Set theoretical analogues of the Barwise-Schlipf theorem

We characterize nonstandard models of ZF (of arbitrary cardinality) that can be expanded to Goedel-Bernays class theory plus $\Delta^1_1$-Comprehension. We also characterize countable nonstandard models of ZFC that can be expanded to Goedel-Bernays class theory plus $\Sigma^1_1$-Choice.


INTRODUCTION
The point of departure of this paper is Theorem 1.1 below, which characterizes recursively saturated models of PA (Peano Arithmetic) as precisely those nonstandard models of PA that are expandable to models of certain subsystems of second order arithmetic. In what follows, ACA 0 is the well-known subsystem of second order arithmetic whose first order part is PA, ∆ 1 1 -CA (respectively Σ 1 1 -AC) is the scheme of ∆ 1 1 -Comprehension (respectively Σ 1 1 -Choice), and Def(M) is the family of first order definable (parameters allowed) subsets of M.
(b) There is X such that (M, X) |= ACA 0 + ∆ 1 1 -CA. (c) (M, Def(M)) |= ACA 0 + ∆ 1 1 -CA + Σ 1 1 -AC. The argument for (a) ⇒ (c) given by Barwise and Schlipf used the machinery of admissible set theory. Not long after, an elementary argument was found by Feferman and Stavi (independently), as reported in Smoryński [Sm]. However, the proof presented for (b) ⇒ (a) by Barwise and Schlipf was shown in [ES] to be impaired by a significant gap, and additionally, a correct proof (using a technique not available to Barwise and Schlipf) was presented. Now, prospering from a 45 year hindsight, we can say that the hard part of Theorem 1.1 is (b) ⇒ (a), and the straightforward part is (a) ⇒ (c) ((c) ⇒ (b) is trivial, of course).
An analogue of Theorem 1.1 in the realm of set theory was presented by Schlipf, as in Theorem 1.2 below, in which o(M) is the ordinal height of the well-founded part of M, and o(HYP(M)) is the ordinal height of HYP(M), where HYP(M) is the least admissible structure over M, as defined in Barwise's definitive text [B] on admissible set theory. Theorem 1.2 implies the analogue of Theorem 1.1 for models of ZF (in which PA is replaced by ZF, and ACA 0 is replaced by GB), using Schlipf's characterization of recursive saturation in terms of o(HYP(M)) = ω.
1.2. Theorem. (Schlipf [Sch]) The following two conditions are equivalent for a nonstandard model M of ZF of any cardinality: (a) There is X such that (M, X) |= GB + ∆ 1 1 -CA. (c) There is X such that (M, X) |= GB + ∆ 1 1 -CA + Σ 1 1 -AC. In a different direction, the paper [En] studies the family of so-called condensable models of ZF, a family that includes all resplendent models of ZF (and in particular, all countable recursively saturated models of ZF). In the terminology of [En], a model M |= ZF is condensable if M ∼ = M(α) ≺ L M M for some "ordinal" α ∈ Ord M , where M(α) := (V(α), ∈) M and L M is the set of formulae of the infinitary logic L ∞,ω that appear in the well-founded part of M. The following theorem gives various characterizations of the notion of condensability (see Section 2 for the definitions of the technical notions used in the statement of Theorem 1.3).
1.3. Theorem. [En] The following are equivalent for a countable nonstandard model M of ZF:    Moreover, without the assumption of countability of M, the following implications hold: The main results of this paper are Theorems A and B below that tie Theorems 1.2 and 1.3 together. The proofs of these results do not rely on machinery from admissible set theory, in particular we obtain a new proof, from first principles, of the equivalence of (a) and (c) of Theorem 1.2 for a countable nonstandard model M of ZFC. Note that if M is ω-nonstandard, then condition (a) in Theorems A and B below is equivalent to recursive saturation of M.
