Invisible Genericity and 0#

M.C. Stanley February 1995

Abstract. 0# can be invisibly class generic.

1. Introduction ...... 1 1.1. Class genericity ...... 2 1.2. H(A) ...... 4 2. Almost anything might be invisibly generic ...... 5 3. Two remarks ...... 12 4. The regularity of o(V) ...... 16 5. Open problems ...... 21 5.1. Is the real 0# generic? ...... 21 5.2. Non-generic reals ...... 23 6. References ...... 23

1. Introduction Roughly, the main theorem of this paper is that some instances of any type of non- constructible object are class generic over L, in one sense. Since small large cardinal properties are inherited in L, let us begin by considering the sense in which 0# can be generic. It is well known that 0# is not set generic over L, or, indeed, over any inner model M such that 0# ∈/ M. This is because, if 0# ∈/ M, then unboundedly many M-cardinals are collapsed in L[0#], a trick no set forcing can perform. It is also known that there can be no L-definable class forcing P such that 0# ∈ L[G] and hL[G]; Gi ² ZFC, for some L-definably generic G ⊆ P. This is because it is known that in such a case the forcing relation on a cone of conditions rooted in G must be L-definable because hL[G]; L, P,Gi ² ZFC; but then ³ ´ ∃p ∃ι p °P “ˇι is an indiscernible” and Lι ² ϕ defines L-truth over L. Further, it is known that there is no L-amenable class forcing property P such that 0# ∈ L[G] and hL[G]; P,Gi ² ZFC, for some hL; Pi-definably generic G ⊆ P. In such a case, again, the forcing relation on a cone of conditions rooted in G must be definable over hL; Pi. Since P is L-amenable (and compatible with 0#), the class P is definable over L[0#], and, in fact, the members of a tail of the indiscernibles are indiscernible in hL; Pi. But then, again, hL; Pi-truth is definable over hL; Pi.

Research supported by NSF grant DMS 9122320. I am grateful to Sy Friedman for pointing out a mistake in the first version, and to David Cook for pointing out an improvement regarding the definability of the forcing relation. Keywords: class forcing, zero sharp AMS Codes: 03E35, 03E40 2 1. INTRODUCTION

Theorem 1. Suppose that 0# exists and that there exists an inaccessible cardinal # in L[0 ]. Then there exists a countable ordinal κ and an Lκ-amenable partial ordering P such that hLκ; Pi ² ZFC and such that if G ⊆ P is hLκ; Pi-definably generic, then # Lκ[G] ² ZFC + “ 0 exists.” This confirms a conjecture of Sy Friedman. # # The subset of ω satisfying “x is 0 ” in Lκ[G] may or may not be the “real 0 ,” that is, the set assumed to exist in the hypothesis of Theorem 1. Note the omission of the predicates P and G in the conclusion of the theorem. It is in this sense that 0# can be “invisibly generic.” As suggested at the outset, the theorem proved in §2 is substantially more general than Theorem 1. For example, any large cardinal incompatible with V =L can be invisibly generic in the same sense. Theorem 2. Suppose that T ⊇ ZFC is a first-order theory in the language of set theory and that T has a countable standard transitive model V. Let κ be the least ordinal not in V and suppose that T ∈ Lκ and that κ is definably regular in Lδ, where δ is the least admissible ordinal greater than κ. Then there exists an Lκ-amenable forcing property PT such that (1) hLκ; PT i is a model of ZFC in the language {∈, P} that includes a predicate symbol for PT ; and (2) if G is PT generic for hLκ; PT i-definable predense classes, then Lκ[G] ² T . In addition to Theorem 2, this paper considers four related matters. The first two, which appear in §3, are remarks on the proof of Theorem 2. One is that if T 2 V =L, ω then T has 2 many non-isomorphic (in the language of set theory) PT -generic models. The other is that the forcing relation for PT is definable. The second two, which appear in §4, regard the hypothesis in Theorem 2 that κ is definably regular in Lδ. Proposition 4.1 states that κ must be regular in Lδ. Proposition 4.2 examines the extent to which κ must lose its regularity in the presence of a generic 0#. Very roughly, it holds that no “good” 0# can be “very generic.” Finally, in §5 some open problems are discussed. The following two subsections provide some definitions and references.

1.1. Class genericity As is customary, we systematically confuse a standard structure hM; ∈ ¹ M 2i with its universe M. Similarly, we confound partial orderings and their fields of conditions. Suppose that M is a transitive set or class. When we say that hM; · · ·i ² ZFC we mean that the standard structure M, augmented by the predicates, functions, or constants ··· , satisfies ZFC as formulated in a language with symbols for these predicates, functions, and constants. Specifically, ZFC in such a language includes all instances of collection and separation in the augmented language. Because of the importance of auxiliary predicates, rather than relying on this notation, the relevant language often will be indicated. Although, as the statements of the theorems above indicate, we shall be working largely with countable standard models of ZFC, in order to make our terminology INVISIBLE GENERICITY AND 0# 3 widely applicable, let us frame as much of it as possible in a second-order set theory (class theory) extending ZFC and sanctioning quantification over classes. Suppose that P is a partial ordering, perhaps a proper class. If G ⊆ P, then G is a filter if G consists of pairwise compatible conditions and G is closed upwards: p ∈ G whenever there exists a q ∈ G such that q 6 p. Suppose that M is an inner model, that is, a transitive class containing all of the ordinals and satisfying each axiom of ZFC. Define the class M P of Shoenfield terms by recursion: ˚a ∈ M P iff ˚a ⊆ M P × P and ˚a is a set of M. For Shoenfield terms ˚a, set n o dom(˚a) = ˚b :(˚b, p) ∈ ˚a, for some p and n o ˚a(˚b) = p ∈ P :(˚b, p) ∈ ˚a , for ˚b ∈ dom(˚a).

If G is a filter on P, define the value of the Shoenfield term ˚a by n o ˚aG = ˚bG :˚b ∈ dom(˚a) and ˚a(˚b) ∩ G 6= ∅ ; and set n o M[G] = ˚aG : ˚a ∈ M P .

Then M[G] is a transitive class. Suppose that x is a set. Then x is set generic over M if there exists a partial ordering P ∈ M and a filter G such that x ∈ M[G]\M and G∩D 6= ∅, for all predense D ⊆ P that are elements of M. Say that a set x is M-amenably generic when there exists an (M-amenable) partial ordering P and a filter G such that (1) x ∈ M[G] \ M; (2) hM; Pi satisfies ZFC in a language with predicate symbols for P; (3) G is hM; Pi-definably generic, that is, G ∩ D 6= ∅, for all D ⊆ P that are predense and definable over hM; Pi; and (4) hM[G]; M, P,Gi satisfies ZFC in a language with predicate symbols for M, P, and G. A set x is weakly generic when x is M-amenably generic over some inner model M. Say that a set x is invisibly generic if, for some inner model M, it satisfies the definition of M-amenable genericity with the following weakened clause (4): (40) M[G] satisfies ZFC in the language of set theory. So x’s genericity is perhaps “invisible” in M[G]. This last notion is dicey in class theory, since ZFC fails in hM[G]; M, P,Gi unless x is in fact weakly generic. Consequently, we shall work with set models when dealing with invisible genericity. Let us recapitulate in this terminology 0#’s status: It is known that 0# is neither set generic nor L-amenably generic. Theorem 1 is that it can be invisibly generic. It remains open whether 0# can be weakly generic. 4 INVISIBLE GENERICITY AND 0#

1.2. H(A) In this section we introduce some notation and state and give references for several standard facts from admissible set theory that figure in later sections. If X is a transitive set, let o(X) = X ∩ OR. Then o(X) is the least ordinal that does not lie in X. Suppose that A is a transitive set and that R1,...,Rn are amenable relations on A. Set A = hA; R1,...,Rni and define Lα(A) by

L0(A) = ∅; S ³ © ª © ª´ ¡ ¢ Lα+1(A) = a∈A cl Lα(A) ∪ Lα(A) ∪ Lα(A) ∩ a ∩ P Lα(A) ; and S Lα(A) = γ<α Lγ (A), if α is a limit ordinal. Here cl(X) is the closure of X under the usual G¨odeloperations, together with the

G¨odeloperations for R1,...,Rn, namely FRi (x) = x ∩ Ri. Say that A is power admissible when it satisfies KP in a language with a function symbol P for the power set operation; that is, A satisfies KP, including instances of Σ1-collection and Σ0-separation involving R1,...,Rn and P, as well as satisfies the axiom ¡ ¢ ∀x ∀y y ∈ P(x) ↔ y ⊆ x . Models of ZFC have natural expansions to power admissible sets. If A is power admissible, then A can be internally resolved into ranks Vα, for α < o(A): V0 = ∅; A¡ ¢ Vα+1 = P Vα ; and S Vα = γ<α Vγ , if α is a limit ordinal. A In other words Vα = Vα . [Our convention is to use blackboard bold for external names of standard transitive structures (and partial orderings). The internal/external distinction is admittedly obscure, since our point of view is continually shifting.] Note that Lα(A) = Vα, for α < o(A). Thus A = Lo(A)(A) and A ∈ Lo(A)+ω(A). Let α > o(A) be least such that Lα(A) is admissible and set

H(A) = Lα(A).

