On Elementary Embeddings from an Inner Model to the Universe.

J. Vickers

Dept. of Mathematics, University of Bristol. Bristol BS8 1TW, England.

P. D. Welch

Graduate School of Science & Technology, Kobe University, Rokko-dai, Nada-ku Kobe 657, Japan.

Abstract

We consider the following question of Kunen:

Does Con(ZF C + ∃M a transitive inner model and a non-trivial elementary embedding

j : M ⟶ V) imply Con(ZF C + ∃ a measurable cardinal)?

We use core model theory to investigate consequences of the existence of such a j : M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of “there exists a proper class of almost Ramsey cardinals”. Conversely, if On is Ramsey, then such a j, M are definable.

We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.

1 Introduction

It is quite natural to study the properties of elementary embeddings j : V ⟶ M for M some inner model, since many such embeddings, if they exist, have first order formulations within ZF C. The question of reversing the arrow and looking at a non-trivial j : M ⟶ V in general does not readily admit of such formulations. So we study in this paper what might be considered the ZF C consequences of the second order statement that there are proper classes j, M such that … (This was the formulation of Kunen’s question as it appears in [7], but of course, if the reader does not like this conceit, then he or she may simply translate everything into properties of elementary submodels of some V_{λ} ⊧ ZF C, by replacing V by V_{λ} for some inaccessible λ. Hence we could equally well be motivated by investigating the possible elementary substructures of some V_{λ} of size λ.)

It is not hard to see that some sort of hypothesis beyond ZFC is needed to generate such situations. If there were a non-trivial j : M ⟶ V, then j ↾ L : L ⟶_{e} L is non-trivial and so O^{#} must exist. Indeed, by considering critical points, one can see that such an embedding could not be definable by class terms, even from parameters, or there would be an infinite regress of ordinals!

Obviously more can be extracted from this, and it is natural to ask how much. As the hypothesis is similar to a statement about On possessing Jónsson or Ramsey like properties, it is not surprising that it implies some kind of large cardinal property. Whatever the “core” of V is, denoting it by K, then j ↾ K^{M} : K^{M} ⟶ K^{V} and we may ask what kind of K must there be for this to happen. Kunen’s question (Question 15.8 of [7]) in this vein was: can we extract an inner model with a measurable cardinal? Or more precisely, does Con(ZF C_{{j̇}} + ∃j : M ⟶ V, j ≠ id, j elementary) ⇒ Con(ZF C + ∃ a measurable cardinal )? (Here ZF C_{{j̇}} means ZF C in L_{{ ˙∈,j̇}} a language with a predicate symbol for j, whilst j : M ⟶ V is considered to be Σ_{1}-elementary in L_{{ ˙∈}} alone. (The latter implies full elementarity.) This question is easily answered however. Perhaps more interestingly, in §1.1 we consider the consequences of the existence of such elementary embeddings from an M into V, varying the basic question by adding closure requirements on j to see what K must be like. From just the existence of such a j we get (Theorem 1.1) that K must contain a proper class of “almost Ramsey” cardinal (defined below). We get full Ramseyness (Theorem 1.3) by asking that dom(j) have a class of regular fixed points. By insisting that dom(j) contain a stationary class of singular fixed points of cofinality ω, then we obtain that K must have a Mahlo class of measurables; if the fixed points have uncountable cofinality, or if M computes correctly countable cofinalities, then O^{sword} exists (Theorem 1.7) (that is, Jensen’s core model for measures of order zero - see [3], is non-rigid). We should like to thank M. Foreman for saving us from a demonstrably inconsistent hypothesis for Theorem 1.7 (we had assumed that M was closed under ω-sequences); and for allowing us to include his counterexample as Theorem 2.3.

In §2 we make a first attempt at some converses by showing that if On is Ramsey, then we may define a j, M with j : M ⟶_{e} V, j ≠ id. We could formulate this as a statement about V_{κ}, where κ is a Ramsey cardinal, and show that definably over ⟨V_{κ}, ∈, I⟩ (where I is a class of good indiscernibles for ⟨V_{κ}, ∈⟩ coming from the Ramsey property) there are classes j, M satisfying our requirements. We thus have Con(ZF C +∃ a Ramsey cardinal ) ⇒ Con(ZF C + j : M ⟶_{e} V, j ≠ id).

Although this is not an exact equiconsistency, it could be considered close. However, overall in this direction we have left several upper bounds open.

We make some comments on results obtainable by Woodin, using the stationary tower forcing. (We should like to thank the referee for prompting us to make such comparisons.) Woodin has shown [11], that if κ is a measurable cardinal, and ℙ_{κ} the full stationary tower forcing, then if G is ℙ_{κ} generic over V, then we do have an embedding j : V_{κ} ⟶ V_{κ}[G]. (Below the measurable κ there is an unbounded sequence of completely Jónsson cardinals, fixed by j.) However in this situation, we cannot have that the structure ⟨V_{κ}[G], j, ∈⟩ ⊧ ZF C_{{j̇}}. That is, we cannot put back the predicate V_{κ}. (For example, note that if V = L[μ] with μ a normal measure on κ, then V_{κ} ⊧ “V = K” (with K the Dodd-Jensen core model), then V [G] is a model of the same, and hence some new “mouse” has been added, that must iterate past all those of V_{κ}!) This, then is slightly different from our requirements that we have ⟨V, j, ∈⟩ ⊧ ZF C_{{j̇}}. If we regard the relationship of V to a forcing extension as being “close”, then perhaps we need the extra strength in order to obtain such a close relationship between the “M” and “V ”. Our results do not require such strength since our “M” and “V ” are not at all close in this sense.