Theorem A. The following are equivalent for a nonstandard model M of ZFC of any cardinality: The following are equivalent for a countable nonstandard model of ZFC: There is X such that (M, X) |= GB + ∆ 1 1 -CA + Σ 1 1 -AC. We suspect that Theorem A can be strengthened by weakening ZFC to ZF. As explained in Remark 4.4, in Theorem B, ZFC cannot be weakened to ZF, and the assumption of countability of M is essential. The proof of Theorem A is presented in Section 3, and Theorem B is established in Section 4.

PRELIMINARIES
In this section we collect the basic definitions, notations, conventions, and results that will be used in the statements and proofs of our main results in Sections 3 and 4.
2.1. Definition. (Models, languages, and theories) Models will be represented using calligraphic fonts (M, N , etc.) and their universes will be represented using the corresponding roman fonts (M , N , etc.). In the definitions below, M is a model of ZF and ∈ M is the membership relation of M.
(d) The well-founded part of M, denoted WF(M), consists of all elements m of M such that there is no infinite sequence a n : n < ω with m = a 0 and a n+1 ∈ M a n for all n ∈ ω. Given m ∈ M, we say that m is a nonstandard element of M if m / ∈ WF(M). We denote the submodel of M whose universe is WF(M) by WF (M). It is well-known that if M is a model of ZF, then WF(M) satisfies KP (Kripke-Platek set theory) [B,Chapter II,Theorem 8.4].
• It is important to bear in mind that we will identify WF(M) with its transitive collapse. (f ) Let L set be the usual vocabulary {=, ∈} of set theory. In this paper we use L ∞,ω to denote the infinitary language based on the vocabulary L set . Thus L ∞,ω is a set-theoretic language that allows conjunctions and disjunctions of sets (but not proper classes) of formulae, subject to the restriction that such infinitary formulae have at most finitely many free variables. Given a set Ψ of formulae, we denote such conjunctions and disjunctions respectively as Ψ and Ψ.
• In the interest of efficiency, we will treat disjunction and universal quantification as defined notions.
(g) L δ,ω is the sublanguage of L ∞,ω that allows conjunctions and disjunctions of sets of formulae of cardinality less than δ. Note that L ω,ω is none other than the usual first order language of set theory, and that in general the language L δ,ω only uses finite strings of quantifiers (as indicated by the ω in the subscript).
(h) We say that F is a fragment of L ∞,ω if F is a set of formulae of L ∞,ω that is closed under subformulae, renaming of free variables, existential quantification, negation, and conjunction.
• A fragment of L ∞,ω that plays a central role in this paper is (i) Given a fragment F of L ∞,ω , and L set -structures N 1 and N 2 , we write N 1 ≺ F N 2 to indicate that N 1 is a submodel of N 2 , and for all ϕ(x 1 , · · ·, x n ) ∈ F and all tuples (a 1 , · · ·, a n ) from N 1 , we have: N 1 |= ϕ(a 1 , · · ·, a n ) iff N 2 |= ϕ(a 1 , · · ·, a n ).
(j) Given a fragment F of L ∞,ω , Th F (M) is the set of sentences (closed formulae) of F that hold in M, and ZF(F) is the natural extension of ZF in which the usual schemes of separation and collection are extended to the schemes Sep(F) and Coll(F) so as to allow formulae in F to be used for "separating" and "collecting" (respectively).
(l) For an ordinal α we use D(α) to denote {ϕ ∈ L ∞,ω : Depth(ϕ) < α}. Within KP, one can code each formula ϕ ∈ L ∞,ω with a set ϕ as in Chapter 3 of [B], but in the interest of better readability we will often identify a formula with its code. This coding allows us to construe statements such as ϕ ∈ L ∞,ω and Depth(ϕ) = α as statements in the first order language of set theory. It is easy to see that for a sufficiently large k ∈ ω, D(α) ⊆ V(ω + kα) for each ordinal α. This makes it clear that (m) Suppose M is nonstandard and W := WF(M). M is W -saturated if for every k ∈ ω and every type p(x, y 1 , · · ·, y k ), and for every k-tuple a of parameters from M, p(x, a) is realized in M provided the following three conditions are satisfied: (n) Every model of GB can be put in the form (N , X) , where N |= ZF and X ⊆ P(N ).

