Archive for Mathematical Logic manuscript No. (will be inserted by the editor)

P. D. Welch

On unfoldable cardinals, ω-closed cardinals, and the beginning of the inner model hierarchy

Abstract. Let κ be a cardinal, and let H_{κ} be the class of sets of hereditary cardinality less than κ; let τ(κ) > κ be the height of the smallest transitive admissible set containing every element of {κ} ∪ H_{κ}. We show that a ZF C-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H_{κ} is as long as τ. (It is known that some weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows:

Theorem Con(ZF C + ω^{2}-Π_{1}^{1}-Determinacy) ⇒ ⇒ Con(ZF C + V = K + ∃ a long unfoldable cardinal ⇒ ⇒ Con(ZF C + ∀C(X^{#} exists) + “∀D ⊆ ω_{1} (D is universally Baire ⇔ ∃r ∈ ℝ(D ∈ L(r)))”, and this is set-generically absolute).

We isolate a notion of ω-closed cardinal which is weaker than an ω_{1}-Erdős cardinal, and show that this bounds the first long unfoldable:

Theorem Let κ be ω-closed. Then there is a long unfoldable λ < κ.

1. Introduction

The study of inner models of ZF C and the natural hierarchy of the canonical inner models L, L[O^{#}], L[O^{##}], …, L[μ], L[O^{†}], …, L[E], … generated by embeddings is a central theme of contemporary set theory. The definitions of the first few models in this series are indeed now venerable. The phenomena of the effect of large cardinals on, e.g. , the constructible power set of ω_{1} was demonstrated initially through Silver’s and Rowbottom’s results on partition properties of cardinals. Some authors (cf. Baumgartner and Galvin [?]) tried reduced even further

P.D.Welch: Institut für Formale Logik, Währingerstr. 25, A-1090 Wien, Austria Dept. of Mathematics, University of Bristol. Bristol BS8 1TW, England. e-mail:

welch@logic.univie.ac.at

The author wishes to gratefully acknowledge support from Nato Grant PST.CLG 975324.

Key words or phrases: large cardinals–constructibility–inner models–descriptive set theory

Mathematics Subject Classification (2000): 03E45,03E15,03E55,03E60

the homogeneity properties required of a partition cardinal to obtain, for example sharps for reals.

Although the modern perspective is that the generating principle for such models is usually the existence of some small iterable structure, such partition properties are nevertheless useful as benchmarks for obtaining consistency results of other principles. (We are thinking here of examples such as those concerning weak Chang’s Conjecture, due to Donder and Koepke [3], and further results of Donder and Lewinsky [2] on flat filters and the like.) There are also phenomena of a descriptive set theoretic nature that seem to bracket, and in turn be bracketed by, the generating iterable structures (the so-called “mice”) of these inner models. These phenomena, (e.g. such as stated as Fact 3.4 below using results from [10], and the consistency strength of a certain level in the difference hierarchy on Π_{1}^{1}, [12]) although of consistency strength greater than that of “For every set X, X has a sharp”, are nevertheless rather weak and seem to fall below all known “(weak) large cardinal” properties. A useful vantage point in this study appears to be the stage where the following ordinals coincide: (i) the first uniform indiscernible for bounded subsets of some uncountable cardinal κ; (ii) the height of the least admissible set containing κ and all sets of herditary cardinality smaller than κ; (iii) the length of the mouse order for mice of size less than κ; (in fact the equality of the first two implies their joint equality with the third). It seems apparent that there is a wealth of detail below say the first measurable cardinal as these canonical inner or “core” models develop, and it is interesting to watch the interplay between these phenomena as the hierarchy unfolds.

It is not the purpose of this article to survey these results in any detail but to contribute further to them, by ascertaining how a small large cardinal definition due to Villaveces [9] fits into this scheme. One can give a description of the strength of the notion in terms of small large cardinals (that is Theorem 1.3 below). Once one has the cardinal notion inside a canonical core model, one can use the hierarchical structure of the model (that is the mouse ordering) to show that in this situation the phenomenon is indeed weak: it seems to require descriptive set theoretic principles alone to find bounds for the consistency strength (this is Theorems 1.4 and 1.6, combined as the first Theorem of the abstract.)

The notion of an ω-closed cardinal is introduced as follows.

Definition 1.1. (I) A structure A = ⟨A, ∈, F ⟩ is called an ω-closed structure if it satisfies the following clauses for some κ:

(i) ⟨A, ∈⟩ is a transitive ZF C^{−}-model, with H_{κ}^{A} ∈ A; (ii) A is amenable and A ⊨ “F is a normal measure on κ”. (iii) F ⊆ A is an ω-closed filter on P(κ)_{A}, that is if ⟨X_{n} | n < ω⟩ is a sequence of sets in F ∩ A then ∩_{n}X_{n} ≠ Ø. (II) Let κ be a cardinal. Then κ is ω-closed if there is a ω-closed structure A with H_{κ} = H_{κ}^{A}.

On unfoldable and ω-closed cardinals  3

This is admittedly a rather technical definition, but inner model theorists will recognise this as deriving from the inductive definition of the core model: an ωclosed cardinal occurs, when we add a filter F onto the extender sequence E we have built, in the inner model L[E, F] the critical point of F is not collapsed and F is an ω-closed filter.

We shall show:

Theorem 1.1. Let κ be ω_{1}-Erdős. Then there is a stationary set of ω-closed cardinals γ < κ.

(We shall recall the definition of ω_{1}-Erdős at 2.2 below.)

Villaveces [9] defines the following notion of unfoldability of a cardinal. We shall investigate this by means of ω-closure.

Definition 1.2. (i) Let κ be inaccessible, and λ > κ. κ is λ-unfoldable ⇔ ∀M(M transitive, |M| > κ, κ ∈ M ⊨ ZF^{−} → → ∃j, N(N transitive, λ ∈ N, j : M → N, crit (j) = κ ∧ j(κ) ≥ λ)).

(ii) κ is unfoldable ⇔ it is λ-unfoldable for all λ.

Actually [9] ’s definition is stated in terms of there being elementary chains of extensions of (V_{κ}, ∈ S) for any S ⊆ κ, but this is easily seen to be equivalent to the above.

Clearly this is a generalisation of weak compactness. Villaveces (see [9]) shows the following: κ’s unfoldability relativises to L; if λ is subtle then V_{λ} ⊨ “∃ unfoldable cardinals”; and in L unfoldables are Π_{m}^{n} indescribable for any m, n < ω. More particularly for us here:

Theorem 1.2. (Villaveces [9]) If κ is weakly compact and κ → (ω_{1})_{2}^{<ω}, then the first unfoldable is less than κ.

We recall that the first cardinal that satisfies κ → (ω_{1})_{2}^{<ω} is ω_{1}-Erdős , but is not weakly compact.