In §3.1 we reprove the Jensen Indiscernibles Lemma [2], originally done using the old-style fine-structure of the Dodd-Jensen K_{DJ}, but now for the modern hierarchy and the K under consideration. The reasons for this are that, corresponding to Jensen’s results for the Dodd-Jensen K, this shows that the α-Erdős (if cf(α) > ω), almost Ramsey, Ramsey etc. properties are downwards absolute between V and any universal weasel. Secondly, this actually also gives a second proof of Theorems 1.1 & 1.3 which are proven in §1 by using a coarse “lift-up” technique.

Our results are all in the region of inner models with measures of order zero, K^{MOZ}, so it is appropriate to use Jensen’s model of [3] in order to discuss them. This model will be K throughout the paper. (Although the Kunen question can be resolved with just the Dodd-Jensen K^{DJ}, or even the Σ_{1}-K of [10].) We assume familiarity with the measures of order zero K, and the techniques of that paper, some of which appear in [4]. We use the strong Covering Lemma for K, [9], [5] on singular cardinals β of V being either singular, or possibly measurable in K if cf(β) = ω.

Mostly however it is the theory of iteration and comparison that is useful, and fine-structural results have been hived off to §3.

In §3.2 we sketch the outlines of Jensen’s Σ^{∗} fine structure theory in a general setting. It is a relatively bare-bones outline, but all the essential definitions are here. What we have omitted is a full discussion of Σ_{1}^{(n)} relations and functions, and a fuller analysis of good parameters, and of witnesses. The intention here is to give an introduction to the theory, and, in §3.3, to that of the Σ^{∗} ultrapower, to provide a modicum of self-containedness for the proof of the fine-structural Upwards Extensions of Embedding Lemma of §3.4 which is needed for Theorem 1.7. This has essentially been extracted from various parts of §3.3 of [3], but we have thought it worthwhile to give a complete version here, as firstly, the reader may not wish to do it for his or herself, (and anyway such arguments are not often in print), and secondly this is a form of “lift-up” which we wish to use elsewhere.

Some prerequisites The set-theoretic notations are standard. [6] gives an account of the basics of general elementary embeddings. Our notation for iterations of mice is that of Jensen’s: an iteration of a mouse M is of the form I = ⟨⟨M_{i}⟩⟨π_{i,j}^{M}⟩⟨ν_{i}, α_{i}⟩ | i < θ⟩ with models M_{i} and indices ⟨ν_{i}, α_{i}⟩ and commuting maps π_{i,j}^{M} : M_{i} ⟶ M_{j}, with ων_{i} ≤ ωα_{i} ≤ On ∩ M_{i} and if E_{i} ≠ ∅ (the ν_{i}’th filter on the E^{M_{i}} filter sequence), then π_{i,i+1}^{M} : M|α_{i} ⟶_{E_{ν_{i}}} M_{i+1} where M|β denotes ⟨J_{β}^{M}, E^{M}, E_{ωβ}^{M}⟩. ( M||β will denote ⟨J_{β}^{E^{M}} , E^{M}, ∅⟩), or when there is no danger of confusion, simply |J_{β}^{E}|.) If E_{ν_{i}} = ∅ then π_{i,i+1}^{M} = id ↾ M|α_{i}. Direct limits are taken as needed. A truncation occurs when ωα_{i} < On ∩ M_{i}. We truncate if need be in order to allow a filter to become a full measure over the subsets of its critical point in M|α_{i}, and thus are able to take an ultrapower.

It is a fundamental fact that in any iteration only finitely many truncations can occur. The coiteration of two mice or “weasels” (= proper class mice) is performed by “iterating away points of least difference”. This results in a terminating double iteration of length some θ ≤ On, with either M_{θ} = N_{θ}, M_{θ} ∈ N_{θ}, or N_{θ} = M_{θ}. We shall use the following well-known lemma:

Basic Coiteration Lemma If (M, N) are premice, and they are coiterated to models (M_{θ}, N_{θ}) with iterations ⟨⟨M_{i}⟩_{i≤θ}, ⟨α_{i}^{M}⟩_{i<θ}, ⟨π_{i,j}^{M}⟩_{i≤j≤θ}⟩⟩, ⟨⟨N_{i}⟩_{i≤θ}, ⟨α_{i}^{N}⟩_{i<θ}, ⟨π_{i,j}^{N}⟩_{i≤j≤θ}⟩⟩ and common indices ⟨ν_{i}|i < θ⟩ and critical points ⟨κ_{i}⟩_{i<θ} where κ_{i} = crit(E_{ν_{i}}^{M_{i}/N_{i}}) then θ < μ^{+} where μ = max{card(M), card(N)} (or if either is a weasel we may need to set θ = On). If θ = On then either:

a) (i) M is a weasel (i.e. On ∩ M = On), α_{i}^{M} = On for i < On, π_{0,θ}^{M}“On ⊆ On and M_{θ} is an “initial segment” of N_{θ} whilst (ii) there is a cub class C so that i < j ∈ C ⇒ π_{i,j}^{N}(κ_{i}) = κ_{j} = j ∧ π_{i,j}^{M}“κ_{j} ⊆ κ_{j}; or

b) the same as a) but with the roles of M and N reversed.

For a premouse M, N, we say that N is a simple iterate of M, if there has been no truncation, or cut down, at any stage of the iteration from M to N. For two premice (set, or class sized) we set M <_{∗} N if in the coiteration of (M, N) to final models (M_{θ}, N_{θ}), then either M_{θ} ∈ N_{θ}, or there is some truncation occurring at some stage in the N iteration. A weasel W is universal if for any set mouse M, M <_{∗} W. (This definition suffices below a strong cardinal.) By O^{sword} (or O^{s}) we mean an iterable mouse of the form N = ⟨J_{ν}^{E}, E_{ων}, F ⟩ with both E_{ων} and F normal measures in N with critical point κ, and if π : N ⟶ M =_{df} Ult(N, F), then M|ν = ⟨J_{ν}^{E}, E_{ων}⟩. It is thus a mouse with a final predicate for a measure “of order one”, and generates embeddings, or a “sharp”, of Jensen’s model K.

Definition 1.1 Let I ⊆ A = L_{κ}[A, ∈ B⃗, …] be a first order structure. Then I is a good set of indiscernibles for A if for any γ ∈ I:

(i) A ↾ γ =_{df} L_{γ}[A ↾ γ, ∈ B⃗ ↾ γ, …] ≺ A; (ii) I\γ is a set of indiscernibles for ⟨A, ⟨ξ⟩_{ξ<γ}⟩.

We take the following as our definition of Ramsey. It is well known to be equivalent to the usual one.

Definition 1.2 κ is Ramsey if any first order structure with κ ⊆ |A| has a good set of indiscernibles of length κ.

Definition 1.3 κ is almost Ramsey if for any first order structure A, with κ ⊆ |A|, has, for any λ < κ, a good set of indiscernibles of length λ.

1.1 Properties of Core Models arising from Embeddings

Theorem 1.1 If ∃j : M ⟶ _{e}V, M a transitive proper class, j ≠ id, then K ⊧ “There exists unboundedly many almost Ramsey cardinals.”

Proof: Assume the theorem false. Then in K there are only boundedly many measurable cardinals. Coiterate (K, K^{M}) to get models ⟨⟨N^{i}, K_{i}^{M}⟩|i ≤ θ⟩ for some θ ≤ On, with indices ⟨ν_{i}|i < θ⟩. Let the critical points of the ultrapowers used be ⟨κ_{i}|i < θ⟩.

Note that as the full measures of K^{M} are bounded in On, and K is universal, π_{γγ+1}^{K^{M}} is only non-trivial for set-many indices γ. Let δ be larger than any such γ.

Lemma 1.2 θ = On and there is a truncation on the K-side of the coiteration, and there is a cub C ⊆ {κ_{i}|i < θ} so that i < j ∈ C → i = κ_{i}, π_{ij}^{K}(κ_{i}) = κ_{j}.

Proof: Suppose that K, K^{M} have a common simple iterate P. As K is universal, so is P, and hence K^{M}. As we are below a strong cardinal, K^{M} is a simple iterate of K, via some iteration map σ : K → K^{M} (cf for example, [8] 8.13). But then j ∘ σ : K → K is non-trivial. Hence O^{s} exists. So there is no such P. Thus θ = On. The Basic Coiteration Lemma above gives the cub class C. If there is no truncation, then for for all i ∈ C, N_{i} is a proper class model of ZF C + “κ_{i} is a measurable cardinal ”. In ZF C measurable implies Ramsey, hence as P(κ_{i})^{N_{i}} = P(κ_{i})^{K_{θ}^{M}} for all δ < i ∈ C, we shall have K_{θ}^{M} ⊧ “There is a proper class of Ramsey cardinals”

By elementarity of π_{0θ}^{K^{M}}, so does K^{M}, and by elementarity of j, so does K and this contradicts our global assumption. Q.E.D.Lemma 1.2

Now let \overline{τ} > δ be sufficiently large so that, if π_{γγ+1}^{K} involves a truncation, then \overline{τ} > γ. Then ρ_{N_{τ}}^{ω} ≤ κ\overline{τ} for τ ≥ \overline{τ}. Indeed we shall assume that ρ_{N_{τ}}^{1} ≤ κ\overline{τ} , and that N_{τ} ⊧ “κ_{τ} is the largest cardinal” since otherwise, if N_{τ} ⊧ κ_{τ}^{+} exists then N_{τ} ⊧ κ_{τ} is Ramsey , or if ρ_{N_{τ}}^{1} = On ∩ N_{τ} then N_{τ} ⊧ “KP + Σ_{1} − separation”. And again this is enough to show that N_{τ} ⊧ “κ_{τ} is Ramsey”. (See the remarks in Claim 5 below). If \overline{K} = KM ; \overline{τ} = K_{On}^{M} is the final model on the K^{M}-side we should again have for all τ ≥ \overline{τ} \overline{K} ⊧ “κ_{τ} is Ramsey”.