Then H(A) is the smallest admissible set with A ∈ H(A). For example, if A = Lκ, then H(A) = Lδ, where δ is the least admissible ordinal greater than κ. If X ⊆ A is first-order definable over A, then X ∈ H(A). (In fact, if A is countable, 1 then this holds for X that are ∆1 definable over A, but we shall not need this fact.) Suppose that C is a binary relation on some set A. Define the well-founded part of C by n ­ ® o wf(C) = b ∈ A : predC(b); C is well-founded , where pred (b) = { a ∈ A : a C b }. Define the rank function ρC on wf(C) as usual: C n o ρC(b) = sup ρC(a) + 1 : a C b . © ª Then set ρ(C) = sup ρC(b)+1 : b ∈ wf(C) . The following can be found in [B, §V.3]: INVISIBLE GENERICITY AND 0# 5

Lemma. Suppose C ⊆ A2 and C ∈ H(A). Then ¡ ¢ (a) ρ(C) 6 o H(A) ; (b) if wf(C) ∈ H(A), then ρ(C) ∈ H¡(A); and¢ (c) if wf(C) ∈/ H(A), then ρ(C) = o H(A) . Because H(A) is the smallest admissible set above A, the admissible set H(A) is projectible into A. A definition of this notion, together with a proof of the following fact can be found in [B, §V.5]: ¡ ¢ n o o H(A) = ρ(C): C ⊆ A2 is well-founded and C ∈ H(A)

2. Almost anything might be invisibly generic The main theorem of this paper is Theorem 2. Suppose that T ⊇ ZFC is a first-order theory in the language of set the- ory and that T has a countable standard transitive model V with o(V) = κ. Suppose as well that T ∈ Lκ and that κ is definably regular in H(Lκ). Then there exists an Lκ-amenable forcing property PT such that (1) hLκ; PT i is a model of ZFC in the language {∈, P} that includes a predicate symbol for PT ; and (2) if G is PT generic for hLκ; PT i-definable predense classes, then Lκ[G] ² T .

Again, let us remind the reader that Lκ[G] is the standard transitive structure G consisting of the values ˚a of the Shoenfield terms in the (Lκ-amenable) forcing lan- PT guage Lκ . We say that κ is definably regular in H(Lκ) when there is no H(Lκ)-definable cofinal function f: λ → κ, for some λ < κ; or equivalently, when κ remains regular in Lδ+1, where Lδ = H(Lκ). It follows from this hypothesis and the hypothesis that κ is o(V), where V is a model of ZFC, that κ is an inaccessible cardinal in H(Lκ). Note also that, for all n, we have that κ is the Σn-projectum of δ, where Lδ = H(Lκ). This observation will be used in the proof of Remark 3.1. The following corollary is immediate from Theorem 2. Theorem 1, that 0# can be invisibly generic, is a special case of it. Corollary 2.1. Suppose that ϕ is a formula in the language of set theory, that the theory ZFC + ∃xϕ(x) ` V 6=L, and that Vλ ² ZFC + ∃xϕ(x), for some inaccessible λ. Then there exists a countable ordinal κ and an Lκ-amenable forcing property P such that

(1) hLκ; Pi ² ZFC + ¬∃xϕ(x) and (2) if G is P generic for hLκ; Pi-definable predense classes, then Lκ[G] ² ZFC + ∃xϕ(x). That is, ϕ(x) is solvable in an invisibly generic extension of Lκ. The forcing property used to prove Theorem 2 will construct a generic Henkin model of the theory T . The work goes into arranging that exactly the elements of this model are denoted by Shoenfield terms. So there are two sorts of terms that we 6 2. ALMOST ANYTHING MIGHT BE INVISIBLY GENERIC shall need, namely, the logical terms needed to construct a Henkin model, and the Shoenfield terms whose denotations constitute the generic extension itself. Our first step is to introduce a one-place function symbol F to provide an adequate supply of logical terms. Let F be a one-place function symbol and let (†) be a formula of the language {∈,F } formalizing the following: ¡ ¢ ∀x ∃α α ∈ OR ∧ F (α) = x ∧ (†) ¡ ¢ ∀α ∀β α, β ∈ OR ∧ F (β) ∈ F (α) → β < α

Of course, any model of ZFC + V =L[0#] has a expansion by definitions to a model of ZFC + V =L[0#] + (†) given by interpreting F as F , where

½ th # x element of L[0 ] under < # , if x is an ordinal; F (x) = L[0 ] ∅, otherwise.

More generally we have Lemma 2.2. Suppose that T ⊇ ZFC is a theory in the language of set theory and that V is a countable standard transitive model of T . Then there exists a V-amenable function F such that hV; F i is a model of T + (†) + ZFC, where ZFC is formulated in the language {∈,F }. Proof: Working in V define a class forcing property F by

f ∈ F iff f is a one-to-one function with dom(f) ∈ OR and rng(f) = Vβ, for some β, and satisfying that β < α whenever f(β) ∈ f(α). f 0 6 f iff f 0 ⊇ f

Then F is <∞-closed. Using the axiom of choice in V, an induction on β shows that any Vβ can be absorbed into the range of a condition extendingS a given condition. If G is F generic for V-definable predense classes, then F = G ∪ { (x, ∅): x∈ / OR } is as required. Note that (†) does not require that F restricted to the ordinals is one-to-one, even though both examples given here (

Conditions will be pairs (r,~s ), where r is a finite piece of the finitary diagram of VG, and ~s is a finite sequence of approximations to the infinitary elementary diagram of VG. The idea is that the class of r’s added by G is accessible to Lκ[G] and so provides an adequate supply of Shoenfield terms, but the ~s ’s fit together to form an amenable ω-sequence that cannot be defined over Lκ[G]. The ~s ’s are used to hide the scratch work needed to construct VG.

Let us begin by setting up some notation and defining PT . Fix a first order theory T ⊇ ZFC in the language of set theory. Fix a countable ordinal κ such that T ∈ Lκ and κ is definably regular in H(Lκ) and T has a standard transitive model V with o(V) = κ. Working in H(Lκ), define X ⊆ κ + 1 by ³ ¡ ¢´ γ ∈ X iff γ 6 κ and Σ2-Skolem HullH(Lκ) γ ∪ {κ} ∩ κ = γ.

Then κ ∈ X and, because κ is definably regular in H(Lκ), the set X ∩ κ is unbounded in κ. Note that if γ ∈ X, then ¡ ¢ ∼ Σ2- Skolem HullH(Lκ) γ ∪ {κ} = H(Lγ ).

It is for this that we take Σ2-Skolem hulls, even though each element of H(Lκ) is Σ1-definable from parameters in κ ∪ {κ}. It follows that there exists a commutative system of Σ2-elementary embeddings { jγγ¯ :γ ¯ 6Sγ in X } such that jγγ¯ : H(Lγ¯) → H(Lγ ); jγγ¯ ¹ γ¯ = id ¹ γ¯; jγγ¯ (¯γ) = γ; and H(Lκ) = γ∈X∩κ rng(jγγ¯ ). Let L be the language with ≈ (equality) having the following symbols: • a two-place relation symbol ∈; • a one-place function symbol F ; and • constant symbols α, for α < κ.