[9] also gives a definition of “On-unfoldable”, which is formalisable in Gödel-Bernays (but not ZFC), and essentially is the following: κ is On-unfoldable, if for any S ⊆ κ, ⟨V_{κ}, ∈, S⟩ has an On-length chain of elementary extensions. He shows that if λ is ineffable then L_{λ} satisifes the second order formula that implies there are “On-unfoldables”. L itself may contain cardinals that are On-unfoldable (for example if 0^{#} exists, every Silver indescernible is On-unfoldable) but, as already mentioned, this can only be expressed in Gödel-Bernays.

As 1.2 involves cardinals less than the first weakly compact uncountable partition cardinal satisfying κ → (ω_{1})_{2}^{<ω}, and the latter relativises to the core model K, the definition of On-unfoldability in the core model invites comparison with the mouse order and this was the initial motivation for this paper. The definition appeared not without independent utility however: for example D.Asperó had shown that ZF C+ Bounded Martin’s Maximum +∃ an ω_{1}-Erdős cardinal ⊢ 2^{ℵ_{0}} = ℵ_{2}. The use of ω_{1}-Erdős can be replaced by ω-closed, as this notion also produces sufficiently long sequences of good indiscernibles. (Subsequently Todorčević ([8]) has removed this hypothesis altogether.)

Villaveces calls “On-unfoldable” long unfoldable, but we shall use that term for a definition that is formalisable in ZFC, and hence somewhat stricter. We ask that there is a class t = {⟨M, T, λ⟩ | φ(M, T, λ)} for some formula Σ_{n} φ(v_{0}, v_{1}, v_{2}) (possibly with other set parameters) in the language of set theory L_{{ ˙∈, ˙=}} that gives the elements of a chain of models under elementary extension.

Definition 1.3. κ is long unfoldable if ∀S ⊆ κ there is a class given by a fixed Σ_{n} formula φ(v_{0}, v_{1}, v_{2}) (possibly containing further set parameters) so that

(i) φ(M, \overline{S}, λ) ⇒ \overline{S} ⊆ λ, On ∩ M = λ, M is a transitive set and ⟨V_{κ}, ∈ S⟩ ≺ ⟨M, ∈, \overline{S}⟩.

(ii) ∀λ, λ′ φ(M_{0}, S_{0}, λ) ∧ φ(M_{1}, S_{1}, λ′) ∧ λ < λ′ →

⟨M_{0}, ∈, S_{0}⟩ ≺ ⟨M_{1}, ∈, S_{1}⟩. (iii) There are arbitrary large λ so that ∃M, S̄ φ(M, S̄, λ).

L cannot contain any long unfoldables according to this definition, as otherwise satisfaction for L would be L-definable. More generally:

Fact 1.1. Suppose κ is long unfoldable, taking S = Ø, and we take a formula φ defining a class as above. Suppose that V_{κ} ≺ M_{λ} ≺ M_{λ′} ≺ … is an elementary chain through On. Then for all sufficiently large λ M_{λ} ≠ V_{λ}.

(Otherwise we should have V_{κ} ≺ V_{λ} ≺ ⋯ ≺ V, and thus be able to define a satisfaction relation for V which is impossible.)

Example If L[μ] is the canonical inner model for a single measurable cardinal and if κ is the measurable cardinal, then κ is long unfoldable using iterated ultrapowers of the structure ⟨H_{κ^{+}}, ∈, μ⟩: For any S ⊆ κ, let φ(R_{i}, S_{i}, κ_{i}) express the following (in the parameters S, μ, H_{κ^{+}}): “Set H_{0} = H_{κ^{+}}, π_{0,i} : H_{0} ⟶ H_{i} with ⟨H_{i}, ∈, μ_{i}⟩ the i’th iterated ultrapower of H_{0} by μ, with ultrapower maps π_{i,j}, and let κ_{i} be the critical point of μ_{i} and R_{i} = (V_{κ_{i}})_{H_{κ_{i}}}; finally set S_{i} = π_{0,i}(S)”.

As the example shows, in L[μ] there will clearly be (stationary many) unfoldable cardinals below the measurable cardinal. This is the canonical example to keep in mind in the sequel.

We improve Villaveces’ theorem 1.2 on the least unfoldable to:

Theorem 1.3. If κ is ω-closed then the first long unfoldable is less than κ.

We thus gain doubly: by obtaining the result for long unfoldables, as well as weakening the large cardinal assumption.

We investigate the consistency strength of long unfoldables a little more closely. By Theorem 1.3 we see that it is less than that of an ω_{1}-Erdős cardinal. However such cardinals fail to exist in L. It is thus instructive to identify some inner core model between L and L[μ] in which such a cardinal can exist. We calibrate this strength, accordingly between two descriptive set theoretic principles, as traditional weak large cardinals are a little thin on the ground in this region. To wit:

Theorem 1.4. ZF C + ω^{2}Π_{1}^{1} - Determinacy ⊢ “There is an inner model of: ZF C + ∃ a Mahlo class of long unfoldables”

Remark: The existence of an ω-closed cardinal suffices to prove (boldface) ω^{2}Π_{1}^{1}- Determinacy. Here the ω^{2}Π_{1}^{1} denotes the level ω^{2} of the Hausdorff difference hierarchy on Π_{1}^{1}. In fact the weaker hypothesis of an “almost < ω_{1}-Erdős ” suffices (Theorem 1.15 of [12].) By “Mahlo” in the conclusion, we mean “definably Mahlo”, that is, stationary with respect to definable closed and unbounded classes.

Theorem 1.5. ZF C + V = K + “∃ a long unfoldable” ⊢ ∀D ⊆ ω_{1} (D is universally Baire ⇔ ∃r ⊆ ω D ∈ L[r]).

The hypothesis of the last theorem proves that it is consistent that the stated characterization of universally Baire subsets of ω_{1}, is set-generically absolute:

Theorem 1.6. ZF C + V = K+ “∃ a long unfoldable” ⊢ “There is an inner model of: ZF C + ∀X(X ⊆ On ⟶ X^{#} exists) +

“∀D ⊆ ω_{1} (D is universally Baire ⇔ ∃r ∈ ℝ(D ∈ L(r)))”, and this is set-generically absolute.”

Universally Baire subsets of ω_{1} are defined and discussed in [5] where it is shown that this characterisation follows from the existence of a measurable cardinal. In [10] this is reduced to an “almost < ω_{1}-Erdös” cardinal (we shall not use this definition here and so refer the reader to this paper for a definition). Thus, by virtue of 1.4 and the remark following it, 1.5 represents an improved upper bound over [10] for this characterisation to be provable. In fact in K the characterisation holds if there is a cardinal κ so that the mouse order, <^{∗}, restricted to H_{κ} stretches out as far as the height of the least admissible set conatining {κ}∪H_{κ}. It is known that ∀X(X ⊆ On ⟶ X^{#} exists) alone is insufficient.