Now let τ > \overline{τ} be such that

(i) τ ∈ C

(ii) τ a limit cardinal, cf (τ) ≥ ω_{2}.

(iii) cf (τ) is greater than any κ_{j} where κ_{j} is the critical point of any measure used on the K^{M} side.

(iv) j“τ ⊆ τ.

Claim 1 K^{M} ⊧ τ is inaccessible. Proof As τ ∈ C, τ = κ_{τ} and so \overline{K} ⊧ τ is inaccessible. Note by (iii) π_{0\overline{τ}}^{K^{M}} (τ) = τ, and the claim follows. QED(Claim 1)

As j ↾ K^{M} : K^{M} \overrightarrow{e} K, we may perform a standard copying of the iteration K^{M} → \overline{K} via j to get an iteration ⟨K̃^{i}|i ≤ \overline{τ}⟩, where K̃^{0} = K with commuting maps j^{i} : K^{M_{i}}⟶_{e}K̃^{i} as follows:

Set j^{0} = j, K̃^{0} = K. Suppose K̃^{i}, j^{i} defined. Set π = π_{ii+1}^{K^{M}}. Suppose π : K^{M_{i}} → K^{M_{i+1}} = Ult(K^{M_{i}}, E_{i})) with crit(E_{i}) = κ_{i}. Set K̃^{i+1} = Ult(K̃^{i}, j^{i}(E_{i})), and σ_{ii+1} : K̃^{i} → K̃^{i+1} the ultrapower map. If π = id, then we define K^{M_{i+1}} = K^{M_{i}}, j^{i+1} = j^{i}, σ_{i,i+1} = id.

Then define j^{i+1}(π(f)(κ_{i})) = σ_{ii+1}(j^{i}(f))(j^{i}(κ_{i})). At limit stages one takes the appropriate direct limits. Let \overline{ȷ} = j\overline{τ} , K̃ = K̃\overline{τ}.

Claim 2 \overline{ȷ}“τ ⊆ τ for τ ∈ C.

Proof This is immediate from π_{0θ}^{K^{M}} (τ) = (τ), as in the last claim and using the definition of \overline{τ}. QED Claim 2

Claim 3 There exists a Σ_{0}, cofinal j̃: ⟨\overline{K}|τ^{+}, F ⟩ ⟶ ⟨M̃, F̃⟩, with j̃ ⊇ \overline{ȷ} ↾ \overline{K}|τ : \overline{K}|τ ⟶ K̃|τ, where τ^{+} = τ^{+K_{τ}^{M}} = On ∩ N_{τ} , and F = E_{τ^{+}}^{N_{τ}} , the final measure of N_{τ} .

Proof We use a standard lift up technique - see for example, [4] Claim 9, forming a pseudoultrapower

D = \{⟨ξ, f⟩ | fa function ∧ domf ⊂ γ < τ, f ∈ \overline{K}|τ^{+} ∧ ξ ∈ \overline{ȷ}(dom(f))\},

with an ∈^{D}-relation

⟨ξ′, f′⟩ ∈^{D} ⟨ξ, f⟩ ⇔ ⟨ξ′, ξ⟩ ∈ \overline{ȷ}(\{⟨x, y⟩ | f′(x) ∈ f(y)\}).

Standard arguments using the fact that cf(τ) > ω show that ∈^{D} is wellfounded, and D is isomorphic to some premouse M̃ ⊇ K̃|τ.

(For wellfoundedness, suppose not and let ⟨ξ_{n+1}, f_{n+1}⟩ ∈^{D} ⟨ξ_{n}, f_{n}⟩ yield an infinite descending chain in the lift-up. Let A_{n} = {⟨x, y⟩ ∈ γ × γ | f_{n+1}(x) ∈ f_{n}(y)} where γ is sufficiently large below τ so that dom(f_{n}) < γ for all n. Let γ < δ < τ be sufficiently large, so that for all n f_{n} ∈ ran π_{δ,τ}^{K}. Let π(\overline{f}_{n}) = f_{n}. Then A_{n} also equals B_{n} = \{⟨x, y⟩ ∈ γ × γ | \overline{f}_{n+1}(x) ∈ \overline{f}_{n}(y)\} using the \overline{f}_{n}’s, all of which are in the model N_{δ} ∩ \overline{K}|(δ^{+})\overline{K}. (So we have pulled down into \overline{K} witnesses to “illfoundedness”.) But now applying \overline{j} to B_{n} we get for all n, \overline{ȷ}(\overline{f}_{n+1})(ξ_{n+1}) ∈ \overline{ȷ}(\overline{f}_{n})(ξ_{n}) - the usual contradiction.)

