesting discussions with Sam Buss. I owe him interesting observations and various corrections. Buss independently proved that condition (i) above implies (ii) and (iii)). He observed that from (i) follows that PH (provably) collapses to Boole( p i+2 ). His result inspired our interpolation theorem 3.1. This work is part of my Ph.D. thesis at the University of Amsterdam. The supervision of Dick de Jongh and Albert Visser has assisted me through the numerous stadia of preparation of this work. 1 Preliminaries. Here we introduce the necessary denitions. Lemma 1.3 provides a smooth bootstrapping. The class of polynomial time computable functions is concisely introduced in section 1.5 in a 3

4 machine independent way. The (standard) comparison of strength of the various fragments are sketched in Section 1.6. In Section 1.7 the relation with Buss' S i 2 is sketched. 1.1 The polynomially bounded hierarchy. We dene the analogue of the analytical hierarchy for nite sets. The language L 2 is the language of second-order arithmetic; it consists of two symbols for constants: 0, 1, two symbols for binary functions: +, and three symbols for binary relations: <, 2,. Moreover, there are two sorts of variables: rst and second-order. Lower case Latin letters x ; y; z ; :: denote rstorder variables and capital Latin letters X ; Y ; Z ; :: second-order variables. First and second-order variables are meant to range respectively over numbers and nite sets of numbers. Terms are constructed from rst-order variables only. The formula x <y is to be read \x is less than y" while X Y stands for \X is a subset of Y ". The intended meaning of X <y is: \all elements of X are less than y". Let t be a term of L 2 in which x does not occur. We adopt the following abbreviations with the usual meaning (Qx <t)', (Qx 2Y )', (QX <t)', (QX Y )', where Q is either 8 or 9. Quantiers occurring in either of these contexts are called (polynomially) bounded quantiers. The class of bounded formulas is denoted by PH. Note that rst-order quantiers range over elements of sets while second-order quantiers range over subsets of sets. Here, rst-order bounded quantiers play the role that sharply bounded quantiers have in rstorder bounded arithmetic (see e.g., [3] or chapter 5 of [7]). A formula is (polynomially) bounded if all of its quantiers are. Counting alternations of second-order quantiers we classify bounded formulas in the (polynomially) bounded hierarchy. We use one of the symbols p or 0 p 0 for formulas containing only bounded rst-order quantiers. We dene inductively p as the minimal class of formulas containing i+1 p i, closed under disjunction, conjunction and bounded existential quantication. The class p i+1 is the minimal class of formulas containing p i, closed under disjunction, conjunction and bounded universal quantication. So, PH equals S i2! p i and S i2! p i. The class p 0( p i ) is the smallest set of formulas containing p i, closed under Boolean operations and bounded rst-order quantication. Sometimes we add to the language L 2 some set F of new symbols for functions. We dene the (relativized) classes of bounded L 2 (F)-formulas: p i (F), p i (F), etc. similarly to those of the language L 2. (We allow terms of L 2 (F) to occur in the bounds of the quantiers.) The domain of an L 2 structure M is composed of two disjoint parts: the numbers and the sets of M. Truth in M is dened as usual but rst-order variables are restricted to range over numbers while second order variables range over sets. To denote elements of a model, we use the same convention as for variables so, we write A2M for `A is a set of M' and a2m for `a is a number of M'. For models we use bold face letters, for the class of rst-order objects of a model M we use the corresponding lower case letter m. The disjoint union of! and P <! (!) constitutes the standard model, functions and relations are interpreted in the natural way. We loosely denote the standard model by!. Computational complexity theory and second-order arithmetic are our main sources of inspiration, concrete intuition and terminology. For our digressions to computational complexity theory it is convenient to think of nite sets as strings i.e., we identify P <! (!) and 2 <!. So, sets of nite 4

5 sets may be identied with languages. The actual form of the isomorphism is immaterial. We stipulate that the length of the string associated to a nite set X! equals (up to some additive constant) the least upper bound of the set X which we henceforth denote by jx j. To begin with, the reader may wish to check that p -formulas dene languages in NP, i.e., if '(X 1 )2p 1 then the language fx :! j= '(X )g is in NP. Vice versa for every language L2 <! in NP there is a formula '(X ) in p such that L is fx : 1! j= '(X )g. In the same way, p 1-formulas coincide with conp languages and, in general, each level of the bounded hierarchy coincides with one of the Meyer-Stockmeyer polynomial time hierarchy (with the only exception of ground level i = 0 which corresponds to uniform-ac 0 languages). When digressing to computational complexity theory, we identify each number x 2! with the set of its predecessors and so, with a string of ones of length x. Therefore, a formula '(x) with one free rst-order variable denes a tally language i.e., a language which is contained in f1g <!. 1.2 The axioms of second-order bounded arithmetic. The theory is axiomatized by the following formulas: a + 1 = b + 1! a = b a<b $ 9x : (a + x) + 1 = b a + 0 = a A<b $ (8x 2A): x <b a + (b + 1) = (a + b) + 1 AB $ (8x 2A)x 2B a 0 = 0 A = B $ : AB ^ BA a (b + 1) = (a b) + a A 6= ;! : (9x 2A)(8y<x) y =2 A a 6= 0 $ 9x : x + 1 = a 9x : A<x : The six axioms on the left are those of Robinson arithmetic. To these we add the dening axioms for the relations < and, the axiom of extensionality, the least number principle and the axioms of niteness (i.e. all sets have an upper bound). The theory p i -comp is axiomatized by and the schema of (nite) comprehension for p i -formulas i.e., for all ' in p i in which X does not occur free, p i -comp : (9X <a)(8x <a): x 2X $ '(x). The theory of second-order bounded arithmetic, BA, is the union of p i -comp for i2!. The theories p i -comp and p 0 (p i )-comp are dened in a similar way and are easily seen to be equivalent to p i -comp. 1.3 Rudimentary functions. In order to keep formulas to a readable size we need to introduce new function symbols. To begin with, let us give some informal denitions. We write jaj for the least upper bound of A and j~a; Aj ~ for the least upper bound of fa 1 ; ::; a n, ja 1 j; ::; ja m jg. It should be clear that p -comp 0 suces to prove the existence of j~a; Aj. ~ We call rudimentary those functions which are obtained by p comprehension or by 0 p 0 minimalization, i.e. those functions denable in either one of the two subsequent ways: F ';p (~a; ~ A) := fx <j~a; ~ Aj p : '(x ; ~a; ~ A)g, f ';p (~a; ~ A) := x <j~a; ~ Aj p '(x ; ~a; ~ A), for some '2 p 0 and p2! (in the denition of F ';p and f ';p, we have stressed that these functions 5