Definition.
Suppose M is a model of ZF, and S ⊆ M .
(a) S is separative (over M) if (M, S) satisfies the separation scheme Sep(S) in the extended language that includes a fresh predicate S (interpreted by S).
(b) S is collective (over M) if (M, S) satisfies the collection scheme Coll(S) in the extended language that includes a fresh predicate S (interpreted by S).
(c) S is amenable (over M) if S is both separative and collective. In other words, S is amenable if (M, S) satisfies the replacement scheme Repl(S) in the extended language that includes a fresh predicate S (interpreted by S). Note that if (M, X) is a model of GB, then each element of X is amenable over M.
(d) For α ∈ Ord M , S is an α-satisfaction class (over M) if S correctly decides the truth of atomic sentences, and S satisfies Tarski's compositional clauses of a truth predicate for D M (α)-sentences (see below for the precise definition). S is an ∞-satisfaction class over M if S is an α-satisfaction class over M for every α ∈ Ord M .
We elaborate the meaning of (d) above. Reasoning within ZF, for each object a in the universe of sets, let c a be a constant symbol denoting a (where the map a → c a is ∆ 1 ), and let Sent + (α, x) be the set-theoretic formula (with an ordinal parameter α and the free variable x) that defines the proper class of sentences of the form ϕ (c a 1 , · · ·, c an ), where ϕ(x 1 , · · ·, x n ) ∈ D(α) (the superscript + on Sent + (α, x) indicates that x is a sentence in the language augmented with the indicated proper class of constant symbols). Then S is an α-satisfaction class over M if (M, S) |= Sat(α, S), where Sat(α, S) is the conjunction of the universal generalizations of axioms (I) through (IV ) below: .
(e) For α < o(M), S is the α-satisfaction class over M, if S is the usual Tarskian satisfaction class for formulae in L M of depth less than α, i.e., the unique α-satisfaction class S over M such that S satisfies: • In the interest of a lighter notation, if S is an α-satisfaction class over M (for a possibly nonstandard α ∈ Ord M ), ϕ(x 1 , · · ·, x n ) is an n-ary formula of D M (α), and a 1 , · · ·, a n are in M, we will often write ϕ (a 1 , · · ·, a n ) ∈ S instead of ϕ (c a 1 , · · ·, c an ) ∈ S.
The following proposition is immediately derivable from the relevant definitions.
In particular, for all sentences ϕ of L M , ϕ ∈ S iff ϕ ∈ Th L M (M).

Remark.
Reasoning within ZF, given any limit ordinal γ, (V(γ), ∈) carries a separative γ-satisfaction class S since we can take S to be the Tarskian satisfaction class on (V(γ), ∈) for formulae of depth less than γ. More specifically, the Tarski recursive construction/definition of truth works equally well in this more general context of infinitary languages since (V(γ), ∈) forms a set. Observe that (V(γ), ∈, S) |= Sep(S) comes "for free" since for any X ⊆ V(γ) the expansion (V(γ), ∈, X) satisfies the scheme of separation in the extended language.
The following generalization of the Montague-Vaught reflection theorem of set theory appears as Proposition 2.8 of [En], in a slightly weaker form, where the class of ϕ-reflecting ordinals (where ϕ ranges over D M (α)) is asserted to be unbounded, as opposed to closed and unbounded. The stronger version below can be readily obtained by putting Proposition 2.6 above together with Proposition 2.8 of [En]. The closed unboundedness of the class of ϕ-reflective ordinals is needed in the proof of Lemma 4.3 of this paper, where it is important to arrange arbitrarily large ϕ-reflective ordinals of countable cofinality.

Then for any
The notion "f is λ-onto X" introduced in Definition 2.8 below, and the corresponding existence result (Proposition 2.9), are adaptations of Definition 2.1 and Lemma 2.2 of Kaufmann and Schmerl's work [KS] on models of arithmetic to the setting of set theory. Lemma 2.2 of [KS] played a key role in the proof presented in [ES] of the direction (b) ⇒ (a) of Theorem 1.1. Proposition 2.9 plays an analogous role in the proof of the direction (b) ⇒ (a) of Theorem A.