We shall make use of admissible fragments LH(Lγ ), for γ ∈ X. (For example, see [B, Chapter III] regarding the syntax and semantics of these languages.) As is customary, we identify formulas with sets that code them. Let us assume that logical symbols, as well as the symbols ∈ and F , are coded by hereditarily finite sets, and that the symbol α is coded by α itself.

The non-logical symbols of the language LH(Lγ ) are ∈, F , and α, for α < γ. The formulas of LH(Lγ ) are those L∞ω formulas formed using these symbols and lying in the admissible set H(Lγ ) (and having only finitely many free variables).

A basic sentence of LH(Lγ ) is an atomic or negated atomic sentence.

For γ ∈ X, let θγ be an LH(Lγ ) sentence that results from conjoining formalizations of the following: • T ; • ZFC in the language {∈,F } ∪ { α : α < γ }; • (†); ¡ W ¢ • ∀x x ∈ β ↔ x ≈ δ , for each β < γ; and ¡ Wδ<β ¢ • ∀x x ∈ OR ↔ δ<γ x ≈ δ . 8 2. ALMOST ANYTHING MIGHT BE INVISIBLY GENERIC

By hypothesis we have that θκ is satisfiable, hence {θκ} is consistent in LH(Lκ). For γ ∈ X, declare that

s ∈ Sγ iff s is a finite set of LH(Lγ ) sentences and

s ∪ {θγ } is consistent in LH(Lγ ).

That is, a finite set s of L sentences is in S when there is no infinitary proof of the ¡ V H(¢Lγ ) γ sentence θγ ∧ s → 0≈1 . By Barwise Completeness, s ∈ Sγ iff H(Lγ ) ² “s ∈ Sγ .” Note also that s ∈ Sγ¯ iff jγγ¯ (s) ∈ Sγ , wheneverγ ¯ 6 γ in X. Declare that (r,~s ) ∈ P iff (r,~s ) meets the following four requirements:

(1) r is a finite set of basic LH(Lκ) sentences;

(2) ~s = h (si, γi): i < n i, for some n ∈ ω, where γi ∈ X ∩ κ and si ∈ Sγi ; and

(3) if i < k < n, then γi 6 γk and sk ⊇ jγiγk (si). Let krk be the least γ ∈ X such that r ⊆ L . Finally, require that γ ¡ ¢ (4) r ∪ jγn−1γ (sn−1) ∈ Sγ , if n = |~s | > 0 and γ ∈ X \ max γn−1, krk ; if |~s | = 0, simply require that r ∈ Skrk. Note that in clause (4) whether r ∪ j (s ) ∈ S does not depend on the choice ¡ ¢ γn−1γ n−1 γ of γ ∈ X \ max γn−1, krk . Order P by declaring that

(r0,~s 0) 6 (r,~s ) iff r0 ⊇ r and ~s is an initial segment of ~s 0.

Note that P is definable over H(Lκ) and so P is Lκ-amenable and hLκ; Pi ² ZFC in a language with a predicate symbol for P.

Lemma 2.3. Suppose that G is P generic for hLκ; Pi-definable predense classes. Then there exists a canonical countable standard transitive model hVG; FG, αiα<κ of θκ such that V hVG; FG, αiα<κ ² r iff (r, ∅) ∈ G, for all finite sets r of basic LH(Lκ) sentences.

Proof: Fix γ ∈ X. First, let us establish, roughly, that Sγ is a “consistency property” in the sense of [B].

(D0) ∅ ∈ Sγ

To see this, note first that since θκ is consistent and jγκ(θγ ) = θκ, we have that H(Lγ ) ² “θγ is consistent.” By Barwise Completeness it follows that {θγ } ∪ ∅ is, in fact, consistent.

Suppose that s ∈ Sγ . The syntax of the infinitary language LH(Lγ ) insures the following: (D1) If ϕ ∈ s, then (¬ϕ) ∈/ s.

(D2) If (¬ϕ) ∈ s, then s ∪ {∼ ϕ} ∈ Sγ . INVISIBLE GENERICITY AND 0# 9

Here ∼ ϕ denotes the result of pushingW theV initial negation in (¬ϕ) inside ϕ one step. (For example, ∼(¬ϕ) = ϕ and ∼ Φ = { ¬ϕ : ϕ ∈ Φ }.) V (D3) If Φ ∈ s, then s ∪ {ϕ} ∈ Sγ , for each ϕ ∈ Φ. (D4) If ∀v ϕ ∈ s, then s ∪ {ϕ(t/v)} ∈ S , for each closed L term t. W γ H(Lγ ) (D5) If Φ ∈ s, then s ∪ {ϕ} ∈ Sγ , for some ϕ ∈ Φ. 0 (D7) If t and t are closed LH(Lγ ) terms, then

(i) s ∪ {t ≈ t} ∈ Sγ ; 0 0 (ii) if (t ≈ t ) ∈ s, then s ∪ {t ≈ t} ∈ Sγ ; and 0 0 (iii) if ϕ(t/v), (t ≈ t ) ∈ s, then s ∪ {ϕ(t /v)} ∈ Sγ .

(D8) If ϕ is a sentence of LH(Lγ ), then either s ∪ {ϕ} ∈ Sγ or else s ∪ {(¬ϕ)} ∈ Sγ . A little argument is need to see

(D6) If ∃vϕ ∈ s, then s ∪ {ϕ(t/v)} ∈ Sγ , for some closed LH(Lγ ) term t.

Work in the real world, where H(Lγ ) is countable. Since {θγ } ∪ s is consistent, it has ¡ W ¢ A A a model A. And A ² ∀x x ∈ OR ↔ δ<γ x ≈ δ . Thus if d ∈ OR , then d = δ , for some δ < γ. Also A ² (†), so F A: ORA → V A is onto. Say A ² ϕ[a], where A A ¡ ¢ © ¡ ¢ª a = F (δ ). Then A ² ϕ F (δ)/v . Thus {θγ } ∪ s ∪ ϕ F (δ)/v is consistent in the real world, hence in H(Lγ ). Suppose now that G is P generic for hLκ; Pi-definable predense classes. Set

Sn o Γ = jγn−1κ(sn−1):(r,~s ) ∈ G and n = |~s | > 0 .

Note first that

(Γ0) θκ ∈ Γ.

Indeed, by (D0) we have that Sγ 6= ∅, for all γ ∈ X ∩ κ. And if s ∈ Sγ , then certainly s ∪ {θγ } ∈ Sγ . Because jγκ(θγ ) = θκ and G is generic, it suffices to see that n o (r,~s ) ∈ P : 0 < |~s | = n and θγ ∈ sn−1, where γ = γn−1

is dense in P. Suppose (r,~s ) ∈ P. We may assume that n = |~s | > 0. Setγ ¯ = γn−1 and ¡ ¢ 0 γ = min X \max(¯γ, krk) . Then r∪jγγ¯ (sn−1) ∈ Sγ . Let ~s be the sequence extending 0 0 0 ~s to length n + 1 such that γn = γ and sn = jγγ¯ (sn−1) ∪ {θγ }. Then (r,~s ) 6 (r,~s ) and lies in the required class of conditions. (Γ1) If ϕ ∈ Γ, then (¬ϕ) ∈/ Γ. 0 0 0 Suppose not. Say ϕ ∈ jγκ(s) and (¬ϕ) ∈ jγ0κ(s ), where (r,~s ), (r ,~s ) ∈ G; γ = γ|~s |−1 0 0 0 0 and s = s ; and γ = γ 0 and s = s 0 . Say γ 6 γ . Then ~s is an initial |~s |−1 |~s |−1 |~s |−1 ¡ ¢ 0 −1 −1 0 segment of ~s because G is a filter. Consequently, j 0 (ϕ) = j 0 j (ϕ) ∈ s . But ¡ ¢ γ κ γγ γκ −1 −1 0 also ¬jγ0κ(ϕ) = jγ0κ(¬ϕ) ∈ s , contradicting (D1). (Γ2) If (¬ϕ) ∈ Γ, then ∼ ϕ ∈ Γ. 10 2. ALMOST ANYTHING MIGHT BE INVISIBLY GENERIC

Say (¬ϕ) ∈ jγκ(s), where (r,~s ) ∈ G and |~s | = n and©s = sn−1 ªand γ = γn−1. We may assume that γ > krk. Then r ∪ s ∈ S , so r ∪ s ∪ ∼ j−1(ϕ) ∈ S by (D2). Let γ γκ γ © ª 0 −1 ~s be the sequence of length n + 1 extending ~s with γn = γ and sn = s ∪ ∼ jγκ (ϕ) . Then (r,~s 0) 6 (r,~s ). It follows from this observation that the hLκ; Pi-definable class n o 0 0 −1 0 (r ,~s ) ∈ P : ∼j (ϕ) ∈ s 0 γ|~s 0|−1κ |~s |−1 n o 0 0 ¡ −1 ¢ 0 0 = (r ,~s ) ∈ P : jγγ|~s 0|−1 ∼jγκ (ϕ) ∈ s|~s |−1 is dense below (r,~s ) in P. By genericity, we have (Γ2). Similarly, using (D3)–(D8) respectively, we have V (Γ3) If Φ ∈ Γ, then ϕ ∈ Γ, for all ϕ ∈ Φ. (Γ4) If ∀v ϕ ∈ Γ, then ϕ(t/v) ∈ Γ, for each closed term t of L . W H(Lκ) (Γ5) If Φ ∈ Γ, then ϕ ∈ Γ, for some ϕ ∈ Γ.