6 are polynomially bounded). Let R be a set of new primitives, one for each (denition of a) rudimentary function. Let R-def be the theory axiomatized by plus the (obvious) dening axioms for the functions in R. Clearly, p 0-comp suces to prove every rudimentary function to be total. So, R-def is a conservative expansion of p 0 -comp. The following lemma ensures us that there is no danger in considering formulas of the expanded language L 2 (R) as abbreviations of L 2 -formulas. In fact, the `translation' does not increase the complexity of the formula. Namely, the following lemma shows that p = 0 p 0 (R) provably in R-def. Lemma 1.3. For every 2 p 0(R), there is 2 p 0 such that R-def ` $. Proof. The lemma is proved by a method which we believe to be well-known to the reader, so, we do not need report it in full detail. One has to unfold the denitions of the rudimentary functions inside the p 0(R)-formulas. We can assume that has only one occurrence of a single rudimentary function F ';p (~a; A) ~ (we also assume this function is a set function; the case of a number function is similar). First, one must rewrite to have all occurrences of rudimentary set functions on the right of the symbol 2. This is done by eliminating the relations < and as is suggested by the axioms of. Then replace each subformula of the form x 2F ';p (~a; A) ~ with x <j~a; ~ Aj p ^ '(x ; ~a; ~ A). Finally, replace subformulas of the form x <j~a; Aj ~ p with an equivalent p 0-formula. The dening axioms of F ';p ensure that the formula obtained is equivalent to the original. In the resulting formula no rudimentary set functions occur. A noteworthy corollary of this lemma is that rudimentary functions are closed under composition. From the Lemma it follows also that p i (R)-comp + R-def is equivalent to p i -comp + R-def and hence a conservative extension of p i -comp. Below, we list a few rudimentary functions that we often use. ha; bi := z : 2z = (a + b)(a + b + 1), the pairing function, A B := fhx ; yi : x 2A ^ y2bg, the cartesian product, A [b] := fy : hb; yi2ag, the b-th row of the `matrix' A, A(b) := z : z 2A [b], the value of the `function' A at b, [x] := fy : y<x g, the set of predecessors of x, fx g := fy : x = yg, the singleton of x. 1.4 Other fragments. In this section we present some other interesting axiomatizations of BA. In the next sections we study the relative strength of their fragments. We agree that all theories we introduce in this section contain, by denition, p -comp. The theories 0 p i -ind and p i -dc (i.e., of induction, 6

7 dependent choice and strong collection for p i -formulas) are axiomatized by the following schemas, for '2 p i. p i -ind : '(0) ^ 8x['(x)! '(x + 1)]! '(a), p i -dc : 8x(8X <b)(9y <b)'(x ; X ; Y )! 9Z (8x <a)'(x ; Z [x ] ; Z [x +1] ) p i -coll : 9Z (8x <a) (9Y <b)'(x ; Y )! '(x ; Z [x ] ) (in the last two schemas Z should not occur free in '). The schema of dependent choices is inspired by second-order arithmetic. We show that dependent choice, induction and strong collection are all equivalent to comprehension. A rather intriguing role is played by the following schema of choice p i -choice : 8x(9X <b)'(x ; X )! 9Z (8x <a)'(x ; Z [x ] ). where ' is in p i. It asserts that the p i -formulas are closed under rst-order bounded quantications. 1.5 Polynomial time computable functions. In this section we introduce the classes of functions P i. These correspond to classes which have been intensively studied in computational complexity theory. For expository reasons we prefer to introduce them in an axiomatic way avoiding direct reference to any model of computation. Formally, our approach is self-contained. A denition of rudimentary function has already been given in Section 1.3. We include here a dierent one. It is easy to see that these two denitions dene the same class of functions. The reader may consult [6] for some details. To begin with, let us work in the standard model, i.e. natural numbers and nite sets of natural numbers. The functions we introduce are of two sorts, number functions and set functions, denoted respectively with lower case and capital letters. Functions take as inputs tuples of numbers and sets and they output either a number (number functions) or a set (set functions). Numbers, as input and/or output are introduced merely as a useful device to express `logarithmically many iterations'. The class R of rudimentary functions is the smallest set of functions closed under composition and under the following schemas 1 (a 0-ary function), F(a) = [a] f (a 1 ; ::; a n ; A 1 ; ::; A m ) = a i + a j for 0<i ; j n and 0 m, f (a 1 ; ::; a n ; A 1 ; ::; A m ) = a i a j for 0<i ; j n and 0 m, F(a 1 ; ::; a n ; A 1 ; ::; A m ) = A i [ A j for 0 n and 0<i ; j m, F(a 1 ; ::; a n ; A 1 ; ::; A m ) = A i n A j for 0 n and 0<i ; j m, f (A) = x (x 2A) (for A = ; this is dened to be 0) F(~a; Y ; ~ A) = S y2y G(~a; y; ~ A) for G in R. 7