Definition.
A set I is an ordinal interval if I = {γ : α < γ < β} for some ordinals α and β. Suppose f : I → X, where I is an ordinal interval, X is some set, and λ is an ordinal. The notion f is λ-onto X is defined by recursion on λ as follows: • For a limit ordinal λ, f is λ-onto X means: ∀γ < λ f is γ-onto X.
2.9. Proposition. (ZFC) Given any set X and any ordinal λ there is some ordinal interval I and a function f : Proof. We use induction on λ. The case λ = 0 is clear since we are working in ZFC. If λ = γ + 1, then for each subset Y of X there is an ordinal interval I Y and some function h Y : Let I α := I Yα and h α := h Yα for each α < κ and choose an ordinal interval I that is order isomorphic to the well-ordering α<κ I α .
More explicitly, let Z = {{α} × I α : α < κ} and let ⊳ be the lexicographic order on Z. Then since ⊳ is a well-ordering, there is an ordinal interval I and an isomorphism F between (Z, ⊳) and (I, ∈) . Note that F ({α} × I α ) ∩ F ({β} × I β ) = ∅ when α and β are distinct elements of κ. Since for each α < κ the function h α : I α → Y α has the property of being λ-onto Y α , the isomorphism F allows us to construct functions f α : This will ensure that α<κ f α is a function from I to X that is γ + 1-onto X.
For limit λ we use a strategy similar to the successor case. By inductive assumption, for each γ < λ there is some ordinal interval I γ and a function f γ : I γ → X such that f γ is γ-onto X. We can therefore find an ordinal interval I and an isomorphism F between the well-ordering γ<λ I γ and I. For each γ < λ we can then construct functions f γ : 2.10. Remark. Recall that Σ 1 k -AC (AC for the axiom of choice) is the scheme consisting of formulae of the form where ψ(x, X) is a Σ 1 k -formula (parameters allowed), and Σ 1 k -Coll (Coll for Collection) is the scheme consisting of formulae of the form where ψ(x, X) is a Σ 1 k -formula (again, with parameters allowed). In the above where x, y is a canonical pairing function.
(a) It is well-known that Σ 1 k -AC implies ∆ 1 k -CA for all k < ω; an easy proof in the arithmetical setting can be found in [Si,Lemma VII.6.6(1)]; the same proof readily works in the set-theoretic context.
(b) Let GBC be the result of augmenting GB with the global axiom of choice. It is well-known that in the presence of GBC, (1) Σ 1 1 -AC is equivalent to Σ 1 1 -Coll, and (2) global choice is provable in GB + Σ 1 1 -AC. For more detail, see, e.g., [Fu,Section 3.1].
Let L ∞,ω be the extension of L ∞,ω based on the extended vocabulary L set = {=, ∈, f }, where f is a function symbol (for a global choice function), and let L N := L ∞,ω ∩ WF(N ), where N |= ZF (N need not be nonstandard, so WF(N ) might be the whole of N ). The following result is the infinitary generalization of the well-known theorem that global choice can be generically added to models of ZFC of countable cofinality [Fe] (and its proof is similar to the proof of the finitary case). A proof of part (b) of Proposition 2.11 can be found in [Sch,Theorem 11].
2.11. Proposition. (Forcing Global Choice) Let N |= ZFC(L N ), and P be the class notion of forcing consisting of set choice functions in N , ordered by set inclusion.
(a) If Ord N has countable cofinality, then there is an L N -generic filter G ⊆ P, in the sense that G is a filter that intersects every dense subset of P that is definable in N by a formula in L N (parameters allowed).

PROOF OF THEOREM A
In this section we establish the first main result of our paper. In part (b) of the following theorem, Def L M is the family of L M -definable subsets of M (parameters allowed).