Theorems 1.5 and 1.6 are actually consequences of what we shall actually prove:

Theorem 1.7. ZF C + V = K + κ long unfoldable ⊢

(i) κ is admissibly measurable (ii) There is an inner model of: ZF C+ “∃ a proper class of admissibly measurable

cardinals”.

(Admissibly measurable is defined at Definition 3.5 below) In [10] it is shown that the conclusions of Theorems 1.5 and 1.6 follow from those of Theorem 1.7 (i) and (ii).

We close this introduction by making a few observations, and mentioning some theorems about the general notion of unfoldability before proceeding to the proofs of the theorems above.

Like weak compacts no large cardinal strength is needed to get the failure of the GCH at an unfoldable:

Theorem 1.8. (Hamkins)[6] Con(ZF C + ∃an unfoldable ) ⇒ ⇒ Con(ZF C + ∃κ unfoldable + 2^{κ} > κ^{+})

Villaveces asked in [9] whether it is possible, over a GCH model say, to destroy the unfoldability of a cardinal but preserve its weak compactness. We were able to do this by some Jensen coding methods. Thus:

Theorem 1.9. Let M ⊨ GCH + κ unfoldable. Then there is a cardinal preserving class forcing extension M[G] ⊨ V = L[r], r ⊆ ω ∧ κ weak compact but not unfoldable.

By brute force one can obtain this effect on a non-stationary class of unfoldables. However Sy Friedman, in response to this query, proved the following theorem that allows one to get this effect everywhere.

Theorem 1.10. (S. Friedman) Let M ⊨ GCH. Then in a coding generic extension M[G] ⊨ V = L[r], r ⊆ ω, cofinalities are preserved, as is Π_{1}^{1}-indescribability. However all instances of Π_{2}^{1}-indescribability are lost.

Since the extension is a model of V = L[r] and Villaveces shows that in L, or L[r], all unfoldables are Π_{m}^{n} indescribable, a full positive answer to Villaveces’ question is a corollary of Friedman’s theorem.

It should be pointed out that the consistency strength of long unfoldability on its own, is rather weak: it does not imply strong indescribability properties in V, only in canonical inner models. Similarly only in inner models of “V = K” does it imply that the mouse order is long. Outside of such models it does not imply even that O^{#} exists. Set models of long unfoldability can be easily constructed. For example:

Theorem 1.11. Con(ZF C + V = L + ∃ an ineffable cardinal) ⊢ Con(ZF C + ∃A ⊆ κ^{+}(V = L[A]) + κ is long unfoldable but Σ_{2}^{2}-describable).

Proof: This is essentially a result of Villaveces stated in our terms plus some coding: if λ is ineffable, then he remarks ([9] p.1121 following Prop.1.4) that over V_{λ} there is κ < λ which is On-unfoldable with respect to V_{λ}; that is for each S ⊆ κ there is a cub class C_{S} ⊆ λ of ordinals, and a class T^{S} = ⟨T_{γ}^{S} | γ ∈ C_{S}⟩ of sets with γ ∈ C_{S} ⟶ ⟨L_{κ}, S⟩ ≺ ⟨L_{γ}, T_{γ}⟩. Let C̃ = ⋂_{S∈P(κ)} C_{S}, and let D = {⟨S, ⟨γ, ξ⟩⟩ | S ∈ P(κ) ∧ γ ∈ C_{S} ∧ ξ ∈ T_{γ}^{S}⟩}. Then D is amenable to L_{λ}. Let ℙ be the Jensen coding conditions for coding over ⟨L_{λ}, ∈, D⟩ the predicate D by a subset A of κ^{+}. Then in ⟨L_{λ}[A], ∈, A⟩ we can decode from A the classes C̃, T. P(κ) is unchanged. Now using A we can define the requisite definable classes t_{S} for S ∈ P(κ). As κ is least with a non-constructible subset of κ^{+} we obtain the above describability of κ. Q.E.D.

(We note that there are versions of the above theorem preserving sharps for all sets using a #-preserving Coding Theorem, providing this holds in the ground model.) On the other hand to prove the existence of such a cardinal such as in Theorem 1.3, seems to require in V more than one requires in K, as shown by Theorem 1.4, where there are lots of iterable structures around to help.

We finally remark here that actually there is rather a lot of “slack” between hypotheses and conclusions in both Theorems 1.4 and 1.6. In particular there is a whole hierarchy of “α-hyper-unfoldable” that one could define and investigate implied by the determinacy hypothesis of 1.4.

Our notations are standard, and any unexplained terms we hope can be found in [7]. H_{κ} will denote the set of sets of hereditary cardinality < κ.

2. ω-closed cardinals and a generalised indiscernibility lemma

Suppose that ⟨A, F ⟩ is an ω-closed structure. By (ii) and (iii) in Definition 1.1 we may form the iterated ultrapowers of A by F. The following are then routinely verified:

(1) H_{κ}^{A} ⊨ ZF C; hence if π_{i,j} : A_{i} ⟶ A_{j} is the iteration map, then π_{i,j}(H_{κ_{i}}^{A_{i}}) = H_{κ_{j}}^{A_{j}} and π_{i,j} ↾ H_{κ_{i}}^{A_{i}} : H_{κ_{i}}^{A_{i}} ⟶_{Σ_{ω}} H_{κ_{j}}^{A_{j}} where A_{j} ⊨ “F_{j} is a normal measure on κ_{j}.”

(2) If H = ⋃_{i} H_{κ_{i}}^{A_{i}} then H ⊨ ZF C+“Every set has a sharp”.

A direct limit argument taken from the proof of the Jensen Indiscernibles Lemma ([1], Ch.16) will be used below to obtain ω-closed structures from good sequences of indiscernibles of uncountable order type.

Definition 2.1. (cf [1] Ch.16) Suppose A = ⟨L_{δ}[A], ∈, A, B⟩ is an amenable structure. I ⊆ δ is a good sequence of indiscernibles for A if for all γ ∈ I:

(i) ⟨L_{γ}[A ∩ γ], ∈, A ∩ γ, B ∩ γ⟩ ≺ A; (ii) I\γ is a set of indiscernibles for the structure ⟨L_{δ}[A], ∈, A, B, ⟨ξ⟩_{ξ≤γ}⟩.

I is said to be remarkable if it satisfies (ii).

Remark: It seems not to have been explicitly stated in the literature known to us, but if the good sequence I is infinite then each γ ∈ I is ineffable in A. This is similar to the argument that Silver indiscernibles are ineffable in L; or see the argument in [7] at p.395 where it is shown that the least ω-Erdős cardinal has ineffables below it.

The following gives the version of α-Erdős we shall use in this paper.