(Γ6) If ∃vϕ ∈ Γ, then ϕ(t/v) ∈ Γ, for some closed term t of LH(Lκ). 0 (Γ7) If t and t are closed LH(Lκ) terms, then (i)(t ≈ t) ∈ Γ; (ii) if (t ≈ t0) ∈ Γ, then (t0 ≈ t) ∈ Γ; and (iii) if ϕ(t/v), (t ≈ t0) ∈ Γ, then ϕ(t0/v) ∈ Γ.

(Γ8) If ϕ is a sentence of LH(Lκ), then either ϕ ∈ Γ or (¬ϕ) ∈ Γ.

In addition to (D3)–(D8)S and the genericity of G, propositions (Γ4), (Γ7i), and (Γ8) use that H(Lκ) = γ∈X∩κ rng(jγκ). Finally, note that

(Γ9) if ϕ is a basic sentence of LH(Lκ), then ϕ ∈ Γ iff ϕ ∈ r, for some (r,~s ) ∈ G. This observation uses that n © ª o (r,~s ) ∈ P : |~s | > 0 and r = s|~s |−1 ∩ basic sentences of LH(Lκ) is dense in P, and that jγκ(ϕ) = ϕ when ϕ is basic. Next, we use Γ to construct a model of θκ. Begin by defining a relation ≡ on closed

LH(Lκ) terms by t ≡ t0 iff (t ≈ t0) ∈ Γ. By (Γ7) we have that ≡ is an equivalence relation. Then there exists a canonical structure A in which elements of the universe are ≡-equivalence classes and

A ² ϕ iff ϕ ∈ Γ,

for all sentences ϕ of LH(Lκ). (This is established by induction on sentences using (Γ1)–(Γ8).) Furthermore, A ² θκ by (Γ0), so there exists a unique standard transitive ∼ model hVG; FG, αiα<κ = A. INVISIBLE GENERICITY AND 0# 11

Finally, suppose that r is a finite set of basic LH(Lκ) sentences. Then V V hVG; FG, αiα<κ ² r iff A ² r V iff r ∈ Γ iff r ⊆ Γ 0 0 0 iff r ⊆ jγn−1κ(sn−1), for some (r ,~s ) ∈ G, where |~s | = n > 0 iff r ⊆ r0, for some (r0,~s 0) ∈ G iff (r, ∅) ∈ G

The next to last equivalence uses that G is generic; the last, that G is a filter.

Lemma 2.4. Suppose that G is P generic for hLκ; Pi-definable predense classes. Then ˚ P ˚ G for each α < κ there exists a Shoenfield term F (α) ∈ Lκ such that FG(α) = F (α) . Thus VG ⊆ Lκ[G].

Proof: Set ½ ¾ ³ ¡ ¢´ F˚(α) = F˚(β), {F (β) ∈ F (α)}, ∅ : β < α .

˚ ˚ P Then F is an Lκ-definable function and F (α) is a Shoenfield term in Lκ, for each α < κ. (Note that { θκ,F (β) ∈ F (α) } is consistent.) G Proceed by induction on α to see that F˚(α) = FG(α). Note that

FG(β) ∈ FG(α) iff hVG; FG, αiα<κ ² F (β) ∈ F (α) ³© ª ´ iff F (β) ∈ F (α) , ∅ ∈ G.

G Thus F˚(α) ⊆ FG(α). For the converse inclusion, note as well that if a ∈ FG(α), then a = FG(β), for some β < α, because hVG; FG, αiα<κ ² (†).

P Lemma 2.5. If ˚a is a Shoenfield term in Lκ and G is P generic for hLκ; Pi-definable G predense classes, then ˚a ∈ VG. Thus Lκ[G] ⊆ VG. Proof: Let us call a Shoenfield term ˚a special when, hereditarily, only conditions of the form (r, ∅) occur in ˚a. That is, ¡ ¢ ˚a is special iff ~s = ∅ and ˚b is special, for all ˚b, (r,~s ) ∈ ˚a.

The following two claims suffice to prove the lemma.

G Claim. If ˚a is a special Shoenfield term, then ˚a ∈ VG.

Proof: Fix a special Shoenfield term ˚a. Because P is Lκ-amenable, n ¡ ¢ o ˚ ˚ b ∈ TC {˚a} : b is a special Shoenfield term ∈ Lκ, 12 2. ALMOST ANYTHING MIGHT BE INVISIBLY GENERIC ¡ ¢ where TC {˚a} is the transitive closure of {˚a}. Now n o ˚bG = ˚cG : ˚c ∈ dom(˚b) and ˚b(˚c) ∩ G 6= ∅ , and for ˚c ∈ dom(˚b),

˚ V b(˚c) ∩ G 6= ∅ iff hVG,FG, αiα<κ ² r, for some finite set r of basic sentences such that (r, ∅) ∈˚b(˚c).

Using that hVG; FG, αiα<κ satisfies ZFC in the language {∈,F } ∪ { α : α < κ }, it G follows by induction on special terms in TC({˚a}) that ˚a ∈ VG.

Claim. Suppose that ˚a is a Shoenfield term and that G is hLκ; Pi-definably generic. Then there exists a special Shoenfield term ˚a∗ such that ˚aG = ˚a∗ G.

Proof: Fix a Schoenfiled term ˚a. It suffices to see that there exists an hLκ; Pi- definable dense class of conditions p00 such that for some special Shoenfield term ˚a∗, 00 G ∗ G if G is hLκ; Pi-definably generic and p ∈ G, then ˚a = ˚a . Pick any condition p = (r,~s ). We may assume that |~s | > 0.¡ Say¢ |~s | = n and set γ¯ = γ . Choose γ ∈ X such that γ > krk, γ¯ and also γ > TC {˚a} ∩ OR. n−1 ¡ ¢ 00 00 00 00 Set p = (r,~s ), where ~s = ~sÄ jγγ¯ (sn−1), γ . Then¡ ¢p 6 p in P. Furthermore, by our choice of γ and definition of ~s 00, if (r0,~s 0) ∈ TC {˚a} is compatible with p00, then 0 00 the sequence ~s is an initial segment¡ ¢ of the sequence ~s . For Shoenfield terms ˚b in TC {˚a} , set

n ¡ ¢ ¡ ¢ o ˚b∗ = ˚c∗, (r0, ∅) : ˚c, (r0,~s 0) ∈˚b and (r0,~s 0) is compatible with p00 .

Then ˚b∗ is a special term, and if q 6 p00, then

q 6 (r0,~s 0) ∈˚b(˚c), for some ~s 0 iff q 6 (r0, ∅) ∈˚b∗(˚c∗). ¡ ¢ ˚ It follows by induction on b in TC {˚a} that if G is hLκ; Pi-definably generic and p00 ∈ G, then ˚bG =˚b∗ G. This completes the proof of Theorem 2.