3.1. Theorem. The following are equivalent for a nonstandard model M of ZF of any cardinality:  (1) makes it clear that GB holds in (M, X). We will use (2) to show that ∆ 1 1 -CA holds in (M, X) . To this end, let U ⊆ M such that U is defined in (M, X) by a Σ 1 1 -formula ∃X ψ + (X, x, A), and M \U is defined in (M, X) by a Σ 1 1 -formula ∃X ψ − (X, A, x), where A ∈ X is a class parameter definable by the L M -formula α(m, v). Here m ∈ M is a set parameter; note that we may assume without loss of generality that the only parameter in ψ + and in ψ − is a class parameter A. Consider the infinitary formulae θ + (x) and θ − (x) defined as follows: In the above ψ + (X/ϕ(y, v), A/α(m, v), x) (respectively ψ − (X/ϕ(y, v), A/α(m, v), x)) is the result of replacing all occurrences of subformulae of the form w ∈ X (where w is a variable) in ψ + (respectively in ψ − ) by ϕ(y, w), and replacing all occurrences of subformulae of the form w ∈ A in ψ + (respectively in ψ − ) by α(m, w), and re-naming variables to avoid unintended clashes. Since each X ∈ X can be written in the form {v ∈ M : M |= ϕ(m 1 , v)} (where m 1 ∈ M is a parameter), U is definable in M by θ + (x) and M \U is definable in M by θ − (x). Therefore we have: Next, we aim to verify (4) below. In what follows D M (α) is as in part (k) of Definition 2.1.
Notice that (4) implies that U is definable in M by θ + α (x), so the verification of ∆ 1 1 -CA will be complete once we establish (4) since θ + α (x) ∈ L M and {a ∈ M : M |= θ + α (a)} ∈ X. To establish (4) we argue by contradiction. Suppose It is easy to see that p(x) ∈ Cod W (M). By (5), for each α ∈ o(M), p(x) ∩ D M (α) is realized in M, so by W -saturation of M, p(x) is realized in M, i.e., M |= ∃x ¬ (θ + (x) ∨ θ − (x)) , which contradicts (3) and finishes the proof of (4) (c) ⇒ (a). This is the hard direction of Theorem 3.1 and will require a good deal of preliminary lemmata. It will be proved as Lemma 3.6. In part (a) of Lemma 3.2, Sat α is as in Proposition 2.5.
3.2. Lemma. If (M, X) |= GB + ∆ 1 1 -CA, then the following hold: Proof. To see that (a) holds we will use induction on α to verify that Sat M α is ∆ 1 1 -definable in (M, X) for each α ∈ o(M). Recall that Sat(α, S) is the first order formula that expresses "S is an α-satisfaction class" (as in Definition 2.2). Suppose Sat M α ∈ X for some α ∈ o(M). Then for each m ∈ M we have: Similarly, for each m ∈ M we have: Thus Sat M α+1 has both a Σ 1 1 and a Π 1 1 definition in (M, X). The limit case is more straightforward since for limit α the following hold for each m ∈ M : This concludes the proof of (a). Note that (b) is an immediate consequence of (a) since the veracity of any L M -instance of replacement in M follows from the amenability of Sat M α over M for a sufficiently large α ∈ o(M).
(Lemma 3.2) The notion of paradefinability introduced in Definition 3.3 below is the set-theoretical analogue of the notion of recursive σ-definability in [ES].
To establish (a), first note that the assumption of the failure of W -saturation in M by Theorem 1.3 implies that there is no S ∈ X such that S is a γ-satisfaction class over M for any nonstandard γ ∈ Ord M . Combined with part (a) of Lemma 3.2, this makes it clear that (a) holds. To verify (b), let A be paradefinable by ϕ α (x, m) : α < o(M) . By replacing ϕ α (x, y) with β≤α ϕ β (x, y), we can assume that Depth(ϕ α (x, y)) < Depth(ϕ β (x, y)) for all α < β < o(M). Let δ be a nonstandard element of Ord M such that ϕ α (x, y) : α < δ is in M and extends ϕ α (x, y) : By part (a) of the lemma, this makes it evident that A is Σ 1 1 -definable in (M, X). (Lemma 3.5) • The proof of Theorem 3.1 will be complete once we verify Lemma 3.6 below, which takes care of the direction (c) ⇒ (a) of Theorem 3.1. The proof of Lemma 3.6 is rather complicated and we therefore beg the reader's indulgence.