Definition 2.2. Let κ be a regular cardinal and α a limit ordinal. Then

(i) κ is α-Erdős if every structure of the form A = ⟨L_{κ}[A], ∈, A, B⟩ has a good sequence of indiscernibles of order type α. (ii) κ ⟶ (α)_{μ}^{<ω} if and only if for every function f : [κ]^{<ω} ⟶ μ there is X ⊆ κ with X homogeneous for f, i.e. , ∀n|f“[X]^{n}| = 1, and ot(X) = α.

The least cardinal satisfying (ii) is additionally α-Erdős .

Proof of Theorem 1.1 Let C ⊆ κ be any cub set. As any α-Erdős cardinal is strongly inaccessible, we may choose E ⊆ κ be such that H_{κ} = L_{κ}[E]. By the ω_{1}-Erdős property, let I ⊆ C be a sequence of good indiscernibles of order type ω_{1}. We shall claim that γ^{∗} =_{df} sup(I) is ω-closed. Standard arguments show that each γ ∈ I is strongly inaccessible, and hence A ↾ γ = ⟨L_{γ}[E], E ↾ γ⟩ ⊨ ZF C. Closely following the original proof of [4], we set Ã_{γ}^{n} = Hull_{A}(γ + 1 ∪ γ) where γ is any member of [I\γ + 1]^{n}; A_{γ}^{n} = Ã_{γ}^{n} ∩ H_{γ^{+}} is a transitive ZF C^{−} model. Set A_{γ} = ∪_{n}A_{γ}^{n} and δ^{n} = On ∩ A_{γ}^{n}. Note that δ^{n} is definable in A_{γ}^{n+1}. Hence:

(1) δ^{n} < δ^{n+1} and A_{γ}^{n} ≺ A_{γ}^{n+1} ≺ A_{γ} ⊨ ZF C^{−}.

For γ < γ′ ∈ I we define π_{γγ′}^{n} : A_{γ}^{n} ⟶_{e} A_{γ′}^{n} by applying shift maps on indiscernibles, to terms in the language appropriate to A:

π_{γγ′}^{n} (t_{A}(ν, γ, γ)) = t_{A}(ν, γ′, γ) for γ ∈ [I\γ′ + 1]^{<ω} and ν ∈ [γ]^{<ω}.

The choice of γ is not important due to their indiscernibility. Define

U_{γ}^{n} = {X ∈ P(γ) ∩ A_{γ}^{n} | γ ∈ π_{γγ′}^{n}(X), for some γ′ > γ, γ′ ∈ I} .

Then U_{γ}^{n} ∈ A_{γ}^{n+1} and, if we set U_{γ} = ∪_{n}U_{γ}^{n}, then we conclude:

(2) ⟨A_{γ}, U_{γ}⟩ ⊨ “U_{γ} is a normal measure on γ”, and is an amenable structure.

Set

⟨⟨A_{γ^{∗}}, U_{γ^{∗}}⟩, ⟨π_{γγ^{∗}}⟩_{γ∈I}⟩ = lim_{⟶}⟨⟨A_{γ}, U_{γ}⟩, ⟨π_{γγ′}⟩_{γ<γ′∈I}⟩

where π_{γγ′} = ∪_{n}π_{γγ′}^{n}. As U_{γ^{∗}} is generated by I as a final segment filter on I, and cf(otp(I)) = ω_{1}:

(3) U_{γ^{∗}} is ω-closed.

Let A_{γ^{∗}} = ⟨L_{β^{∗}}[E^{∗}], E^{∗}⟩. Clearly E^{∗} ↾ γ^{∗} = E ↾ γ^{∗}, and, using (2), ⟨L_{β^{∗}}[E^{∗}], U_{γ^{∗}}⟩ is amenable, and, using (3), iterable. Further E ↾ γ^{∗} ∈ A_{γ^{∗}} ⊨ ZF C^{−}; hence, using (i) of the definition good indiscernibles as L_{γ}[E ↾ γ] ≺ L_{κ}[E] for any γ ∈ I ∪ {γ^{∗}}:

(4) H_{γ^{∗}} = |L_{γ^{∗}}[E^{∗}]| ∈ A_{γ^{∗}}.

But then this together with (3) and the comments after it show that γ^{∗} is ω-closed. Q.E.D.

Note then that an ω-closed cardinal need not be regular (in V), but only have uncountable cofinality. Note also in the above construction we do not claim that ⟨A_{γ^{∗}}, U_{γ^{∗}}⟩ ⊨ KP as this is a much stronger property. But the concept of ω-closure is what is needed to get back an uncountable sequence of good indiscernibles:

Theorem 2.1. Let H_{κ} = L_{κ}[B] where B ⊆ κ and L_{κ}[B] ∈ A where A = ⟨A, ∈ , F ⟩ witnesses the ω-closure of κ. Then there is I ⊆ κ, ot(I) ≥ ω_{1}, with I a good set of indiscernibles for ⟨L_{κ}[B], ∈, B⟩.

Proof: Let C ⊆ κ be closed and unbounded such that γ ∈ C ⟶ L_{γ}[B ∩ γ] ≺ L_{κ}[B] with C ∈ A. The argument is standard, but has to be conducted without using Σ_{1}-Collection (which in general will not hold over A), and we simply check below that these details can be done. Let ⟨φ_{m} | m ∈ ω⟩ be a recursive enumeration of the formulae of L_{{ ˙∈, ˙=,Ḃ}}. Let f : [C]^{<ω} ⟶ κ be defined by:

f({α_{1}, …, α_{2n}}) = μλ < α_{1}(∃δ_{1} < ⋯ < δ_{k}, ∃m < ω λ =≺ m, δ_{1}, … δ_{k} ≻ ∧

L_{κ}[B] ⊭ φ_{m}(δ, α_{1}, …, α_{n}) ↔ φ_{m}(δ, α_{n+1}, …, α_{2n}) ) if such exists,

= 0 otherwise.

Then f ∈ A and is regressive. A routine argument shows that if I ⊆ C is infinite and is homogeneous for f then f must take value 0 everywhere, and hence I is a good sequence of indiscernibles of the required type. The construction of I is by induction on n defining homogeneous sets I_{n} for f^{n} =_{df} f ↾ [C]^{n}. We assume as inductive hypothesis that, for n = m and for any sequence ⟨g_{α} | α < κ⟩ ∈ A where each g_{α} : [C]^{m} ⟶ κ is regressive, there is a sequence ⟨X_{α} | α < κ⟩ ∈ A with X_{α} ∈ F homogeneous for g_{α}. (Thus X_{α} has type κ as it has F -measure 1. For the base case of n = 0 i.e. for functions of zero variables, we of course set X_{α} = κ.)

Thus let ⟨f_{β} | β < κ⟩ ∈ A be a sequence of regressive functions on [C]^{n} with n = m + 1. Define

f_{β}^{α}(x) = f_{β}({α} ∪ x) if α < min(x)
= 0 otherwise.