We define as usual F̃ = ⋃_{X∈\overline{K}|τ^{+}} j̃(F′ ∩ X). Then ⟨M̃, F̃⟩ is a premouse. QED Claim 3

We note that F is determined by the iteration points ⟨κ_{i}|i ∈ C ∩τ⟩, and so F is ω −complete. Another standard argument (for example, cf [4] Claim 10) shows that F̃ is determined by ⟨j̃(κ_{i})|i ∈ C ∩ τ⟩. Hence

Claim 4 F̃ is ω − complete.

Lastly:

Claim 5 M^{+} = ⟨M̃, F̃⟩ is a mouse and ρ_{M^{+}}^{ω} = ρ_{M^{+}}^{1} = τ.

Proof As j̃is Σ_{0} −cofinal, it is easily seen that all proper initial segments M̃|ν for ν < On∩M̃ are mice.

(1) Γ = {ν < On ∩ M̃ | ⟨M̃|ν, F̃ ∩ M̃|ν⟩ is amenable} is bounded in On ∩ M̃.

Proof The usual proof that τ measurable ⇒ τ Ramsey, for any f ∈ M̃|ν, with ⟨M̃|ν, F̃∩M̃|ν⟩ amenable, proceeds by taking the given function f : [τ]^{<ω} ⟶ τ, and building by induction on n a sequence X_{n} ⊃ X_{n+1} of homogeneous sets so that f“[X_{n}]^{n} has cardinality 1. One may do this over an amenable M̃|ν (which is a model of Σ_{0}-comprehension), and get such a sequence ⟨X_{n} | n < ω⟩ Σ_{1}(M̃|ν, F̃ ∩ M̃|ν). If ν < On ∩ M̃ then X = ∩_{n}X_{n} ∈ F̃ ∩ M̃. If Γ were unbounded in τ^{+}\overline{K}, then τ would be Ramsey in M̃ and so in \overline{K}, and so in N_{τ}. Again then all κ_{i} are Ramsey in \overline{K}, contradicting our global assumption. QED (1).

But the boundedness of Γ shows there is a failure of Δ_{1}-comprehension over ⟨M̃|ν, F̃ ∩ M̃|ν⟩. But this implies ρ_{M^{+}}^{1} ≤ τ, as M^{+} ⊧ τ is the largest cardinal. QED (Claim 5)

As M̃|τ = K̃|τ these conditions show via an easy comparison argument that a code, A_{M^{+}}^{1}, for M^{+} is in K̃, and so M^{+} ∈ K̃. Now suppose A = ⟨A, (f_{n})_{n<ω}⟩ is a first order structure on τ in \overline{K}. Then \overline{ȷ}(A) is a structure with domain \overline{ȷ}(τ) in K̃ whose natural restriction to τ, \overline{ȷ}(A) ↾ τ is j̃(A) ↾ τ. But j̃(A) ↾ τ ∈ M̃ and F̃ is an ω-complete measure on P (τ) ∩ M̃. Hence definably over M^{+} there is a homogeneous set X for j̃(A) ↾ τ of order type τ. (This is the set X = ∩_{n}X_{n} constructed as indicated in the proof of (1) above. Each X_{n} ∈ F̃ and so contains a final segment of the ⟨j̃(κ_{i})|i ∈ C ∩ τ⟩, and thus has order type τ, guaranteeing the same for X.) But M^{+} ∈ K̃, and hence ∀λ < κ X ∩ λ ∈ (H_{τ})_{K}. As \overline{ȷ}(A) ↾ τ is an elementary subalgebra of \overline{ȷ}(A), X ∩ λ are good indiscernibles for \overline{ȷ}(A). Hence for any λ < τ

(2) K̃ ⊧ “\overline{ȷ}(A) has a good set of indiscernibles of order type ≥ λ”

Hence by elementarity \overline{K} thinks the same. As A ∈ \overline{K} was arbitrary, \overline{K} ⊧ “τ is almost Ramsey” and again by elementarity of π_{0τ}^{K^{M}} and j, K thinks the same. Hence all τ ∈ C, τ ≥ \overline{τ}, are almost Ramsey in K. Q.E.D.Theorem 1.1

Theorem 1.3 If ∃j : M \overrightarrow{e} V, j ≠ id with unboundedly many regular cardinals fixed by j, then

K ⊧ “∃ unboundedly many Ramsey cardinals”.

Proof: We use the machinery and notation of the last theorem. Again we assume the theorem false, and hence that there are only boundedly many measurable cardinals in K.

Claim 1 Unboundedly many regular cardinals fixed by j are inaccessible.