8 The functions dened by the rst seven schemas are called basic rudimentary. We shall refer to the last schema as rudimentary collection. The class P is closed also under the following schema of second-order (polynomially bounded) recursion F(0; ~x ; ~ X ) = G(~x ; ~ X ); F(y + 1; ~x ; ~ X ) = [jy; ~x ; ~ X j p ] \ H (y; ~x ; ~ X ; F(y; ~x ; ~ X )) for any G, H in P and p2!. The recursion schema introduced above is polynomially bounded for two reasons. We bound both the size of the output and the depth of the recursion. So, no more than polynomially many nested iterations of functions are possible. The class P is also denoted P 0. In general, the classes P i are obtained by adding to P Turing oracles for p i -formulas and closing under rudimentary collection and second-order recursion. Turing oracles for p i -formulas are functions of the form ( f0g if '(~a; A) ~ F(~a; ~ A) = for ' in p i. ; otherwise, Now, going back to theories of second-order arithmetic, let us use R and P i to indicate also three sets of symbols for functions, a dierent symbol for each denition of a function in the corresponding class. Let L 2 (R) and L 2 (P i ) be the corresponding expansion of L 2. Let P i -def be the theories axiomatized by and the dening axioms of the functions in P i. We choose the obvious dening axioms of functions obtained by the basic schemas. Namely, for Turing oracles the dening axioms are those given above. If F is (the symbol of the function) obtained by rudimentary collection from G2R we take as dening axiom of F x 2F(~a; Y ; ~ A) $ (9y2Y ): x 2G(~a; y; ~ A). If F is obtained by second-order recursion from G; H 2P and p2!, then the dening axiom is F(0; ~a; ~ A) = G(~a; ~ A) ^ F(y + 1; ~a; ~ A) = H (y; ~a; ~ A; F(y; ~a; ~ A)) \ [jy; ~a ; ~ A)j p ]. 1.6 Relations among fragments. We assume the reader to be familiar with fragments of rst-order arithmetic (see e.g., [7]), so, we merely sketch proofs. It is easy to see that the comprehension schemas for p i, p i and p 0 (p i )- formulas are equivalent. Also, we may contract quantiers, so, p -dc and i+1 p i+1-choice are respectively equivalent to p i -dc and p i -choice (these last two theories are dened in the obvious way). The theory p -choice proves that i+1 p i+1-formulas are closed under rst-order bounded quantication. In the schemas of p i -choice, p i -dc and p i -coll we can in addition require the set Z to be a subset of [a + 1][b] without strengthening the schema. The easy proofs of these facts are left to the reader. Lemma 1.6. For all i2!, (i) p i+1 -ind =) p i+1 -choice =) p i -comp 8

9 (ii) p i+1 -comp () p i+1 -ind () p i+1 -dc () p i+1 -coll (iii) p i+1 -comp =) P i -def =) p i -comp We understand the rst inclusion of (iii) as: every model of p i+1-comp has a unique expansion to a model of P i -def Proof of (i). For the rst inclusion, it is sucient to prove p i -choice. This is proved in a straightforward manner. By the observation above, the quantier 9Z in the schema of choice can be bounded. So, assuming the antecedent of the implication one can prove the consequent by induction on the parameter a. The second implication is proved by induction on i. Assume that p -choice proves i+1 p i -comp (this is true by denition if i = 0, for i >0 it follows from our induction hypothesis), we show that p -choice proves i+2 p -comp. Reason in a model of i+1 p i+2-choice. Let '(x)2 p. For some b and some i+1 2p i the formula ' is equivalent to (9X <b) (x ; X ). We have (*) (8x <a): (9X <b)(8y <b)[ (x ; X ) _ : (x ; Y )]. We may apply the axiom of choice to get a set Z [a][b] such that So, (8Y <b)[ (x ; Z [x ] ) _ : (x ; Y )]. (x ; Z [x ] ) $ '(x). Hence, p i -comp suces to prove the existence of the set fx <a : '(x)g. Proof of (ii). Since all three theories above prove p i+1-choice, in the following proof we use without explicit mention that p i+1-formulas are closed under bounded rst-order quantication. It is immediate that p -comp contains i+1 p -ind. i+1 For the converse inclusion, reason in a model of p -ind; let i+1 '2p i+1 and choose a parameter a. We want a set X <a such that x 2X $ '(x) for all x <a. We are done if we can nd a set of maximal cardinality among those such that x 2X! '(x) for all x <a. In fact, for such an X, also the converse implication holds. Formally, we write Y : [c],! X for the p 0-formula saying that Y is an injection of [c] into X or, in other words, that the cardinality of X is at least c. By p 1-ind, there exists a largest c<a such that (9X <a)(9y <hc; ai): (Y : [c],! X ) ^ (8x 2X )'(x). The X, witnessing the existential quantier for c maximal, is the required set satisfying x 2X $ '(x) for all x <a. This completes the proof of the rst equivalence. To prove that p -ind implies i+1 p -dc it is convenient to derive i+1 p i -dc. This is done by straightforward induction as for the schema of choice in previous lemma. The converse implication is proved by induction on i. Reason in a model of p -dc. We show that for every i+1 2p, i+1 (*) (0) ^ (8x <a)[ (x)! (x + 1)]! : (a). Without loss of generality, we may assume that (x) is equivalent to (9X <b) '(x ; X ) for some '(x) in p i and some parameter b. Assume the antecedent of (*), then (8x <a)(8x <b)(9y <b)['(x ; X )! '(x + 1; Y )]. The formula between square brackets is equivalent to a p i+1-formula, so, (after few manipulations) 9