Then ⟨f_{β}^{α} | α, β < κ⟩ ∈ A and by the inductive hypothesis there is ⟨X_{β}^{α} | α, β < κ⟩ ∈ A with each X_{β}^{α} ∈ F and homogeneous for f_{β}^{α}. As A ⊨ ZF C^{−} if we set X_{β} =_{df} △_{α<κ}X_{β}^{α} then ⟨X_{β} | β < κ⟩ ∈ A, and by normality of F, each X_{β} ∈ F.

Suppose β < κ and {α_{1}, α_{2}, …, α_{n}}, {α_{1}, α′_{2}, …, α′_{n}} ∈ [X_{β}]^{n}. Then define

γ_{β}^{α_{1}} =_{df} f_{β}^{α_{1}}({α_{2}, …, α_{n}}) = f_{β}^{α_{1}}({α′_{2}, …, α′_{n}}) < α_{1}.

Then for each fixed β, α → γ_{β}^{α} is a function, h_{β}, in A with domain X_{β} and together ⟨h_{β} | β < κ⟩ ∈ A. If we set Y_{β}^{γ} =_{df} {α ∈ X_{β} | h_{β}(α) = γ} then U =_{df} {Y_{β}^{γ} | γ, β < κ} is in A. By amenability of A U ∩ F ∈ A and so by normality for each β < κ ∃Y ∈ U ∩ F, ∃γ < κ Y = Y_{β}^{γ}. Thus, using a wellorder of U ∩ F in A, we may define a function G : κ → U, G(β) = some Y ∈ U ∩ F on which h_{β} is constant. Let Z_{β} = G(β) ∩ X_{β}. Then ⟨Z_{β} | β < κ⟩ ∈ A, each Z_{β} ∈ F and is homogeneous for f_{β}. This completes the inductive step.

Now returning to our original regressive f : [C]^{<ω} ⟶ κ we may inductively define for each n a sequence of sets ⟨I_{m} | m ≤ n⟩ ∈ A homogeneous for ⟨f^{m} | m ≤ n⟩. In general a requisite sequence n ⟶ I_{n} may not exist in A. However we do have that each I_{n} ∈ F and thus by ω-closure I =_{df} ∩_{n}I_{n} ≠ Ø. By considering tail segments it is easy to see that I is unbounded in κ - as cf(κ) ≥ ω_{1} we get our desired conclusion. Q.E.D.

Putting these last two together gives us an equivalence:

Corollary 2.1. Let κ be a cardinal. Then the following are equivalent: (i) κ is the least cardinal which is ω-closed; (ii) κ is the least cardinal so that ∃B ⊆ κ∧H_{κ} = L_{κ}[B]∧∃I ⊆ κ, ot(I) = ω_{1},

I cofinal in κ, with I a good set of indiscernibles for ⟨H_{κ}, ∈, B⟩.

We remark that the proof of 1.1 does not actually require the height of A to be a cardinal, or for the domain of A to be an initial rank of the V -hierarchy. It in fact shows the following (using the notation of that proof):

Theorem 2.2. Suppose A = ⟨L_{δ}[A], ∈, A, B⟩ is a structure with A, B ⊆ δ, and let I ⊆ δ be a good sequence of indiscernibles for A of limit order type with supremum γ^{∗} where cf(γ^{∗}) > ω, then there is an iterable, amenable structure ⟨A_{γ^{∗}}, U⟩ extending A ↾ γ^{∗} = ⟨L_{γ}[A], A ∩ γ^{∗}, B ∩ γ^{∗}⟩.

3. The consistency strength of long unfoldables

3.1. Preliminaries

We recall first some basic facts about Q-structures in inner core models. The reader may refer to [12] where this is done for models with many measurables, or in a more abbreviated form for the Dodd-Jensen K in [10]. By “K” we shall mean the core model built under some anti-large cardinal assumption, but in practice as we are proving some consistency results which are weaker than that of the existence of a measurable cardinal we shall really be working with the internal structure of a core model above the largest measurable of its extender sequence E, and below its dagger E^{†}. As this level of generality would yield essentially nothing new we simply work below O^{†}, and use the Dodd-Jensen K (however using modern fine structure, not that of [1].).

We shall adopt the mouse notation and concepts of [12]. A premouse is a fully amenable, acceptable J-model of the form M = ⟨J_{γ}^{E^{M}} , E^{M}, F^{M}⟩ with E^{M} = {⟨X, ν⟩|ν < γ ∧ X ∈ E_{ν}^{M}} such that any proper initial segment M|β =_{df} ⟨J_{β}^{E^{M}} , E^{M}, E_{β}^{M}⟩ is sound. If E_{β}^{M} ≠ Ø then there is κ < β with E_{β}^{M} an M|βnormal measure on P(κ)^{M|β}. If π : M|β ⟶_{E_{β}} Ult(M|β, E_{β}) = N, then E^{N} ↾ β = E^{M} ↾ β, and (κ^{+})^{N} = β. If E_{γ}^{M} = F^{M} ≠ Ø then we call M active, otherwise it is inactive. A mouse is an iterable premouse, that is one in which iterated ultrapowers of M (by its topmost measure F, or any initial extenders on the E-sequences arising, truncating the model when required) are wellfounded. We let <^{∗} denote the canonical wellordering of active mice under coiteration.

Definition 3.1. Let M = ⟨J_{γ}^{E}, E, F ⟩ be an active mouse. K_{M}, the inner model generated by M is

K_{M} =_{df} ⋃_{i<On} H_{κ_{i}}^{M_{i}}

where

M_{i} = ⟨J_{γ_{i}}^{E_{i}}, E_{i}, F_{i}⟩ ⊨ “crit (F_{i}) = κ_{i}”

is the i’th iterate of M by the top extender F_{i}. We set C_{M} = {κ_{i} | i ∈ On}.

K_{M} is then an inner L[E] model, with C_{M} a cub class of indiscernibles for it.

Definition 3.2. (i) For κ a cardinal, let F_{κ} be the cub filter on κ. (ii) Let W be an inner model, κ a cardinal of W , F_{κ} = (F_{κ})_{W}, then:

(a) set θ_{κ}^{W} = maximal θ so that ⟨J_{θ}^{F_{κ}}, ∈ F_{κ}⟩ ⊨ “F_{κ} is a normal measure on κ”.

(b) Then also set:

Q_{κ}^{W} =_{dt} ⟨J_{θ_{κ}^{W}}^{F_{κ}} , ϵ, F_{κ}⟩.

If W = K_{M}, then we replace W and write θ_{κ}^{M}, Q_{κ}^{M} … etc.

Fact 3.1. Suppose there is no inner model with a measurable cardinal. If μ_{κ} = ot{{M ∈ H_{κ}}, <^{∗}}, the length of the mouse ordering restricted to H_{κ}, then θ_{κ} = κ(1 + μ_{κ}).

For b a set of ordinals, we set

I^{b} = class of Silver indiscernibles for L[b] if b^{#} exists;
= On otherwise.