Proof Note first that as, again, N\overline{τ} iterates past \overline{K}, for each regular cardinal η > \overline{τ}, there is C_{η} ⊆ η with π_{i,k}^{N}(κ_{i}) = k = κ_{k} for i < k ∈ C_{η}, and N_{η} ⊧ “η is measurable”. So there is F = F_{η}, on the N_{η} sequence, with F a normal measure on η in N_{η}, and with F the final segment filter determined by C_{η}. Hence η is inaccessible in \overline{K}. Now assume that only V - successor cardinals are fixed unboundedly in On by j. Then for any V -successor cardinal satisfying j(η) = η we see that j(β) = β where β^{(n)+} = η, β the maximal limit cardinal below η. By assumption β is singular in M (if it were regular in M, it would be so in V, contradicting our assumption that no limit regular cardinals of V are fixed by j). By the Covering Lemma in M and elementarity β^{+M} = j(β^{+M}) = β^{+V}. Continuing with the notation of the last theorem, in the coiteration of K^{M} with K, by induction on α < \overline{τ} one may check that β^{+K^{M}} = β^{+K_{α}^{M}} = β^{+\overline{K}}. So β^{+} is not inaccessible in \overline{K} - a contradiction.

We then have the following strengthening of Claim 2 of Theorem 1.1:

Claim 2 \overline{ȷ}(η) = η for η any sufficiently large inaccessible fixed point of j.

Proof As all ultrapowers for γ < \overline{τ} taken on the K^{M} side are by meansures in KM ; \overline{τ}, any such ultrapower fixes η. (Recall that \overline{τ} > η is above any γ with π_{γ,γ+1}^{K^{M}} ≠ id.) Hence the same is true of the direct limits. Hence ∀α ≤ \overline{τ}π_{0,α}^{K^{M}} (η) = η. Similarly by induction on i < \overline{τ} it is easy to see that the maps completing the commutative diagram of the i’th stage of the copy iteration, j^{i} : K^{M_{i}} ⟶ K̃^{i}, also satisfy j^{i}(η) = η, and hence j\overline{τ} (η) = η. \overline{ȷ} was defined as j\overline{τ}. QED Claim 2

(Although we do not make use of these facts, note that the linear iteration on the K^{M} side can be embedded into a universal iteration definable inside K^{M}, all of the action of which takes place below KM ; \overline{τ}. This implies that for η as in Claim 2 (η^{+})^{K^{M}} = (η^{+})^{K_{i}^{M}} = (η^{+})\overline{K} for all i < \overline{τ}. Further (η^{+})\overline{K} is fixed by the “completion maps” j^{i} for i ≤ \overline{τ}. As cf((η^{+})\overline{K}) = cf(On∩ N\overline{τ}) ( = ω by our assumption on N\overline{τ}) we also have that (η^{+})^{M} = (η^{+})^{K^{M}} = (η^{+})\overline{K} < (η^{+})^{K} by the fact that N\overline{τ} goes past \overline{K} and by the Covering Lemma applied inside M.)

Hence we may repeat the proof of Theorem 1.1, using F = F_{η} as F (and η for τ) there, but because j^{η}(η) = η we argued at (2) of the last theorem above, that K̃^{η} ⊧ “j(A) = \overline{ȷ}(A) has a good set of indiscernibles I ⊆ η of size η”, and ultimately that K ⊧ “∃ unboundedly many Ramsey cardinals”. Q.E.D.Theorem 1.3

Theorem 1.4 If ∃j : M \overrightarrow{e} V, j ≠ id with stationary many fixed points of cofinality ω then {α | K ⊧ “α is measurable”} is a Mahlo class in K.

Proof: Suppose not. It will suffice to show that K^{M} is universal by Lemma 1.5.

Lemma 1.5 If there exists a stationary class S such that for all τ ∈ S τ is singular in a weasel W then W is universal.

Proof: Suppose W is not universal. Then there exists a mouse M which iterates past W. By the Basic Coiteration Lemma of the introduction there is a cub class C of cardinals left behind as inaccessibles in the model W_{On}. But for any sufficiently large τ ∈ C∩S, we have π_{0,τ}^{W} “τ ⊆ τ. By induction on γ < τ one sees that (τ singular)^{W} ⇒ (τ singular)^{W_{γ}}. This therefore holds in the direct limit at W_{τ} and as crit(π_{τ,On}^{W}) > τ this means that (τ singular)^{W_{On}} - a contradiction.

Q.E.D.Lemma 1.5

But for τ ∈ S, by the Covering Lemma, τ is singular or measurable in K. Since j(τ) = τ, τ is singular or measurable in K^{M}. If the class of τ ∈ S that is measurable in K is non-stationary, by Lemma 1.5, K^{M} is universal. As we are below O^{¶}, K iterates simply up to K^{M} without K^{M} moving, with iteration map π_{KK^{M}}. But then j ∘ π_{KK^{M}} is a non-trivial elementary embedding from K to K which implies O^{s}.

Q.E.D.Theorem 1.4

A similar argument shows that if we weaken the assumption of the last theorem to that of a class of singular cardinals fixed by j, then K contains a proper class of measurable cardinals.

Theorem 1.6 If ∃j : M \overrightarrow{e} V j ≠ id, with stationary many fixed points of cofinality > ω then O^{s} exists.