By our supposition there is some β ∈ Ord M such that: (2) There is no α ∈ Ord M above β with M(α) ≺ L M M.
• We now distinguish between the following two cases, and will show that each leads to a contradiction, thus proving Lemma 3.6 (recall that the proof of Lemma 3.6 starts with "Suppose not"). Our proof was inspired by the proof of [ES,Theorem 4], which the reader is highly advised to review before reading the proof below, especially since the argument for Case B below is a more complex version of the argument for the "tall case" in the proof of [ES,Theorem 4]. One of the reasons for this increase in complexity has to do with the fact that in nonstandard models of arithmetic (and in ω-nonstandard models of set theory) it is easy to find an ill-founded subset A of the nonstandard ordinals of M that is paradefinable since we can choose A to be {c − n : n ∈ ω}, where c is any nonstandard finite ordinal. The existence of such an ill-founded A plays a key role in the proof of [KS,Lemma 2.4] since in conjunction with the arithmetical analogue of Proposition 2.9, it allows one to deduce that if some recursive type is omitted, then a recursive type consisting of formulae describing an ordinal interval is omitted. However, in a nonstandard model M of set theory that is ω-standard, the existence of an ill-founded paradefinable subset of the nonstandard ordinals of M takes far more effort to establish (with the help of additional assumptions, as indicated in the proof of Case B below).
Therefore Γ ∈ X, which implies that Γ is amenable over M, so coupled with the fact that Γ is cofinal in Ord M we conclude that there is some f ∈ X such that f is an isomorphism between Γ and Ord M (both ordered by ∈ M ). This contradicts the fact that Γ is well-founded and Ord M is ill-founded, and thus shows that Case A is impossible.
It is easy to see that δ α s are well-defined for each α ∈ o(M). More specifically, let X α = ξ ∈ δ : V(ξ) ≺ D(α) V . Then by the choice of δ, X α is nonempty for each α ∈ o(M); and by part (b) of Lemma 3.2, X α is coded in M, so sup (X α ) is well-defined, and by Proposition 2.6 (Elementary Chains) sup (X α ) ∈ X α , so max(X α ) is well-defined. It should also be clear that: Next, we observe: To see that (8) is true, note that δ α > β by (7), so if (8) is false, then V(δ α ) ≺ D(β) V for all β ∈ o(M), which contradicts (2). Thus (8) implies that {δ α : α ∈ o(M)} is ill-founded when viewed as a subset of Ord M . Moreover, for any α ∈ o(M), {δ β : β < α} is finite. To verify this, first note that there is a fixed natural number k such that the depth of the L M -formula that defines δ β for any β ∈ o(M) is at most β + k. Therefore in light of (5) and (6) and the fact that Sat M α (which is present in X by part (a) of Lemma 3.2. and is therefore amenable over M) can evaluate the defining formulae of {δ β : β < α} , the well-foundedness of Ord M as viewed in M implies that {δ β : β < α} is finite from the point of view of M. Therefore {δ β : β < α} is finite in the real world (this is clear if M is ω-standard; if M is ω-nonstandard, then it is trivial since α would have to be a finite ordinal since α ∈ o(M)). Putting all this together, we conclude: (9) The order type of {δ α : α ∈ o(M)} under ∈ M is ω * (i.e., the reversal of ω).