3. Two remarks This section contains two remarks on Theorem 2. The first of these is that under the ω hypotheses of Theorem 2, the theory T + θκ has 2 many distinct invisibly generic models iff T + θκ 2 V =L. The reader may recall that T is part of θκ. Nevertheless, we shall write ‘T + θκ’ where ‘θκ’ would suffice to render vividly that we are working with a certain infinitary extension of T . Our second remark is that the forcing relation for PT is uniformly definable over hLκ; PT i. INVISIBLE GENERICITY AND 0# 13

Suppose that the countable ordinal κ is definably regular in H(Lκ), that T ∈ Lκ is a theory in the language of set theory including ZFC, and that T + θκ is satisfiable. It is not difficult to see that the second order predicate “G is PT generic for hLκ; PT i- 1 definable¡ ¢ predense classes” is Σ1 over H(Lκ). It follows that either every such G lies in H H(Lκ) or else there exist perfectly many such G’s, c.f. [B, Theorem IV.4.4]. Using an idea of H. Friedman’s [HF, Theorem 6.2], though, we can prove a sharper result.

Remark 3.1. Suppose that the countable ordinal κ is definably regular in H(Lκ), that T ∈ Lκ is a theory in the language of set theory including ZFC, and that T + θκ is satisfiable.

(1) If T +θκ ² V =L, then any two models of T +θκ are isomorphic in the language of set theory. ω (2) If T + θκ 2 V =L, then there exist 2 many pairwise non-isomorphic (in the language of set theory) invisibly generic models of T + θκ.

Proof: Claim (1) is clear. For (2), let V be a model of T + θκ + V 6=L. Say V satisfies ϕ, where the sentence ϕ formalizes “F (α) ⊆ λ and F (α) ∈/ L,” for some α and λ.

Claim. Suppose that (r,~s ) ∈ PT ¡is such that n = |¢~s | > 0¡ and ϕ ∈ sn−1. Then¢ there exists a δ < λ such that both r ∪ { δ ∈ F (α) },~s and r ∪ { δ ∈/ F (α) },~s are conditions extending (r,~s ) in PT . Proof: Otherwise, set n ¡ ¢ o x = δ < λ : r ∪ {δ ∈ F (α)},~s ∈ PT . © ª Set γ = γn−1. Then δ ∈ x iff the infinitary theory T ∪{θκ}∪r∪jγκ(sn−1)∪ δ ∈ F (α) is consistent in LH(Lκ). Thus x is Π1-definable over H(Lκ) and x ⊆ λ, which is less than κ, the Σ1-projectum. Consequently x ∈ H(Lκ). Again, since x ⊆ λ < κ, it follows that x ∈ Lκ. VG Now if G is PT generic and (r,~s ) ∈ G, then F (α) = x and so VG ² “F (α) ∈ L,” contradicting that VG ² ϕ. ω It follows that for each of 2 many different x ⊆ λ, there exists an hLκ; PT i-definably generic Gx such that x ∈ Lκ[Gx]. Furthermore, each cell is countable when the set of such Lκ[Gx]’s is partitioned by isomorphism in the language of set theory, because ω each Lκ[Gx] is countable. Consequently, there exist 2 many pairwise non-isomorphic such Lκ[Gx].

We turn now to our second remark, namely, that the forcing relation for PT is definable. This is perhaps surprising since predensity reduction fails for PT and, in fact, generic classes G blatantly ω-cofinalize the ordinals of Lκ. It also shows that what is at issue in deciding whether 0# can be weakly generic is not the definability of the forcing relation. Originally, I proved that the forcing relation for PT is definable over hLκ; PT i, when it restricted to formulas of bounded logical complexity. David Cook extended that argument to show that, in fact, the forcing relation is uniformly definable for all sentences of the forcing language, noting that this does not immediately imply 14 3. TWO REMARKS that L[G]-truth is uniformly definable over L[G], because PT and G are not definable over L[G]. Even at bit more is true. In the sense made precise by Remark 3.2, the forcing relation is uniformly definable for all sentences of an infinitary forcing language with formulas lying in H(Lκ).

Fix T and set P = PT . The ordinary forcing language for P is the finitary first order language without equality having the following non-logical symbols: two-place relation P symbols ∈ and ⊆; and constant symbols ˚a, for each Shoenfield term ˚a ∈ Lκ. As usual, we systematically confuse symbols and formulas with sets coding them in Lκ. In the forcing language we treat equality as a defined relation; that is, t ≈ t0 abbreviates (t ⊆ t0) ∧ (t0 ⊆ t).

In contrast to visible genericity, the standard structure Lκ[G] cannot uniformly define satisfaction for atomic sentences of the forcing language, because the classes P ˚ Lκ and G are not available. Working outside of Lκ[G], declare that Lκ[G] ² b ∈ ˚a ˚ ˚G G G ˚G and Lκ[G] ² ˚a ⊆ b when b ∈ ˚a and when ˚a ⊆ b , respectively. Then extend ² to non-atomic sentences as usual. Recall from Lemma 2.5 that special Shoenfield terms are terms that, hereditarily, contain only conditions of the form (r, ∅). Fix any γ ∈ X and work in H(L ). Say ¡ ¢ γ ˚ that ˚a is a special pseudo-term when ˚a ∈ Lγ and ˚a is a set of pairs b, (r, ∅) , where r ˚ is any finite set of basic LH(Lγ )-sentences and b is a special pseudo-term. In particular, consistency of r with θγ is not required, so special pseudo-terms are not Shoenfield terms, in general. Of course, every special© term is a special pseudo-term.ª The advan- tage that this notion enjoys is that ˚a ∈ Lγ : ˚a is a special psuedo-term is always a set in H(Lγ ).

Next, let us define an infinitary forcing language KH(Lγ ). Let K comprise the symbols ∈, ⊆, and ˚a, for all special pseudo-terms ˚a. Let KH(Lγ ) be the usual infinitary language, namely, the intersection of K∞ω with H(Lγ ). Then KH(Lγ ) is ∆1-definable over H(L ). Consequently, j (ϕ) is a formula in K whenever ϕ is a formula γ γκ S H(Lκ) in KH(Lγ ). Furthermore, KH(Lκ) = γ∈X∩κ jγκ”KH(Lγ ).

Say that a formula ϕ of KH(Lγ ) is legitimate when every special pseudo-term occur- P∩Lγ ring in ϕ is a genuine (special) Shoenfield term in Lγ .

Remark 3.2. There exists an hLκ; Pi-definable three-place relation ° (defined on triples (p, γ, ϕ) such that p ∈ P, γ ∈ X ∩ κ, and ϕ is a legitimate KH(Lγ )-sentence) having the property that if G is an hLκ; Pi-definably generic filter on P, then

Lκ[G] ² jγκ(ϕ) iff ∃p∈G p °γ ϕ, whenever γ ∈ X ∩ κ and ϕ is a legitimate KH(Lγ )-sentence.

Of course, if ϕ is a finitary legitimate KH(Lκ)-sentence, then ϕ ∈ Lγ , for any sufficiently large γ ∈ X ∩ κ, and jγκ(ϕ) = ϕ. Because in a generic extension each Shoenfield term is forced equal to some special Shoenfield term (in the sense of the second Claim in Lemma 2.5), it follows that the forcing relation for first order sentences of the ordinary forcing language is uniformly definable. INVISIBLE GENERICITY AND 0# 15

Proof: Fix any γ ∈ X. Working in H(Lγ ), define by recursion an operation associ- ating a Boolean combination of basic LH(Lγ )-sentences to each KH(Lγ )-sentence: W ³³W V ´ ´ ˚ γ ˚ γ [[b ∈ ˚a]] = ˚c∈dom(˚a) (r,∅)∈˚a(˚c) r ∧ [[b ≈ ˚c]] V ³³W V ´ ´ ˚ γ ˚ γ [[˚a ⊆ b]] = ˚c∈dom(˚a) (r,∅)∈˚a(˚c) r → [[˚c ∈ b]] V γ V γ [[ Φ]] = ϕ∈Φ[[ϕ]] W γ W γ [[ Φ]] = ϕ∈Φ[[ϕ]] [[¬ϕ]]γ = ¬[[ϕ]]γ V γ ˚ γ [[∀v ϕ]] = α<γ [[ϕ(F (α)/v)]] , where F˚ is as in Lemma 2.4.