10 one can apply p i+1-dc to get a set Z [a + 1][b] such that Z [0] = A ^ (8x <a)['(x ; Z [x ] )! '(x + 1; Z [x +1] )], where A is any set such that '(0; A). Since p i -ind holds (by induction hypothesis if i >0 or, by denition, if i = 0), we can apply induction on x to the formula '(x ; Z [x ] ) to prove '(a; Z [a] ) and hence (a). This completes the proof of the second equivalence. We leave the proof that p i -comp is equivalent to p i -coll to the reader. Proof of (iii). The second implication is true by denition if i = 0. For i >0 this holds because P i contains p i Turing oracles and is closed under rudimentary collection. For the rst implication, consider rst the case i = 0. Given a model M of p 1 -comp we show that there is a unique way of dening new functions on M which satisfy the axioms of P-def. We proceed by induction on the denition of F 2P. The new primitives are added in order to have that for some p -formula 1 ' M j= F(~x ; ~ X ) = Y $ '(~x ; ~ X ; Y ) M j= 8~x; ~ X 9!Y '(~x ; ~ X ; Y ) The proof is actually standard and need not be reported here in detail. The key step is when F is obtained by recursion. In this case p -dc is used. For i >0 let T i 1 be the set of Turing oracles. Clearly there is a unique way to add to a model M new primitives for T i -functions and having them satisfy their dening axioms. Now, it is easily seen that, if M models p i+1-comp, then it models p (T i 1 )-comp too. At this point the proof proceeds as in the case i = 0. We sum the lemma up in the following table. An arrow means provability. Next to the arrow we write the partial conservativity we shall prove in sections 2.2 and

11 p p f P 1 -def + -choice 89 f P 1 -def -comp ( -ind -dc -coll) f p 1 89 p 1 p f P 0 -def + -choice f 89 f P 0 -def p 0 -comp 1.7 Relations with Buss' bounded arithmetic. In the introduction we mentioned that p i -comp coincides with Buss' S i 2 by a suitable translation of formulas. This translation has been found independently by many authors (see e.g., [14], [15], [10]). It is not necessary to include full details here, but, to give some clue to the reader, we quickly show how to transform a model of S i 2 into a model of p i -comp and vice versa. Let M 1 be a model of S i 2. Let M 2 be the second-order structure having as rst-order objects the elements a of M 1 such that 2 a exists and as second-order objects those nite subsets of M 1 which are coded in the usual way by elements of M 1. I.e., for every a2m 1 we add the set A to M 2 such that a = P x 2A 2x Functions and relations of M 2 are dened in the natural way. Note that multiplication of rstorder elements is a total operation in M 2. In fact if 2 a and 2 b exist in M 1 then 2 ab exists too, since it is equal to 2 a #2 b. It is easy to see that M 2 models p i -comp. In fact, it is sucient to note that for every second-order formula '(x ; X )2 p i there is a rst-order formula ' (x ; y)2 b i such that for every a; A2M 2 M 2 j= '(a; A) () M 1 j= ' (a; P x 2A 2x ). 11

12 To see the other direction, we apply the inverse procedure. Let M 2 be a model of p i -comp. We think of sets of M 2 as representing numbers, i.e., we think of the set X as the number n(x ) := P x 2X 2x Clearly, in general such a number need not exist in M 2. Still, formalizing the natural algorithm for addition and multiplication of binary numbers, we may dene in M 2 some set functions X Y and X Y such that n(x Y ) = n(x ) + n(y ) and n(x Y ) = n(x ) n(y ). It is well known that such an algorithm is computable in polynomial time, so, X Y and X Y are total functions in every model of P-def. Let X #Y be the set f jx j jy j g which exists because M 2 models p 0 -comp. Also, all other functions of the language of S 2 can be dened in a similar way. Now, one can construct a model of S i 2 having as its domain the second-order elements of M 2 and as functions and relations the ones just dened. The reader may check that the 32 axioms of BASIC hold in M 1. Because M 2 is a model of p i -ind, it is not dicult to see that M 1 satises logarithmic inductions for b i -formulas. Hence M 1 is a model of S i 2. To translate the results of section 3.3 in the language of S 2 note that second-order models of p i+1 -choice correspond to rst-order models of BBb i+1 (cf. chapter 5 of [7]), i.e., the schema (8x <jtj)(9y<s)'(x ; y)! : 9w(8x <jtj)'(x ; (w) x ). where ' is in b i+1. 2 Witnessing theorems and conservativity results. Buss was the rst to give an extensive characterization of complexity classes as classes of functions denable and provably total in some weak fragment of arithmetic. However, the very idea of the proofs we report here goes back to the Mints-Parsons' famous partial conservativity result of I 1 over PRA [12], [13]. Buss', Parsons' and Mints's proofs are proof-theoretical. Wilkie gave a model-theoretic proof (unpublished) of Buss' theorem (see [7]). Here we adapt a modeltheoretical proof of Mints-Parsons' theorem given by Albert Visser (unpublished). 2.1 Closures Let M be a model of p 0-comp and let W be a subset of M. We say that W is closed under R-functions if F(~c; C ~ ); f (~c; C ~ )2W for every ~c; C ~ 2W and F ; f 2R. The R-closure of W in M is the minimal R-closed subset of M containing W, i.e., hhwii R := ff(~c; ~ C ); f (~c; ~ C ) : ~c; ~ C 2W and F ; f 2Rg. We interpret R-closed subsets as substructure in the canonical way. The functions and relations of N are the restriction of those of M. In the same way we dene P i -closed sets in models of P i -def. We say that NM is a p i -elementary substructure of M, if for every p i -formula ' and every 12