Let I(x, y) := {z : x ∈ z ∈ y}. In M, define X as the ordinal interval I(β, δ), and apply Proposition 2.9 to get hold of a function f and some ordinal interval I such that f : I → X and f is δ-onto X. Let I 0 be the ⊳-first ordinal interval I that supports such a function, where ⊳ is a canonical well-ordering of all ordinal subintervals. Then define s 0 and t 0 so that I 0 = I(s 0 (β, δ), t 0 (β, δ)). For α > 0 we define s α and t α by recursion on α : • If α is a successor ordinal λ + 1, then s α (β, δ) and t α (β, δ) are respectively the left and right end points of the ⊳-first ordinal subinterval I of the ordinal interval I(s λ (β, δ), t λ (β, δ)) such that f ↾ I is δ α -onto {x ∈ I(β, δ) : ψ α (x, β, δ)}.
Next we will show: Naturally, we use induction on α to verify (10). Proposition 2.9 and part (b) of Lemma 3.2 make it clear that the induction smoothly goes through for the base case and the successor case. The limit case requires the additional fact that if α is a limit ordinal, then by (9) there is some λ 0 < α such that the tail δ λ : λ 0 ≤ λ < α is a constant sequence.
To verify (11) recall that within M, f maps each interval I(s α (β, δ), t α (β, δ)) into {x : ψ α (x, β, δ)}. Therefore if some element m of M realizes p(x, β, δ), then f (m) realizes p(x, β, δ), which contradicts (4). We are now finally ready to wrap up the proof. Let It is evident that I is paradefinable in M. The complement of I can written as: which makes it clear that M \I is also paradefinable in M. Therefore by part (b) of Lemma 3.5 both I and its complement are Σ 1 1 -definable in (M, X) and thus I ∈ X, which implies that the supremum of I exists in M (since each element of X is separative over M). This contradicts (11) and concludes the demonstration that Case B is impossible.

PROOF OF THEOREM B
In this section we establish the second main result of this paper. (b) There is X such that (M, X) |= GB + ∆ 1 1 -CA + Σ 1 1 -AC. Proof. Suppose M is a nonstandard model of ZFC. By Theorem 3.1 (b) ⇒ (a) holds, so we will focus on establishing (a) ⇒ (b). This will be done in two stages.
Stage 1. We use forcing with set choice functions (as in Proposition 2.11) to expand M to a model (M, f ) that satisfies the following properties: (1) f is a global choice function over M, and (M, f ) |= ZF(L M ).
Part (b) of Proposition 2.11 assures us that (1) holds. The verification of (2) involves a careful choice of the generic global choice function. For this purpose we first verify Lemmas 4.2 and 4.3 below. In Lemma 4.2 the expression "α is a Beth-fixed point" means that α = (α), where is the Beth function. It is well-known that α is a Beth-fixed point iff V(α) is a Σ 1 -elementary submodel of the universe V of sets. Proof. Fix a nonstandard δ ∈ Ord M and consider the type p(x, δ) (where δ is treated as a parameter) consisting of the formula together with formulae of the form Ref ϕ (x) as in Theorem 2.7 (Reflection), where ϕ ranges in L M . It is easy to see that p(x, y) satisfies conditions (m1) and (m2) of part (m) of Definition 2.1. Moreover, by Proposition 2.7 (Reflection), p(x, δ) also satisfies condition (m3) of the same definition (since each closed and unbounded subset of ordinals has unboundedly many members of countable cofinality). Therefore by the assumption of W -saturation of M, p(x, δ) is realized in M by some γ, which makes it clear that γ is nonstandard and M(γ) ≺ L M M.