Claim. Suppose that G is an hLκ; Pi-definably generic filter on P. Then κ Lκ[G] ² ϕ iff [[ϕ]] ∈ Γ, for all legitimate sentences ϕ of KH(Lκ). Proof: Proceed by induction on the complexity of ϕ to unwind definitions, using G properties (Γ1)–(Γ8) and that every element of Lκ[G] is of the form F˚(α) , for some α < κ. Let us illustrate this with two cases:

˚ ˚G G Lκ[G] ² b ∈ ˚a iff b ∈ ˚a ³ ´ iff ∃˚c∈ dom(˚a) ˚a(˚c) ∩ G 6= ∅ and ˚bG = ˚cG ³ ´ iff ∃˚c∈ dom(˚a) ∃(r, ∅)∈˚a(˚c)(r, ∅) ∈ G and ˚bG = ˚cG ³ V ´ iff ∃˚c∈ dom(˚a) ∃(r, ∅)∈˚a(˚c) r ∈ Γ and [[˚b ≈ ˚c]]κ ∈ Γ W ³³W V ´ ´ ˚ κ iff ˚c∈dom(˚a) (r,∅)∈˚a(˚c) r ∧ [[b ≈ ˚c]] ∈ Γ iff [[˚b ∈ ˚a]]κ ∈ Γ. ¡ ¢ Lκ[G] ² ∀v ϕ iff Lκ[G] ² ϕ F˚(α)/v , for all α < κ ¡ ¢ iff [[ϕ F˚(α)/v ]]κ ∈ Γ, for all α < κ V ¡ ¢ ˚ κ iff α<κ[[ϕ F (α)/v ]] ∈ Γ iff [[∀v ϕ]]κ ∈ Γ.

γ Using that [[ϕ]] is defined over H(Lγ ) by Σ1-recursion, note that if γ ∈ X ∩ κ, then

¡ γ ¢ κ jγκ [[ϕ]] = [[jγκ(ϕ)]] .

It follows that if ϕ is a legitimate sentence in KH(Lκ), then

κ −1 γn−1 [[ϕ]] ∈ Γ iff [[jγn−1κ(ϕ)]] ∈ sn−1, for some (r,~s ) ∈ G with 0 < |~s | = n. 16 3. TWO REMARKS

Working in hLκ; Pi define the three-place relation ° by declaring that p °γ ϕ iff p ∈ P and γ ∈ X ∩ κ and ϕ is a legitimate sentence of KH(Lγ ) and n o ¡ γ ¢ (r,~s ) : 0 < |~s | = n and γn−1 > γ and jγγn−1 [[ϕ]] ∈ sn−1 is predense with respect to p. Then ° is as required.

4. The regularity of o(V)

Theorem 2’s most curious hypothesis is that κ is definably regular in H(Lκ); the most disappointing feature of its proof is that G collapses κ. It is natural to ask whether these are essential features of Theorem 2, or merely accidental ones of the proof given in §2. The first of these questions can be given a fairly satisfactory answer without much trouble. In the proof only Σn-definable regularity, for sufficiently large n, is used. On the other hand, κ must be regular in H(Lκ) in the usual sense that no function that is an element of H(Lκ) renders κ singular: Proposition 4.1. Suppose that κ is a countable ordinal and that there exists a real # x such that Lκ[x] satisfies ZFC + “x is 0 .” Then κ is regular in H(Lκ). Proof: Fix such a real x and let “0#” denote it. Let δ be the least admissible ordinal # greater than κ, so Lδ = H(Lκ). Let A be the Ehrenfeucht-Mostowski (0 , κ+ω)- model. Identify wf(A), the well-founded part of A, with the transitive set to which it¡ is isomorphic.¢ Now wf(A) must be an admissible set [B, Truncation Lemma] and o wf(A) > κ¡because¢ a well ordering of type κ lies in A. Thus Lδ ⊆ A, though possibly δ = o wf(A) . Suppose now that f: λ → κ, for some λ < κ, and that f ∈ Lδ, hence f ∈ A. A We maintain that the range of f is bounded in κ. Say f = t (ι1, . . . , ιn, γ1, . . . , γk), A A where t is a term and ι1 < ··· < ιn < κ and κ = γ1 < ··· < γk are indiscernibles in the sense of A. (Of course, some of the γi’s need not be genuine ordinals.) Let γ¯1 < ··· < γ¯k be indiscernibles in the sense of A such that λ, ιn < γ¯1 < κ. Set ¯ A ¯ ¯ f = t (ι1, . . . , ιn, γ¯1,..., γ¯k). Then A ² f: λ → γ¯1. But A ² f(α) = f(α), for each α < λ by indiscernibility. Hence rng(f) ⊆ γ¯1 < κ. We cannot give as satisfactory an¡ answer regarding¢ the extent to which κ must lose its regularity. Note that G ∈ H hL ; P i [G] in the proof of Theorem 2, since ¡ ¢ κ T ¡ ¢ ¡ ¢ G˚ = { (ˇp, p): p ∈ PT } ∈ H hLκ; PT i . [Incidentally, H hLκ; PT i = H H(Lκ) = Lδ, where δ is the second¡ admissible¢ ordinal above κ.] It follows¡ that κ has¢ definable cofinality ω in H hLκ; PT i [G], indeed, has cofinality ω, if H hLκ; PT i [G] satisfies Σ0-separation. Only with an additional hypothesis—and some effort—can we give an upper bound on how long o(V) can remain regular. Say that a filter G ⊆ P is hyper-generic over the countable standard transitive¡ ¢ model hV; Pi when G ∩ D 6= ∅, for all predense D that are definable over H hV; Pi . Of course if G is hyper-generic over hV; Pi, then G is hV; Pi-definably generic. ¡ ¢ # # # V[G] If V[G] ² ZFC + “0 exists,” let 0G be 0 . INVISIBLE GENERICITY AND 0# 17

Proposition 4.2. Suppose that hV; Pi is a countable standard transitive model of # ZFC + “0 does not exist,” that G is hyper-generic over hV; P¡i, and that¢ V[G] satisfies ZFC + “0# exists.” Then o(V) is not definably regular in H hV; Pi [G]. The proof will be recycled in the next section to prove Proposition 5.1. ¡ ¢ ¡ Proof: Set λ = o(V). Note that¡ λ, V, ¢P ∈ H hV; Pi . Because P ∈ H hV; Pi) and G is hyper-generic, we have that H hV; Pi [G] is admissible. (This is the only use made of G’s hyper-genericity.) We also have that G ∈ H(hV; Pi)[G]. The proof¡ comes¢ in three parts. Assuming for a contradiction that λ is definably regular in H hV; Pi [G], Part I reflects a long definable well-founded relation down into the universe V[G]. Part II uses this to show that then 0# must lie in V. The argument in Part II uses that if κ is a cardinal of uncountable cofinality in L[0#], then any club subset of κ that is eventually contained in every constructible club subset of κ constructs 0#. This is proved in Part III. ¡ ¢ Part I. Suppose for a contradiction that λ is definably¡ regular¢ in H hV; Pi [G]. Then there exists a Σ2-elementary substructure X of H hV; Pi [G] such that λ, V, P,G ∈ X and X∩V[G] = Vκ[G], where κ = X∩λ is some limit cardinal of uncountable cofinality in V[G]. Note that κ is inaccessible in V, since V ² “0# does not exist.” 0 ∼ 0 0 Set A = hVκ; P ∩ Vκi and G = G ∩ Vκ. Then X = H(A)[G ] and H(A)[G ] is admissible. [Checking this requires some¡ calculations.¢ Note that H(A) is Σ1 definable over H(A)[G0] and that H(A)[G] ² ∀x∃˚a x = ˚aG .] Working in Vκ, define n o ORVκ · ω = (i, α): i ∈ ω and α is an ordinal and let ORVκ · ω be ordered lexicographically (in type κ · ω). Set n o Vκ B = t(b1, . . . , bn): t is a definable term of ZFC + V =L and b1, . . . , bn ∈ OR ·ω .