13 ~a; ~ A2N N j= '(~a; ~ A) =) M j= '(~a; ~ A) We write N p M if N is a p i -elementary substructure of M. Similar notation is used also for i other classes of formulas. Lemma 2.1. (Denability of Skolem functions) (i) R-closed substructures of models of p 0 -comp are p 0 -elementary. (ii) For every model M of P i -def and every W M the P i -closure N of W is a p i (P i )- elementary substructure of M. In particular N is a model of P i -def. Proof. For (i), observe rst-order Skolem functions for p 0-formulas are in R. This suces. The proof of (i) when i = 0 is obvious as in the previous lemma. For i >0 it suces to show that among the P i -functions there are Skolem functions for p i (P i )-formulas. I.e., for every p i (P i )-formula ' there is a function F in P i such that 9Y <j~a; ~ Aj p '(~a; ~ A; Y )! '(~a; ~ A; F(~a; ~ A)). To see this we shall dene a function F that, by binary search, produces the minimal (in the lexicographic order) set Y <j~a; ~ Aj p satisfying '(~a; ~ A; Y ). Let us dene the function G by recursion in the following way (omitting parameters and bounds) G(0; ~a; ~ A) = ; G(y + 1; ~a; ~ A) = ( G(y; ~a; ~ A) if (9Y <j~a; ~ Aj p )[(G(y; ~a; ~ A)Y ) ^ '(~a; ~ A; Y ) ^ y =2 Y ] G(y; ~a; ~ A) [ fyg otherwise (recall that P i is closed under denition by p i (P i )-cases since it contains the characteristic functions of p i -formulas and is closed under composition) Finally, we dene F(~a; ~ A) = G(j~a; ~ Aj p + 1). We leave to the reader the verication that F produces a witness, if one exists, of 9Y <j~a; ~ Aj p '(~a; ~ A; Y ). The class of P i -functions is closed under p i -denition by cases, so, an easy compactness argument proves the following witnessing theorem for P i -def. Corollary 2.1. (Witnessing theorem for P i -def.) Each 89 p i+1 sentence provable in P i -def has a witnessing function in P i. Proof. We have to prove that, for all '2 p i+1, there is a function F in P i such that P i -def ` 8 ~ X ; ~x 9Y '(~x ; ~ X ; Y ) =) P i -def ` 8 ~ X ; ~x '(~x ; ~ X ; F(~x ; ~ X )). By contraction of quantiers it suces to show that the implication above holds for p i -formulas. So, let ' be a p i i -formula such that for no F 2P (*) P i -def ` 8 ~ X ; ~x '(~x ; ~ X ; F(~x ; ~ X )). Let ~c; ~ C be fresh constants and consider the theory 13

14 (**) P i -def + f:'(~c; ~ C ; F(~c; ~ C )) : F 2P i g This theory is consistent. Otherwise by compactness, for a nite set of functions ff 1 ; ::; F n g in P i, P i -def ` 8~x; ~ X : '(~x ; ~ X ; F 1 (~x ; ~ X )) _ :::: _ '(~x ; ~ X ; F n (~x ; ~ X )). So, since P i -functions are closed under denition by p i -cases, one can combine F 1; ::; F n together to nd a function F 2P i satisfying (*). Now, choose a model M of the theory (**) and let N be the P i -closure of ~c; C ~. By the previous lemma N is a model of P i -def. The same lemma excludes the possibility of having in N a set Y such that '(~c; C ~ ; Y ). Thus P i -def does not prove 8~x; X ~ 9Y '(~x ; X ~ ; Y ) and the corollary follows. 2.2 A model-theoretical version of Buss' witnessing theorem. We derive our version of Buss witnessing theorem from the following lemma. Lemma 2.2. Every model M of P i -def has an 9 p i+1-elementary extension to a model N of p -comp such that for every i+1 p i -formula ' there is a function F 2P i with (undisplayed) parameters from N such that (*) below holds (*) N j= 8X 9Y '(X ; Y )! 8X '(X ; F(X )). Proof. We claim that, if we succeed in satisfying condition (*), we obtain also that N models p i+1-comp. To prove the claim it is sucient to check that in N the schema of dependent choices holds for p i -formulas. Assume in N holds 8x 8X 9Y '(x ; X ; Y ) where a bound b on X and Y is implicit in '. Let a2n, we want to nd a Z such that (8x <a) '(x ; Z [x ] ; Z [x +1] ). By (*), for some F 2P i and for all x and X, '(x ; X ; F(x ; X )). Dene the following function G by second-order recursion (F can be bounded by b): G(0) = ;, G(x + 1) = F(x + 1; G(x)). Finally, Z is obtained by rudimentary collection: S x fx g G(x). This proves our claim. <a+1 Now, let M be a model of P i -def. The required model N is constructed as the union of an 9 p i+1 -elementary chain of models of P i -def, M = M 0 p 9 M 1 p 9 M 2 p 9 :::. i+1 i+1 i+1 The chain is constructed by stages. Each link of the chain is constructed using a model W as intermediate step, as in the following diagram W 6 9 p i+1 9 p 9 p i+1 i+1 - M - s M - s+1 p i 14