(Lemma 4.3) By Proposition 2.7 (Reflection) we can fix a sequence α n : n < ω that is cofinal in Ord M such that M(α n ) ≺ L M M and M |= cf(α n ) = ω. Then we build an L M -generic choice function f over M by recursively building a sequence of conditions p n : n < ω , as we shall explain. Thanks to Lemma 4.2 (applied within M) we can get hold of a condition p 1 whose domain is M (α 1 ) such that p 1 is L M(α 1 ) -generic over M(α 1 ). Generally, given a condition p n in M whose domain is M (α n ) and which is L M(α n ) -generic over M(α n ), we can use Lemma 4.2 to extend p n to a condition p n+1 whose domain is M (α n+1 ), and which is L M(α n+1 ) -generic over M(α n+1 ). Then by the choice of α n : n < ω , the union f of these conditions p n : n < ω will be L M -generic over M. Moreover, f will have the key property that f ↾ M (α n ) is L M(α n ) -generic over M(α n ) for every n < ω (and thus truth-and-forcing holds for each of these approximations). Then thanks again to truth-and-forcing, together with the fact that M(α n ) ≺ L M M for each n < ω, we can conclude: More explicitly, suppose (M(α n ), f ↾ M (α n )) |= ϕ(a) for some L M -formula ϕ(x) and some a ∈ M (α n ). Then for some condition p ∈ f ↾ M (α n ), we have M(α n ) |= [p ϕ(a)], and thus by elementarity M |= [p ϕ(a)], which by genericity of f assures us that (M, f ) |= ϕ(a). Note that ( * ) guarantees that the adjunction of the global choice function f to M preserves W -saturation and concludes Stage 1 of the proof.
Stage 2. Let f be the global choice function constructed in Stage 1, and let X = Def L M (M, f ), i.e., the family of subsets of M that are definable in (M, f ) by some L M -formula (parameters allowed).
We will treat f as a binary predicate so that variables are the only terms in L M (this will slightly simplify matters in the argument below). By part (b) of Proposition 2.11, (M, f ) |= ZF(L M ), which makes it clear that GBC holds in (M, X) . Recall from part (a) of Remark 2.10 that ∆ 1 1 -CA is provable in GB + Σ 1 1 -AC, and that in the presence of GBC, Σ 1 1 -AC is equivalent to Σ 1 1 -Coll. Hence in light of the fact that GBC holds in (M, X) the proof of (b) will be complete once we verify that Σ 1 1 -Coll holds in (M, X). For this purpose, suppose for some parameter A ∈ X we have (1) (M, X) |= ∀x ∃X ψ(x, X, A).
Let α(m, v) be the L M -formula that defines A, where m ∈ M is a set parameter. Then (2) (M, X) |= ∀x θ(x), where θ(x) := ϕ(y,v)∈L M ∃y ψ(x, X/ϕ(y, v), A/α(m, v)), and ψ(X/ϕ(y, v), A/α(m, v), x) is the result of replacing all occurrences of subformulae of the form w ∈ X (where w is a variable) in ψ by ϕ(w, v), and replacing all occurrences of subformulae w ∈ A in ψ by α(w, v). In these replacements, we will assume that some variables will be renamed to avoid unintended clashes.
Let D M (α) consist of all formulae of L ∞,ω of depth less than α that appear in M. We claim that (3) below holds.
Consider the L M -type p(x) := {¬θ α (x) : α ∈ o(M)} . It is easy to see that p(x) ∈ Cod W (M). By the assumption that (3)  is the class of sets that are hereditarily ordinal definable from the parameter x. More explicitly, it is well-known that ZFC + ∀x(V = HOD(x)) is consistent, assuming that ZF is consistent. 2 On the other hand, Kaufmann [K] showed, using the combinatorial principle ✸ ω 1 , that every countable model M 0 of ZF has an elementary end extension M that is recursively saturated and rather classless, and later Shelah [Sh] used an absoluteness argument to eliminate ✸ ω 1 . Here the rather classlessness of M means that if X is a subset of M that is piecewise coded in M, then X is parametrically definable in M (X is piecewise coded in M means that for every α ∈ Ord M , V M (α) ∩ X is coded by an element of M), then X is parametrically definable in M. Therefore if M is a recursively saturated rather classless model of ZFC + ∀x(V = HOD(x)), then by recursive saturation of M, M satisfies condition (a) of Theorem 4.1, but it does not satisfy condition (b) of Theorem 4.1 since if M expands to a model (M, X) of GB + Σ 1 1 -AC, then as pointed out in part (b) of Remark 2.10, there is a global choice function F coded in X. But the veracity of GB in (M, X) implies that F is piecewise coded in M and therefore F is parametrically definable in M, which contradicts the fact that ∀x(V = HOD(x)) holds in M.