0 In Vκ[G ] define a relation E on B by 0 0 0 0 # t(b1, . . . , bn) E t (b1, . . . , bm) iff pt(vi1 , . . . , vin ) ∈ t (vj1 , . . . , vjm )q ∈ 0G0 , where the sequence of natural numbers hi1, . . . , in, j1, . . . , jmi is ordered just as the 0 0 Vκ 0 sequence hb1, . . . , bn, b1, . . . , bmi is ordered in OR · ω. Then hB,Ei is a Vκ[G ]- ¡ # ¢ definable version of the Ehrenfeucht-Mostowski 0G, κ · ω -model. ∼ th Working in V[G] we have that hB,Ei = Lδ, where δ is the (κ · ω) Silver indis- # cernible provided by 0G. Hence hB,Ei is well-founded. Because¡ hB,E¢ i is definable 0 0 0 over Vκ[G ], we have that hB,Ei ∈ H(A)[G ] and so δ < o H(A)[G ] , since hB,Ei is actually well-founded and δ is its rank. Furthermore, the isomorphism of hB,Ei and 0 Lδ lies in H(A)[G ], by ∆1-separation. Thus we have ¡ ¢ ¡ ¢ • o H(A)[G0] = o H(A) ; • κ = o(A) is inaccessible in H(A) and is a limit cardinal of uncountable cofi- nality in H(A)[G0]; and ¡ ¢ ∼ 0 • hB,Ei = Lδ, for some δ < o H(A) , and this isomorphism lies in H(A)[G ]. In fact, H(A)[G0] ² “κ is inaccessible,” but we don’t need this. 18 4. THE REGULARITY OF O(V)

Part II. Choose C ∈ H(A) to be a well-founded relation on Vκ having rank δ. Let ρC: Vκ → δ be the rank function for C: n o ρC(a) = sup ρC(b) + 1 : b C a

Then ρC ∈ H(A) by ∆1-separation. th Define h Ca : a ∈ Vκ i by letting Ca be the ρC(a) element of Lδ under

Set K = 4α<κ Kα. Then the club set K ⊆ κ lies in H(A) and is eventually contained in each club C ⊆ κ such that C ∈ Lδ. 0 0 Work in H(A)[G ]. Because the isomorphism of hB,Ei with Lδ lies in H(A)[G ], there exist Silver indiscernibles I for Lδ having order type κ · ω. Let h ιξ : ξ < κ · ω i enumerate I monotonically and let the elementary embedding j: Lδ → Lδ be induced by j(ιξ) = ικ+ξ, for ξ < κ · ω.

If C¯ is a club subset of ι0 and lies in Lδ, then j(C¯) is club in ικ = κ and lies ¯ Lδ in Lδ¡. Consequently, K is eventually contained in j(C). Now P (ι0) is countable in H A)[G0] and κ has uncountable cofinality in H(A)[G0]. Thus there exists a β < κ such that Tn o K \ β ⊆ j(C¯): C¯ ∈ Lδ is club in ι0 .

It follows by Lemma 4.3.5 below that K \ β ⊆ I ∩ κ. Hence K \ β is an unbounded set of indiscernibles for Lκ. Since κ is inaccessible in V and since K ∈ H(A) ⊆ V, it follows that V ² “0# exists.”

Part III: Fast closed unbounded sets This final part of the proof of Proposition 4.2 is independent of the rest of this paper. It is devoted to proving the following elementary fact, which was used above in Part II:

Proposition 4.3. Suppose that 0# exists and that κ is a cardinal of uncountable cofinality in L[0#]. Suppose that C is a closed unbounded (club) subset of κ that is eventually contained in every constructible club subset of κ. Then 0# ∈ L[C].

Lemma 4.3.1. Suppose that κ is inaccessible in L and that U is an L-ultrafilter on κ. Suppose that f: κ → κ is a constructible function such that { α < κ : f(α) > α } ∈ U. Then there exists a constructible club C ⊆ κ such that { α < κ : f(α) ∈/ C } ∈ U.

Proof: Set C = { β < κ : f”β ⊆ β }. Then C is constructible club subset of κ and { α < κ : f(α) ∈ C } ⊆ { α < κ : f(α) 6 α } ∈/ U. INVISIBLE GENERICITY AND 0# 19

Lemma 4.3.2. Suppose that j: L → L is elementary with critical point ¯ι and that j(¯ι) = ι. If C ⊆ ι is a constructible club set, then there exists a constructible club C¯ ⊆ ¯ι such that j(C¯) is eventually contained in C.

Proof: Let I be¡ the class of Silver¢ indiscernibles provided by j and suppose that C ∈ Skolem HullL ι ∪ {ι, δ1, . . . , δk} , where δ1, . . . , δk ∈ I \ (ι + 1). Let h D¯ α : α < ¯ι i be a constructible enumeration of n ¡ ¢ o D¯ ∈ Skolem HullL ¯ι ∪ {¯ι, δ1, . . . , δk} : D¯ ⊆ ¯ι is club . ¡ ¢ ¯ ¯ ¯ ¯ Set C = 4α<¯ι Dα. Then j(C) = 4α<ι Dα, where h Dα : α < ι i = j h Dα : α < ¯ι i is a constructible enumeration of n ¡ ¢ o D ∈ Skolem HullL ι ∪ {ι, δ1, . . . , δk} : D ⊆ ι is club .

Corollary 4.3.3. Suppose that j: L → L is elementary and that ι = j(¯ι). Suppose that f: ι → ι is constructible and such that { α < ι : f(α) > α } ∈ U, where U is an L-ultrafilter on ι. Then there exists a constructible club C¯ ⊆ ¯ι such that { α < ι : f(α) ∈/ j(C¯) } ∈ U. Proof: If j is trivial, then the corollary just restates Lemma 4.3.1. Otherwise, use Lemma 4.3.2, as well.

Let us next introduce some notation. First of all, if ¯ι < ι are indiscernibles, let j¯ιι be the embedding induced by omitting I ∩ [¯ι, ι). That is, let h ιξ : ξ ∈ OR i enumerate the Silver indiscernibles I monotonically; if ¯ι < ι in I, say ¯ι = ια and ι = ιβ, let j¯ιι: L → L be the elementary embedding induced by ½ ιξ, if ξ < α; j¯ιι(ιξ) = ιβ+ζ , if ξ = α + ζ.

Next, let us introduce some notation regarding L-ultrapowers. If j: L → L is any elementary embedding and κ is the critical point of j, then define a normal L-ultrafilter on κ by

X ∈ Uj iff X is a constructible subset of κ and κ ∈ j(X).

Let Ult(L, Uj) be the usual definable class ultrapower of L by Uj. Then e: x 7→ [idx] is an elementary embedding L → Ult(L, Uj), where [idx] is (a Scott representative of) the equivalence class of the function idx with domain κ, taking the value x identically.

Also as usual, let σ: Ult(L, Uj) →Σω L by ¡ ¢ σ [f] = j(f)(κ).

∼ Finally, let µ: σ” Ult(L, Uj) = L be the transitive collapse of the range of σ. Define ˆ to be the compositions of these functions: ˆ = µ ◦ σ ◦ e. Then ˆ: L →Σω L. 20 4. THE REGULARITY OF O(V)

Lemma 4.3.4. Suppose that 0# exists and that ι is an indiscernible. Let ι∗ be the least indiscernible greater than ι and set j = jιι∗ . Then ˆ = j.

Proof: First note that for any non-trivial j: L →Σω L we have that j = σ◦e. Indeed, if the critical point of j is κ, then ¡ ¢ (σ ◦ e)(x) = σ [idx]

= j(idx)(κ) = j(x).

Thus when j = jιι∗ it suffices to see that µ is the identity function, that is, that rng(σ) = L. Now rng(j) ⊆ rng(σ) by the above calculation. Thus I \{ι} ⊆ rng(σ). It suffices to see that ι ∈ rng(σ). For this, we need a constructible f: ι → ι such that j(f)(ι) = ι. The diagonal function f(α) = α is such a function. Lemma 4.3.5. Let I be the class of Silver indiscernibles. Suppose that j: L → L is non-trivial and elementary with critical point ¯ι. Set ι = j(¯ι) and Tn o E = j(C¯): C¯ ⊆ ¯ι is a constructible club . Then E = I ∩ [¯ι, ι). Proof: If C¯ ⊆ ¯ι is a constructible club, then ¯ι ∈ j(C¯). It follows by indiscernibility that I ∩ [¯ι, ι) ⊆ E. ¡ ¢ Given δ < ¯ι, the interval (δ, ¯ι) is club in ¯ι and j (δ, ¯ι) = (δ, ι). Consequently δ∈ / E. Finally, we must see that if δ ∈ (¯ι, ι) \ I, then δ∈ / E. Suppose that ¯ι 6 ι0 < δ < ι00 6 ι, where ι00 is the least indiscernible greater than ι0.