15 Suppose M s has already been constructed. Let ' s be the s-th p i -formula of an enumeration (to be specied below) of p i -formulas with parameters in M s. Let (*) s be (*) with ' s for '. We shall construct a model M s+1 realizing (*) s for some function F 2P i. Observe (*) s is a 98 p -formula, i+1 so, its truth is preserved upwards in the chain and nally inherited by the union N. It is easy to choose the enumeration such that eventually all p i -formulas with parameters in N are considered. The details of the enumeration are as follows. At each stage s we x an arbitrary enumeration s f t g t2! of all p i -formulas with parameters in M r s. Finally, let ' s be t for s = hr ; ti. To dene M s+1 proceed as follows. If (*) s holds for ' s, we do nothing, i.e. we dene M s+1 := M s. Otherwise, we try to make the antecedent of (*) s false in M s+1. We construct M s+1 and C 2M where 9Y ' s (C ; Y ) fails. Since (*) s does not hold in M s, the following theory has a model W s Diag(M s ) + f:' s (C ; F(C )) : F 2P i with parameters in M s g, where C is a fresh constant and Diag(M s ) is the elementary diagram of M s (to check the consistency, argue by compactness). W s is elementary equivalent to M s, so, in particular, it is a model of P i -def. Dene M s+1 := hhm s + C ii P i Closure to be taken in W. Clearly M s+1 is a p i -elementary substructure of W s which is elementary equivalent to M s, so, every 9 p i+1 -formula true in M s+1 will be true in W s and hence in M s. In M s+1 there is no witness of 9Y ' s (C ; Y ). This completes the proof of the lemma. Corollary 2.2. (89 p -conservation and witnessing theorem for i+1 p -comp.) i+1 p i+1-comp is 89 p -conservative over P i i+1 -def, therefore every 89 p sentence provable in i+1 p i+1-comp has a witnessing function in P i. Proof. Immediate from the previous lemma and from Lemma A model theoretical characterization of choice. We introduce the concept of R-extension. This is an extension where all second-order objects are constructible relative to the extended model. This notion may be viewed also as a second-order generalization of conal extension. It will be used to give a model theoretical characterization of SP i+1 -choice over p i -comp. An useful application of this notion is given in the proof of the Corollary below. (The conservativity result in Corollary 2.3 (b) will nd application in the following sections to characterize the collapse of BA.) Let M and N be models of p 0 -comp. Recall that their rst-order parts are denoted respectively by m and n. We say that N is an R-extension of M if (o) m is conal in n, i.e., for all a2n there is b2m, such that a<b. (i) M p 0 N. (ii) N = hhm + nii R, i.e., for every A2N there are a2n such that N j= A = F(a) for some F 2R with parameters in M. We write M R N when N is an R-extension of M. 15

16 Fact 2.3. Let N be an R-extension of M. (a) M 9 p 1 N. (b) M j= p i+1 -choice =) M 9 p i+2 N. (c) M j= p i -comp () N j= p i -comp. (d) M j= P i -def () N j= P i -def. Proof of (a). If N j= 9Y '(Y ) for some p 0-formula ' with parameters in M, then for some b2m, and some F 2R with parameters in M N j= (9x <b)'(f(x)), so, by p 0-elementarity, this holds in M too. This proves (a). Proof of (b). Let M be a model of p -choice. Let a2m and i+1 '2p i with parameters in M and suppose N j= 9Y (8X <a)'(x ; Y ). It suces to show that the same formula holds in M too. As induction hypothesis we assume 9 p i+1-elementarity. Since N is an R-extension, for some F 2R with parameters in M, N j= 9y (9x <y)(8x <a)'(x ; F(x)). Then, clearly, N j= 9y (8Z ha; yi)(9x <y)'(z [x ] ; F(x)). So, by 9 p i+1 -elementarity, M j= 9y (8Z ha; yi)(9x <y)'(z [x ] ; F(x)). Finally, by p i+1 -choice, M j= 9y (9x <y)(8x <a)'(x ; F(x)). This proves (b). Proof of (c). The `left to right' direction of Fact (c) is true by denition when i = 0. For i >0 it follows from (b) and Lemma 1.6. In fact, these imply that N is an 9 p i+1-elementary extension of M. So, let ' be any p i -formula with parameters in N. By (ii), we can assume that all secondorder parameters ~c of ' belong to M. Let a2n be arbitrary. Choose in M a b>a;~c. Since M j= p i -comp there is a set A2M such that M j= (8x ; ~y<b)[hx ; ~yi2a $ '(x ; ~y)] where the variables ~y replace all rst-order parameters of '. By 9 p i+1-elementarity, A satises the same property in N too. So, the set of N, veries, B = fx <a : hx ;~ci2ag N j= (8x <a)[x 2B $ '(x ;~c)] This proves that N is a model of p i -comp. 16