Proof: As in the proof of the last theorem, it would suffice for a contradiction to show that K^{M} is universal. But j(τ) = τ so τ is singular in K by the Covering lemmas of [9] and [5]. So K^{M} ⊧ τ is singular. The result follows by Lemma 1.5. Q.E.D.

Theorem 1.7 If ∃j : M \overrightarrow{e} V, j ≠ id and ∀α(cf(α) = ω ⇒ cf^{M}(α) = ω) then O^{s} exists.

Proof: We first note that the assumption implies that there is a stationary class S of cardinals τ with cf(τ) = ω which are fixed points of j. Consider the comparison of K with K^{M} with resulting coiteration ⟨⟨N_{i}, K_{i}^{M}⟩|i ≤ θ⟩ for some θ ≤ On, with indices ⟨ν_{i}|i < θ⟩. Let the critical points of the ultrapowers used be ⟨κ_{i}|i < θ⟩. Let α = crit(j). Clearly cf(α) > ω.

(1) On the K side of the coiteration, the very first index used, ν_{0}, is less than α^{+} =_{df} α^{+}^{K}, and hence involves a truncation to a set mouse N_{0} = K|ν_{0}.

Proof: Suppose the first index used on the K side were ν_{0} ≥ α^{+}. Then α^{+} = α^{+}^{K^{M}} and by our assumption, and the Covering Lemma in M, cf(α^{+}) > ω. Then P(α)^{K} = P(α)^{K^{M}} and if U is the derived measure from the embedding j (meaning that U = {X ⊆ α | X ∈ K ∧ α ∈ j(X)}), then consider π : K ⟶ K̃ =_{df} Ult(K, U). K̃ is wellfounded as α^{+} = α^{+}^{K}, and U ω-complete. As we are below O^{¶}, π is an iteration map, π_{0,μ} for an iteration by full measures ⟨E_{ν}|ν ≤ μ⟩. The first measure, E_{0}, used in this iteration must have critical point α = crit(U), and so E_{0} = E_{α^{+}}^{K}; and it agrees with U on P(α) ∩ K; so U must be E_{α^{+}}^{K}. Let σ : K^{M} ⟶ K′ =_{df} Ult(K^{M}, U), then E_{α^{+}}^{K′} = ∅, by coherence. Defining k : K′ ⟶_{Σ_{0}} K by k([f]_{U}) = j(f)(α) we have as usual that k ↾ α^{+} +1 = id ↾ α^{+} +1, whilst E_{α^{+}}^{K} ≠ ∅, E_{α^{+}}^{K′} = ∅ - a contradiction. QED (1)

We have as usual a cub class C and N_{1} iterating past K^{M}, with π_{N_{i}N_{k}}(κ_{i}) = κ_{k} for i < k ∈ C, whilst π_{K_{0}^{M} ,K_{i}^{M}} “κ_{i} ⊆ κ_{i}.

However C ∩ S cannot form a stationary class of cardinals singular in K^{M}, as then we should have K^{M} universal - a contradiction as in the last theorem. Again the only other possibility is that it forms a class of measurables of K^{M}.

Let ω_{2} < κ ∈ C ∩ S with κ greater than the last truncation point on the iteration on the K side. Let κ′ = κ^{+}^{K_{κ}^{M}}. Then K_{κ}^{M}|κ′ is an initial segment of N_{κ}, and the latter is sound above κ. Then κ′ = κ^{+}^{N_{κ}} = sup π_{K^{M} ,K_{κ}^{M}} “κ^{+}^{K^{M}}. Note

(2) ρ_{N_{κ}}^{ω} < κ, and κ > cf(κ′) > ω. Proof: By (1) there is at least one truncation on the K-side, and from this the first clause follows: since from the last truncation point i_{0} say, we have ρ_{N_{io}}^{n} ≤ κ_{i_{0}} for some n. For this n we have ρ_{N_{κ}}^{n} < κ. As M calculates countable cofinalities correctly, by the Covering Lemma

in M, cf(κ^{+}^{K^{M}}) = cf(κ^{+}^{M}) > ω. QED (2)

Let the iteration ⟨K_{i}^{M} | i ≤ κ⟩ with indices ⟨ν_{i}|i < κ⟩ be copied (as in Claim 1 of Theorem 1.1) via j = σ_{0} to an iteration of K = K̃^{0} as ⟨K̃^{i} | i ≤ κ⟩ with commuting maps σ_{i} : K_{i}^{M} ⟶_{e} K̃^{i} and indices ⟨σ_{i}(ν_{i})|i < κ⟩.

Let δ = sup σ_{κ}“κ′. Then σ_{κ} : K_{κ}^{M}|κ′ ⟶_{Σ_{0}} K̃^{κ}|δ. Note that σ_{κ}(κ′) = κ^{+_{K^{κ}}} = κ^{+K} = κ^{+V}

by the Covering Lemma in V, and the fact that the copy iteration below κ fixes κ and κ^{+K}.

So we see δ < κ^{+_{K^{κ}}}.