Definition 3.3. For κ a cardinal

(i) C(κ) =_{df} ⋂_{a⊆γ<κ} I^{a};

(ii) u_{2}(κ) =_{df} least element of C(κ) which is greater than κ.

If all bounded subsets of κ have sharps, u_{2}(κ) = sup{κ^{+L[a]}|a ∈ H_{κ}}.

Definition 3.4. (i) Let A = A_{κ} be the smallest transitive admissible set with H_{κ} ∪ {κ} ⊆ |A_{κ}|.

(ii) τ(κ) =_{df} On ∩ A_{κ} - the height of the smallest transitive admissible set containing κ and every element of H_{κ}.

Definition 3.5. κ is admissibly measurable if there is a transitive, iterable, and amenable structure ⟨A, F ⟩ ⊨ “KP_{{ ˙∈,Ḟ }} + F is a normal measure on κ” with H_{κ} ⊆ |A|.

Notwithstanding the appearance of KP in the language including Ḟ, the notion of admissibly measurable is considerably weaker than ω-closure. This is due to the fact that H_{κ} is only required to be a class, and not a set of |A|.

Fact 3.2. In the core model below (or including) a single measurable cardinal, if κ is a cardinal, then θ_{κ} ≤ u_{2}(κ) ≤ τ(κ).

Remark: In K,θ_{κ} < u_{2}(κ) < τ(κ), θ_{κ} = u_{2}(κ) < τ(κ), as well as θ_{κ} = u_{2}(κ) = τ(κ) are all possible (even assuming that H_{κ} is closed under #’s). (Example 1 of [10], using L^{#}, yields an example of the first possibility here; the third possibility occurs in L^{μ}, - see Example 3 of [10]. Lastly we sketch how the second possibility can arise: consider the least inner model K_{0}, of V = K where for some κ we have u_{2}(κ) = τ(κ); then by Fact 3.3 (ii) to come, we have that the structure A = ⟨Q_{κ}, ∈, F_{κ}⟩ is admissible; then the mouse order <^{∗}↾ H_{κ} has order type θ_{κ} = τ(κ). Let ⟨M^{β}|β < τ(κ)⟩ be the enumeration of these mice in ascening <^{∗} order. Let F : τ(κ) ⟶ τ(κ) be the function defined by F (β) = the critical point of the κ + 1 iterate of M^{β}. Then ran(F) ⊆ (κ, τ(κ)); further F is Δ_{1}^{Q_{κ}}. Hence there is some some proper initial level J_{ν}^{F_{κ}} of Q_{κ} which is closed under the function F. This ν becomes “θ_{κ} = u_{2}(κ)” of the inner model W = L[J_{ν}^{F_{κ}}], which is then a proper inner model of K_{0}. By the leastness in the choice of K_{0}, we have that “ν < τ(κ)” holds in W.)

(Note: Assuming there is no inner model with a measurable cardinal, by Σ_{3}^{1}- correctness arguments for κ ≥ ω_{2}, u_{2}(κ)_{V} = u_{2}(κ)_{K}, and, for κ > ω_{2}, by the Covering Lemma for K, θ_{κ} = θ_{κ}^{K} .)

Fact 3.3. [10] (Lemma 4) Suppose V = K, H_{κ} is closed under #’s, and u_{2}(κ) = τ(κ). Then

(i) τ(κ) = θ_{κ} = u_{2}(κ); (ii) In K, A = ⟨Q_{κ}, ∈, F_{κ}⟩ ⊨ KP and so κ is admissibly measurable.

Let Ψ be the sentence “∀D ⊆ ω_{1} (D is universally Baire ⟺ ∃r ∈ ℝ (D ∈ L[r]))”.

Fact 3.4. ([10] Theorem 1, Lemma 4) (i) If there exists an admissibly measurable, then Ψ_{K}. (ii) In K the following are equivalent:

(a) “There exist arbitrarily large admissibly measurable cardinals”,

(b) “Ψ is set generically absolute.”

To obtain the conclusions of Theorems 1.5 and 1.6 , it thus suffices to show that in K there is an admissibly measurable cardinal, or an inner model with a proper class of them.

To prove Theorem 1.7 we shall use the equivalence of ω^{2}-Π_{1}^{1}-Determinacy from [12]. It is shown there (Main Theorem 3) that this amount of determinacy is equivalent to the existence of a “clever” mouse. This is essentially a mouse with the critical point of its top measure κ that reflects a Σ_{1}-Rowbottom property of its measure into the Q-structure of the generated model at κ: ⟨Q_{κ}^{M}, ∈ F_{κ}^{M}⟩. However we shall take as our definition here another equivalent notion, that of “inwardly reflecting.” ([12] Lemma 1.10 show they are equivalent.)

Definition 3.6. Let M = ⟨J_{α}^{E^{M}} , E^{M}, F^{M}⟩ be an active mouse with crit(F^{M}) = κ = κ_{M}. M is inwardly reflecting if for any p ∈ H_{κ}^{M}, for any φ ∈ Σ_{1}∩L_{{ ˙∈, ˙=,Ḟ }}, there is N ∈ H_{κ}^{M} so that ∀κ ∈ [C_{M}]^{r} (where F V (φ) ⊆ {v_{0}, …, v_{r−1}}), and ∀λ ∈ C_{M}, max κ < λ

Q_{λ}^{M} ⊨ φ(κ, p) ⟺ Q_{λ}^{N} ⊨ φ(κ, p)

An interpretation of this is that below an inwardly reflecting mouse, given any inner model KM̃ generated by a mouse M̃, there is a parameter p ∈ HM̃ ; κ and a Σ_{1}-formula φ so that φ is true of p on the indiscernibles for KM̃ in some (or any) sufficiently large Q-structure, but this fails to be true in any smaller inner model (meaning precisely, that if <^{∗} is the mouse ordering and N <^{∗} M, and K_{N} is generated by N, then φ is not true of p on indiscernibles given by the class of iteration points of N, C_{N}). Once one has an inwardly reflecting mouse, such inner models cannot be characterised by sets of Gödel numbers of Σ_{1}-formulae, with parameters, true on indiscernibles in Q-structures. We have

Fact 3.5. ([12, Theorem 3, Lemma 1.10]) ZF C ⊢ ω^{2}-Π_{1}^{1}-Determinacy ⟷ there exists an inwardly reflecting mouse.