17 The converse direction (`right to left') is also true by denition when i = 0. So, assume it true for i and let us prove it for i + 1. Let N be a model of p i+1-comp. By induction hypothesis, M is a model of p i -comp and by Fact (b), a 9p i+1-elementary substructure of N. Let '(x ; Y ) be a p i -formula with parameters in M and an implicit bound on Y. It sucies to nd in M a set A such that (8x <a)[x 2A $ 9Y '(x ; Y )]. By Lemma 1.6, N models p i+1-coll so, for some set Z N j= (8x <a)[9y '(x ; Y ) $ '(x ; Z [x ] )]. Then, for some b2n and function F 2N with parameters in M, N j= (8x <a)[9y '(x ; Y ) $ '(x ; F(b) [x ] )]. We claim that the set A2M such that A = fx <a : 9y '(x ; F(y) [x ] )g is the required one. We only need to show that (8x <a)[9y '(x ; Y )! x 2A], because the converse is obvious. If 9Y '(x ; Y ) holds in M for some x <a, then this will be also true in the p i -elementary extension N. Then 9y '(x ; F(y)[x ] ) holds in N and, again by p i - elementarity, is true in M too. Therefore x 2A. Proof of (d). To prove the `left to right' direction we use Lemma 2.2. This lemma characterize models of P i -def as those having an 9 p i+1 -elementary extension to a model M0 of p i+1 -comp. So, it suces to show that there exists a model W satisfying the following diagram of (restricted) elementary extensions M 9 p i+1 - M 0 j= p i+1 -comp R 9 p i+1 N - W. Consider the theory Diag(M 0 ) + Diag p (N). i+1 This theory has a model, otherwise, let, for a contradiction, for some '2 p i, (8X <j~cj p )'(X ;~c)2diag p (N) and i+1 Diag(M 0 ) ` :8X '(X ;~c) where we assume that a bound on X is implicit in '. Also, since M R N, we can assume that all other parameters of ' but ~c are in M. Let a2m be such that ~c<a. Replacing the constants 17

18 ~c2/m 0 with variables and quantifying we obtain M 0 j= (8~x<a)9X :'(X ; ~x). We may apply p i+1-choice to get, M 0 j= 9Z (8~x<a):'(Z [~x] ; ~x) so, by 9 p i+1 -elementarity, M j= 9Z (8~x<a):'(Z [~x] ; ~x). Recall that M models p i -comp, so, by (b), R-extensions of M are 9p i -elementary. So, N j= 9Z (8~x<a):'(Z [~x] ; ~x). Therefore, N j= (8~x<a)9X :'(X ; ~x). A contradiction since we assumed that 8X '(X ;~c)2diag p (N). i+1 Let W 0 be a model of the theory above and let W := fa; A2W 0 : a; A<b for some b2mg Clearly W 0 is a model of p i+1 -comp and consequently also W. To prove N 9 p i+1 W, it suces to observe that N p W 0, that W PH W 0 and that N is conal in W. This proves the left i+1 to right direction of (d). For the converse, assume N is a model of P i -def. By Lemma 2.2, there is a model N 0 such R 9 N 9 p i+1 - N 0 j= p i+1 -comp where the diagonal arrow follows from (b) since, by (c), M is a model of p i. Lemma 2.3. Every model M of p i -comp has an R-extension to a model N of p i+1 -choice. Proof. The proof is similar to that of Theorem 2.2. The model N is constructed as the union of a chain M = M 0 R M 1 R M 2 R ::: By the Fact above we have that we actually construct a 9 p i+1-elementary chain of models 18

19 of p i -comp. Let f' sg s 2! be an enumeration with innitely many repetitions of all formulas with parameters in N, such that all parameters of ' s are in M s (see Theorem 2.2 for details on this enumeration). The chain is constructed so that for all ' s 2 p i, either (1) or (2) below holds. (1) For every a2m there is a Z 2M such that M s j= (8x <a)' s (x ; Z [x ] ). (2) There is a c2m s+1 such that M s+1 j= 8Y :' s (c; Y ). Each link of the chain is constructed using a model W as intermediate step, as in the following diagram R W 6 p 0 R R - M - s M - s+1 Suppose M s has already been constructed. If (1) holds in M s then let M s+1 := M s. Otherwise, let W be any model of Diag(M s ) + (c<a) + f:' s (c; F(c)) : F 2R with parameters in Mg. Such a model exists, otherwise, for some n M s j= (8x <a) W n m=0 ' s(x ; F n (x)). Using p i -comp one can dene a set Z in M such that for all (x <a), Z [x ] = F m (x) for the minimal m<n such that ' s (x ; F m (x)) holds. But we assumed such a set does not exist. Clearly W is, up to isomorphism, an elementary superstructure of M s. Let M s+1 := hhm + cii R (closures to be taken in W). To check that M s R M s+1 note that M s+1 is a p 0 substructure of W. Also, observe that all elements of M s+1 are generated by elements of M s and the rst-order element c<a, so, conditions (o), (i) and (ii) in the denition of R-extension are fullled. To check that (2) holds, suppose not, for a contradiction. If 9Y ' s (c; Y ) held in M s+1, then we would have ' s (c; F(c)) for some F 2R with parameters in M s. We will reach a contradiction by showing that instead ' s (c; F(c)) must fail in M s. By construction, we have :' s (c; F(c)) in M s+1. To pull this back to M s we reason as follows. Since M s is a model of p i -comp, for some A2M s, (8x <a)[x 2A $ ' s (x ; F n (x))]. So, by the elementary equivalences proved above, this holds also in W and in M s+1. In W we have c2/a and, by p 0 equivalence this holds in M s+1. So, :' s (c; F(c)), a contradiction. Finally, N is a model of p i+1-choice, since the truth of both formulas in (1) and in (2) is preserved along the 9 p -chain. i+1 19