Now j: L →Σω L is induced by the order-preserving shift of the indiscernibles j ¹ I. It follows that there exists an elementary j0: L → L such that j0 ¹ ι = id ¹ ι and 0 00 0 00 0 0 00 j = j ◦ j¯ιι. In turn, j¯ιι = jι ι ◦ jι ι ◦ j¯ιι . Set U = Ujι0ι00 . Then jι ι = σ ◦ e, where

e σ L −→ Ult(L, U) −−−→ L. onto ¡ ¢ Choose f ∈ L such that f: ι0 → ι0 and δ = σ [f] . Since ι0 < δ, n o α < ι0 : f(α) > α ∈ U.

By Corollary 4.3.3, there exists a constructible club C¯ ⊆ ¯ι such that n o 0 α < ι : f(α) ∈/ j¯ιι0 (C¯) ∈ U.

Then ¡ ¢ Ult(L, U) ² [f] ∈/ e j¯ιι0 (C¯) , hence ¡ ¢ L ² δ∈ / σ e(j¯ιι0 (C¯)) = j¯ιι00 (C¯). ¡ ¢ 0 00 00 0 Now j ◦ jι00ι ¹ ι = id ¹ ι . Thus δ∈ / j jι00ι(j¯ιι00 (C¯)) = j(C¯) and so δ∈ / E. INVISIBLE GENERICITY AND 0# 21

Proof of Proposition 4.3: Work in L[0#]. Note that κ is an indiscernible, since # κ is a cardinal in L[0 ]. Let ¯ι be the least indiscernible and set j = j¯ικ. By hy- pothesis the club C ⊆ κ is eventually contained in j(C¯), for each constructible club C¯ ⊆ ¯ι. But there are only countably many such C¯’s and κ has uncountable cofinality. Consequently, there exists β < κ such that

Tn o C \ β ⊆ j(C¯): C¯ ⊆ ¯ι is a constructible club = I ∩ [β, κ).

It follows that 0# ∈ L[C].

5. Open problems 5.1. Is the real 0# generic? In order to make such a question precise—lacking some greater metaphysics—we should seek to isolate some internally accessible property that decides whether 0# is generic. By “the real 0#,” then, we mean the 0# of the ambient universe, perhaps assuming ancillary axioms. Making sense of this project may well require that “internal accessibility” be under- stood generously. Class genericity is at best a second-order property. Furthermore, by Theorem 2, there is no first-order statement dictating that 0# is not invisibly generic. On the other hand, it is conceivable that the real 0# is generic for first-order reasons:

Question A. Suppose that 0# exists. Do there exist definable classes M, P, and G such that • hM; Pi is an inner model of ZFC and 0# ∈/ M; • G ⊆ P is hM; Pi-definably generic; and • 0# ∈ M[G]?

Say that a real x is a “0# supported by V,” where V is a countable standard transitive model of ZFC, when there exists a countable standard transitive model V0 ⊇ V such that o(V0) = o(V), x ∈ V0 \ V, and V0 ² ZFC + “x is 0#.” Call a 0# supported by a countable V a “countable 0#.”

Question B. Under suitable hypotheses, is there a countable weakly generic 0#?

That is, does there exist a countable standard transitive model V and a V-amenable partial ordering P such that • hV; Pi ² ZFC + “0# does not exist;” • G ⊆ P is hV; Pi-definably generic; and • hV[G]; V, P,Gi ² ZFC + “0# exists?” Affirming A affirms B. In fact, if A is affirmed, then every countable 0# is weakly generic. “Every countable 0# is weakly generic” might be seen to provide second-order grounds for claiming that the real 0# is generic. 22 5.1. IS THE REAL 0# GENERIC?

Question C. Suppose that V is a countable standard transitive model of ZFC + “0# does not exist.” Is every 0# supported by V invisibly generic?

Again, affirming A affirms C. In fact, “every countable 0# is weakly generic” affirms C. Invisible genericity is not a second-order property (in set theory) because, as in the case of Theorem 2, ZFC may fail in hV[G]; V, P,Gi. Nevertheless, affirming Question C would provide some grounds for claiming that the real 0# is generic. “Good” countable 0#’s are counterexamples to a strengthened C in which genericity is replaced by hyper-genericity:

Proposition 5.1. Suppose that (1) hV; Pi is a countable standard transitive model of ZFC + “0# does not exist;” (2) G is hyper-generic over hV; Pi; (3) V[G] satisfies ZFC + “0# exists;” (4) the forcing relation for P, when restricted to bounded formulas, is definable over hV; Pi; and that ¡ ¢ (5) o(V) has uncountable cofinality in H hV; Pi [G]. ¡ # ¢ Then the Ehrenfeucht-Mostowski 0G, o(V) · ω -model is not well-founded.

+ L # In fact, if κ = o(V), then (κ ) as calculated by 0G is not well-founded. Hypothesis (4) can be replaced by “hV[G]; V, P,Gi satisfies ZFC,” because this implies the definability of the forcing relation on a cone of conditions lying in G. Proof: Set A = hV; Pi and κ = o(V). We may assume that P ° “0# exists” and that o(V) has uncountable cofinality in H(A)[G] whenever G is hyper-generic over hV; Pi. We may also assume that P has a weakest condition, so that ordinary canonical terms aˇ for ground model sets a are defined. Note first that no countable 0# supported by V lies in H(A). Indeed, if a ∈ H(A) is a countable 0# supported by V, then we can reach a contradiction as follows. Working in V, let ˚x be a term such that P ° “˚x is 0#”. We maintain that a = { n : P ° nˇ ∈ ˚x }, in contradiction to hypothesis (1). Otherwise, let p ° nˇ ∈ ˚x, for some n∈ / a, or p ° nˇ ∈ / ˚x, for some n ∈ a. Let G be hyper-generic with p ∈ G. Since G is a fortiori definably generic over hV; Pi, we have that ˚xG 6= a and that ˚xG is a countable 0# supported by V. Furthermore, both ˚xG and a lie in H(A)[G], contradicting that o(V) has uncountable cofinality there. # Fix a hyper-generic G and, working in A[G], define the (0G, κ · ω)-model hB,Ei as in the proof of Proposition 4.2. If hB,Ei is well-founded, then we have that ¡ ¢ ¡ ¢ • H(A)[G] is admissible and o H(A)[G] = o H(A) ; • κ = o(A) has uncountable cofinality in H(A)[G]; and ¡ ¢ ∼ • hB,Ei = Lδ, for some δ < o H(A) , and the isomorphism lies in H(A)[G].

0 # By Part II of the proof of Proposition 4.2 (with G = G), it follows that 0G ∈ H(A), contradicting that no countable 0# supported by V lies in H(A). INVISIBLE GENERICITY AND 0# 23

5.2. Non-generic reals Whether or not 0# itself turns out to be weakly generic, we can ask Question D. Suppose that 0# exists. Can there exist a non-constructible real x that is not weakly generic. # Sy Friedman [SF] has found a real x such that 0

6. References

[B] Jon Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic, Springer-Verlag, New York (1975). [HF] Harvey Friedman, Countable models of set theory, Cambridge Summer School in Math. Logic, Lecture Note in Math., vol. 337, Springer-Verlag, New York (1973), pp. 539–573. [SF] Sy Friedman, The genericity conjecture, Jour. Sym. Logic 59 (1994), pp. 606–614. [S1] M.C. Stanley, A non-generic real incompatible with 0 # , Annals of Pure and Applied Logic (to appear).

Added in proof. Subsequent work has shed some light on two of these questions. The answer to Question C is Yes, if the definition of “invisibly generic” is modified. Say that V [G] is generic if, whenever V [G] ² ϕ also V [G0] ² ϕ, for comeagerly many G0 in a neighborhood of G in the Stone Space of the forcing partial ordering. An observation on the result of [4] largely affirms Question E. The real of [4] is not invisibly generic over any inner model for any admissibility preserving forcing. “Admissibility preserving” means that V [G], the class of values of Shoenfield terms, is a model of KP whenever G is definably generic.

Math Department San Jose State San Jose, CA 95192 e-mail: [email protected]