3.2. The conclusions

Proof of Theorem 1.5 We assume V = K and that κ is long unfoldable. Note that H_{κ} is closed under #’s: otherwise V = L[B] for B a bounded subset of κ. However this is impossible by the argument of Fact 1.1, that there would then be a definable satisfaction relation for L[B]. By Fact 3.4 it suffices to show that κ is admissibly measurable, that is, to show that Theorem 1.7 (i) holds. Suppose not, then by Fact 3.2 & 3.3 θ_{κ} ≤ u_{2}(κ) < τ(κ) and Q_{κ} = ⟨J_{θ_{κ}}^{F_{κ}}, ∈, F_{κ}⟩ is inadmissible. Hence there is a parameter p ∈ H_{κ} = H_{κ}^{Q_{κ}} and a Σ_{1} formula φ(v_{0}, v_{1}, p) ∈ L_{{ ˙∈, ˙= Ḟ }} defining a map Σ_{1} over Q_{κ}, G : κ ⟶ θ_{k}, which is cofinal and monotone. The latter can be coded by a subset of κ, S ⊆ κ say, and then by definition of long unfoldability, there are ⟨M_{λ}, ∈ S_{λ}⟩ with λ ⊆ M_{λ} and ⟨V_{κ}, ∈, S⟩ ≺ ⟨M_{λ}, ∈ S_{λ}⟩. By elementarity S_{λ} codes Q_{λ}^{M_{λ}} - a level of the J^{F_{λ}}-hierarchy which is accordingly an initial segment of the full Q_{λ}. Further φ defines a Σ_{1} cofinal map of λ into On ∩ Q_{λ}^{M_{λ}}, in the parameter p.

If W is the tower obtained by taking the union of all the models M_{λ} ⊨ V = K in a long chain, we have a proper inner model W of V = K, in which φ with p witnesses the inadmissibility of Q_{λ}^{W} on cofinally many λ. Let N be the <_{∗} -least mouse not in W. Then N generates W: W = K_{N}.

Claim C = {λ | φ defines a cofinal map fromp Σ_{1} over Q_{λ}^{W} into θ_{λ}^{W} }

is closed and unbounded. Proof: (of claim) As we already have unboundedness, suppose λ is a limit point of C. Then λ is a limit cardinal of W, and Q_{λ}^{W} is the union of the λ’th iterates of the structures Q_{μ}^{W} for μ < λ (taking iterated ultrapowers of Q_{μ}^{W} by the measures F_{μ}^{W} and their images). But then, as the ultrapower maps are Σ_{1}-preserving

∀γ < θ_{λ}^{W} ∃δ > γ ∃μ < λ ∃δ̄ < θ_{μ}^{W} ∃ξ < μ (φ_{Q_{μ}}(ξ, δ̄, p) ∧ δ = i_{Q_{μ},(Q_{μ})_{λ}}(δ̄))

shows that φ_{Q_{λ}}(v_{0}, v_{1}, p) defines a function that cannot be bounded in its range below θ_{λ}^{W}. Q.E.D.(Claim)

Consequently this property of φ, p characterises Q_{κ}^{N} at each of its iteration points, since it will have an iteration point in the cub class C.

As p ∈ H_{κ} with κ inaccessible, a Löwenheim-Skolem argument shows that such an N can be found with N ∈ H_{κ}.

Suppose N = ⟨J_{γ}^{E^{N}} , E^{N}, F^{N}⟩ with crit(F^{N}) = δ < κ. If i : N ⟶ N_{κ} is the iteration map obtained by taking κ-many iterated ultrapowers by the top measure F^{N}, i(Q_{δ}^{N}) = Q_{κ}^{N_{κ}}, and i ↾ Q_{δ}^{N} : Q_{δ}^{N} ⟶ Q_{κ}^{N_{κ}} is elementary. Hence φ(v_{0}, v_{1}, p) defines a map G : κ ⟶ On ∩ Q_{κ}^{N_{κ}} which is cofinal and Σ_{1}(Q_{κ}^{N_{κ}}). But N ∈ H_{κ}, and so Q_{κ}^{N_{κ}} is a proper initial segment of Q_{κ}. But φ is supposed to define an unbounded map G : κ ⟶ θ_{κ}! This is a contradiction.

Q.E.D.(Theorem 1.5)

Proof of Theorem 1.6. Using the notation of the proof of the last theorem, as we have established that Q_{κ} is admissible, then S_{λ} codes an admissible Q_{λ}^{M_{λ}}. Thus in the model W we have arbitrarily large λ with Q_{λ}^{W} admissible. Then W is a model satisfying the conclusion of Theorem 1.7 (ii). Q.E.D.(Theorem 1.6) .

Proof of Theorem 1.4 As mentioned in §2.1, ω^{2}-Π_{1}^{1}-Determinacy implies there is an inwardly reflecting mouse M. Let W = K_{M}, the model generated by M. let C_{M} be the class of critical points of the top measure F^{M_{i}} of the iterates of M. We shall show that each κ_{i} ∈ C_{M} is long unfoldable in W. Let κ = κ_{0} = min C_{M}. Let S ⊆ κ, S ∈ W. Consider the ω’th iterate of M beyond κ_{0} with critical point κ_{ω}. Let π_{i,j}^{M} : M_{i} ⟶_{Σ_{1}} M_{j} be the iteration maps. Then π_{i,j}^{M} extends to a map π_{i,j}^{M} : W ⟶_{Σ_{ω}} W. Applying π_{0,2}^{M}(⟨W_{κ_{0}}, S⟩) = ⟨W_{κ_{2}}, S_{κ_{2}}⟩. We have then ⟨W_{κ_{0}}, S⟩ ≺ ⟨W_{κ_{2}}, S_{κ_{2}}⟩. Let p = ⟨W_{κ_{0}}, S⟩ ∈ H_{κ_{1}}^{M_{1}} be our parameter in the definition of the inwardly reflecting property as applied to the active mouse M_{1}. We see that

Q_{κ_{ω}}^{M} ⊨ ∃M̄, S̄ (On ∩ M̄ = κ_{2} ⊇ S̄ ∧ ⟨W_{κ_{0}}, S⟩ ≺ ⟨M̄, S̄⟩).

This is a Σ_{1} statement about p and κ_{2} ∈ C_{M}, φ(p, κ_{2}) say. By the definition of inwardly reflecting, there is N ∈ H_{κ_{1}}^{M_{1}} so that Q_{κ_{ω}}^{N} ⊨ φ(p, κ_{2}). If ⟨M′, S′⟩ ∈ K_{N} witnesses this, and if π_{i,j}^{N} : N_{i} ⟶ N_{j} are the iteration maps associated to N, we may use these to construct a formula defining elementary extensions of ⟨M′, S′⟩. Specifically, as N ∈ H_{κ_{1}}^{M_{1}}, N <^{∗} M, and hence C_{M}\{κ_{1}} ⊆ C_{N}. Let λ be a cardinal of W > κ_{2}; then set ⟨M_{λ}, S_{λ}⟩ = π_{κ_{2},λ}^{N}(⟨M′, S′⟩). Putting this together, we have inside W, a formula defining an On-length chain of elementary extensions of ⟨W_{κ_{0}}, S⟩ of the form ⟨M_{λ}, S_{λ}⟩. As S was arbitrary, we have that κ_{0} is long unfoldable in W. As κ_{0} = min C_{M}, and the latter is a cub class of indiscernibles for W, we have in W that the class of long unfoldables is definably Mahlo. Q.E.D.(Theorem 1.4)

As shown in [12] Lemma 1.11, an inwardly reflecting mouse is rather weak, even in the Σ_{1}-mouse hierarchy: if M is an active mouse, κ = crit(F^{M}), and ρ_{M}^{1} (the first projectum of M) = κ, then M is inwardly reflecting. Consequently in any inner model of V = K, in which there is an active sound mouse with crit(M) = κ, and κ an actual cardinal of the model, we shall have that ρ_{M}^{ω} ≥ κ for all n < ω and ω^{2}Π_{1}^{1}-Determinacy.