20 Now we can easily prove the announced characterization. Theorem 2.3. For every M j= p i -comp the following are equivalent (i) M j= p i+1 -choice (ii) Every R-elementary extension of M is 9 p i+2 -elementary. Proof. That (i) implies (ii) has already been observed in the fact above. For the converse, suppose M j= (8x <a)(9y <b)'(x ; Y ) for some '2 p i. Let N be the R-elementary extension of M to a model of p i+1-choice as guaranteed by the Lemma above. Then, by the assumed 9 p -elementarity i+2 N j= (8x <a)(9y <b)'(x ; Y ). Let Z in N be such that (8x <a)'(x ; Z [x ] ). By the denition of R-extension, N j= (9y<c)(8x <a)'(x ; F(y) [x ] ) for some c2m and some F 2R with parameters in M. So, by 9 p i+2-elementarity this formula holds also in M. The following conservativity results are consequences of the lemma above. Corollary 2.3. (a) p i+1 -choice is 89p i+1 -conservative over p i -comp. (b) P i -def + p i+1 -choice ` p i+1 -comp =) P i -def ` p i+1 -comp. (c) p i+1 -choice ` p i+1 -comp =) p i -comp ` p i+1 -comp. Proof. The proof of (a) is obvious. The proof of (b) and (c) are similar. Let us prove (c). Assume P i -def + p i+1 -choice proves p i+1 -comp. Let M be any model of P i -def. In particular, M is a model of p i -comp, so, by the Lemma above, it has an R-extension N to a model of p i+1 -choice. By Fact (d) above, N is also a model of P i -def. So, by our assumption, N models p i+1 -comp. By Fact (c) above, M is also a model of p i+1 -comp. 2.4 Ultrapowers In this section we present an ultrapower construction corresponding to the construction in Lemma 2.3. Let M be a model of p 0 -comp and a2m. Let U a be an ultralter on the set fb2m : M j= b<ag. We dene on M the following equivalence relations A 1 B i fc2m : M j= c<a ^ A(c) = B(c)g 2 U a 20

21 A 2 B i fc2m : M j= c<a ^ A [c] = B [c] g 2 U a The denitions of A(c) is given in the end of Section 1.3. The 1 equivalence class of A is denoted with A =1, the 2 equivalence class with A =2 (the lter U a is usually clear from the context). We shall systematically confuse equivalence classes with their representative. Let M=U a be the model whose rst-order elements are the 1 equivalence classes and whose second-order elements the 2 equivalence classes. The relations and the functions of M=U a are dened in the canonical way. To every element c; C 2M we associate the elements id(c) and id(c ) of M=U a in the usual way, id(c) = S x <a fx g fcg and id(c ) = S x <a fx g C. The map id is an embedding of M into M=U a. So, as usual, we consider M as a substructure of M=U a. Fact 2.4. (a) If M is a model of p i -comp then for every p 0 (p i )-formula ' M=U a j= '( ~ B =1 ; ~ C =2 ) () fd2m : M j= (d <a) ^ '( ~ B(d); ~ C [d] )g 2 U a. (b) M=U a is an R-extension of M. Proof of (a). For atomic ' the lemma holds by denition. The inductive step for the Boolean connectives is immediate. We show that of the existential quantiers. The =) direction is again immediate. For the converse let us rst consider rst-order quantiers. Suppose the lemma holds for '2 p 0( p i ) and let us show that it is holds also for 9y ', where the bound on y is implicit in '. Let us write ' x for '( B(x); ~ C ~ [x ] ) and assume fx 2M : M j= (x <a) ^ 9y' x (y)g 2 U a. Then by p i comprehension there is a set A in M such that M j= 8y (8x <a)[' x (y) $ hx ; yi2a] so, fx 2M : M j= (x <a) ^ ' x (A(x))g 2 U a, and, by induction hypothesis, M=U a models '(A =1 ) and hence 9y'(y) For second-order quantiers the proof is similar. nothing to prove. Suppose the lemma holds for '2 p i 9Y ', where the bound on Y is implicit in '. If We can assume i >0 otherwise there is and let us show that it is holds also for fx 2M : M j= (x <a) ^ 9Y ' x (Y )g 2 U a, by p i comprehension, (since i >0) there is a set A in M such that M j= 8Y (8x <a)[' x (Y ) $ ' x (A [x ] )] so, fx 2M : M j= (x <a) ^ ' x (A [x ] )g 2 U a, and, by induction hypothesis,m=u a models '(A =2 ) and hence 9Y '(Y ). 21