The argument of the Indiscernibles Lemma provides just such mice given good indiscernibles over K of sufficient length. In particular, if in K κ is ωclosed, thence taking a good sequence I of indiscernibles for ⟨K_{κ}, ∈, E^{K}⟩ of order type ω_{1}, with sup I = γ^{∗} ≤ κ say, then each γ ∈ I is inaccessible in K, and the lemma’s proof provides an ω-closed mouse, a sound mouse M = ⟨J_{β}^{E^{M}} , E^{M}, F^{M}⟩ with crit(F^{M}) = γ^{∗}, E^{M} ↾ γ^{∗} = E^{K} ↾ γ^{∗} and therefore ρ_{M}^{n} ≥ γ^{∗} for all n < ω. The core model “below an ω-closed mouse” has been partially investigated in [11]. For Theorem 1.3, if we additionally assume V = K, all that is needed is some such ω-closed F from the K-sequence on some K-cardinal κ, in order for there to be long unfoldables below κ.

Proof of Theorem 1.3 We assume that κ is ω-closed. By Theorem 2.1 we have a sequence I whose order type is of uncountable cofinality, of good indiscernibles for ⟨V_{κ}, ∈, B⟩ where B ⊆ κ codes H_{κ}: that is, for any strongly inaccessible λ ≤ κ we have that V_{λ} = L_{λ}[B ∩ λ]. Each γ ∈ I is ineffable (see the remark above after Definition 2.1). Fix now γ as min(I). By Villaveces’ argument the ineffability of γ implies that there are λ < γ which are On-unfoldable over V_{γ}. that is for any S ⊆ λ there is a chain E^{γ}, of transitive ZF C models M_{δ} ∈ V_{γ}, and subsets S_{δ} ⊆ δ with ⟨V_{λ}, ∈, S⟩ ≺ ⟨M_{δ}, ∈, S_{δ}⟩ ≺ ⟨M_{δ′}, ∈, S_{δ′}⟩ for λ ≤ δ ≤ δ′ < γ (We may set ⟨M_{λ}, S_{λ}⟩ = ⟨V_{λ}, S ⊕ B⟩.) Note that we may define a final model for this chain ⟨M_{γ}, ∈, S_{γ}⟩ with On ∩ M_{γ} = γ and ⟨M_{γ}, S_{γ}⟩ = ⋃_{δ<γ}⟨M_{δ}, S_{δ}⟩.

Let γ^{∗} = sup I. By the goodness of the indiscernibles we have that γ < γ′ in I implies that V_{γ} ≺ V_{γ′} ≺ V_{γ^{∗}} ≺ V_{κ}. We thus have extensions of the chain E^{γ} to E^{γ^{∗}}.

Let A = ⟨A_{γ^{∗}}, ∈, U_{γ^{∗}}⟩ be the iterable structure of the proof of Theorem 1.1. Let H = L[B̃] be the inner model that it generates (in the sense of (2) at the beginning of §2), we see that there is an embedding π : H ⟶ H, with crit(π) = γ^{∗}. Hence A is a model of “There is a chain E^{γ} with last model ⟨M_{γ}, ∈, S_{γ}⟩” for γ < γ^{∗}”. But in H we have V_{γ} ≺ V_{γ^{∗}}^{H} ≺ V_{κ_{i}}^{H}, for any i where the i’th model A_{i} has critical point κ_{i}.

Hence using iterations of A we can stretch this chain to have length On, and have κ_{i}’th models ⟨M_{κ_{i}}, S_{κ_{i}}⟩ for i < ∞. Thus, using A as a parameter, we then can construct a formula, just as in the canonical example of L[μ], defining a proper class, which gives us a tower of extensions of ⟨V_{λ}, ∈, S⟩ as demanded by the definition of long unfoldability.

Q.E.D.(Theorem 1.3)

In the above proof if we could have told in advance which λ it was that would be On-unfoldable over V_{γ} we could use a weaker partition property - for example if κ satisfied κ ⟶ (< ω_{1})_{2^{λ}}^{<ω}, (we refer the reader to [2] for the precise definition) this should be more than enough. However Villaveces’ argument is nonconstructive. So we have to leave the matter open.

References

1. A.J. Dodd, The core model, London Mathematuical Society Lecture Notes in Mathematics, vol. 61, Cambridge University Press, Cambridge, 1982.

2. H-D. Donder and J.-P. Lewinski, Some principles related to Chang’s conjecture, Annals of Pure and Applied Logic 45 (1989), 39–101.

3. H-D. Donder and P.Koepke, On the consistency strength of ‘accessible’ Jónsson cardinals and of the weak chang conjecture, Annals of Pure and Applied Logic 25 (1983), 223–261.

4. H-D. Donder, R.B.Jensen, and B. Koppelberg, Some applications of K, Set theory and Model theory (R.Jensen and A.Prestel, eds.), Springer Lecture Notes in Mathematics, vol. 872, Springer Verlag, 1981, pp. 55–97.

5. Q. Feng, M. Magidor, and W.H.Woodin, Universally Baire sets of reals, Set Theory of the Continuum (W. Just H. Judah and W. H. Woodin, eds.), MSRI Publications, Springer Verlag, 1992.

6. J. D. Hamkins, Unfoldable cardinals and the GCH, Journal for Symbolic Logic (to appear).

7. T. Jech, Set theory, Pure and Applied Mathematics, Academic Press, New York, 1978.

16  P. D. Welch: On unfoldable and ω-closed cardinals

8. S. Todorčević, Sigma-one forcing absoluteness and the continuum, unpublished (April 2002).

9. A. Villaveces, Chains of elementary extensions of models of set theory, Journal of Symbolic Logic 63 (1998), 1116–1136.

10. P. D. Welch, Characterising subsets of ω_{1}, Journal of Symbolic Logic 59 (1994), no. 4, 1420–1432.

11. , Countable unions of simple sets in the core model, Journal of Symbolic Logic 61 (1996), no. 1, 293–312.

12. , Determinacy in the difference hierarchy of co-analytic sets, Annals of Pure and Applied Logic 80 (1996), 69–108.