We now have to do a fine-structural lift-up argument, as this time we know that ρ_{N_{κ}}^{n+1} < κ < ρ_{N_{κ}}^{n} for some n ≥ 0. Fix this n.

(3) There is a lift-up map σ̃ : N_{κ} ⟶_{Σ^{∗}} M̃ for some mouse M̃ with σ̃ ⊇ σ_{κ} ↾ K_{κ}^{M}|κ′ where M̃ has the following properties: (i) M̃|δ = K̃^{κ}|δ; (ii) ρn+1 ; _{M} < δ ≤ ρn ; _{M}; (iii) M̃ is the Σ_{1}^{(n)}-closure

of κ ∪ {p} for some p ∈ Rn+1 ; _{M}.

Proof: This is a somewhat standard argument (cf [3] §3.3.2) exploiting the fact that cf(κ′) > ω to get a wellfounded fine-structural “pseudo-ultrapower”. Instead of the coarse pseudoultrapower of Claim 3 of Theorem 1.1 we define

D = {⟨a, f⟩ | dom(f) ⊂ τ < κ′ and either f ∈ Σ_{1}^{(m)}(N_{κ})(m < n) ∨ f ∈ H_{ρ_{Nκ}^{n}}^{N_{κ}} }

and then set

⟨a, f⟩{e \atop I}⟨b, g⟩ ⟺ ⟨a, b⟩ ∈ σ_{κ}(\{⟨u, v⟩ | f(u){∈ \atop =}g(v)\}

and

Ġ(⟨a, b⟩) ⟺ a ∈ σ_{κ}({u | f(u) ∈ G}) (for G ∈ {F, E}).

We then define D = ⟨D, I, e, Ė, Ḟ ⟩ and proceed to prove a Łos Theorem for Σ_{0} formulae. The fact that cf(κ′) > ω results in e being wellfounded, and then D is isomorphic to M̃ for some premouse M̃. The canonical embedding σ̃ : N_{κ} ⟶ M̃ is defined by σ̃(x) = [⟨0, {⟨x, 0⟩}⟩]_{I}. That M̃ is iterable, and so a mouse, is an amplification of the argument that it is wellfounded (see [3] §3.3.3). Using that κ′ is the N_{κ}-successor of κ enables one to prove that σ̃ is a Σ_{1}^{(n)} preserving embedding, and ultimately properties (ii) and (iii) ((i) holds by construction). The full proof is in §3.4.

QED (3)

(4) M̃ ∈ K̃^{κ}; thus ρω ; _{M} = ρn+1 ; _{M} = κ.

Proof: There is a Σ_{1}^{(n)}(M̃) subset of κ that codes M̃. As K̃^{κ} is universal (being just a simple iterate of K), in their coiteration the first point of difference between M̃ and K̃^{κ} is not less than δ, so we see that this code is in K̃^{κ}. As κ is obviously a cardinal of K̃^{κ} we must have

ρω ; _{M} = ρn+1 ; _{M} = κ. QED (4)

(5) core(M̃) is an initial segment of K̃^{κ}. core(M̃)|δ + 1 = M̃|δ + 1; hence E_{M} ; δ = E_{K^{κ}} ; δ.

Proof: Let P =_{df} core(M̃). Then P ≅ Σ_{1}^{(n)_{M}}Hull(ωρn+1 ; _{M} ∪ {p_{M}}). (Where p_{M} is the standard

parameter of M̃.) But as E_{M} ; δ ≠ ∅, M̃ is an iterate of P above δ. Let γ ≥ δ be least so that ρ_{_{K^{κ}}|γ}^{ω} ≤ κ. Let N = K̃^{κ}|γ. Then N||δ = M̃||δ = P ||δ. But over both P and N there are Σ^{∗} definable subsets of κ coding wellorders of type δ. As δ = κ^{+} (or is the height of the ordinals) in both models, neither model contains such a code. So a simple comparison argument shows that P and N coiterate without truncation to the same model. But then by soundness, in fact P = N. QED (5)

Remark: With a little more work on parameters, one can show that in fact M̃ is itself sound and an initial segment of K̃^{κ}.

Let \overline{F} = E_{κ′}^{K_{κ}^{M}}. Let \overline{M} = |K_{κ}^{M}||κ′| = J_{κ′}^{E^{K^{M}_{κ}}}. Let F be defined by: F = ⋃_{X∈\overline{M}} σ_{κ}(X ∩ \overline{F}). Then:

(6) σ_{κ} extends to: σ_{κ} ↾ ⟨\overline{M}, \overline{F} ⟩ ⟶_{Σ_{0}} ⟨K̃^{κ}|δ, E_{K^{κ}} ; δ⟩ ∧ F = E_{K^{κ}} ; δ.

Proof: σ_{κ} is a cofinal map into |K̃^{κ}||δ|. And F = |K̃^{κ}||δ| ∩ E_{K^{κ}} ; σ_{κ}(κ′). By the initial segment

condition F = E_{K^{κ}} ; δ. QED (6)

(6) shows, as σ̃ ⊇ σ_{κ}, that E_{K^{κ}} ; δ = E_{M} ; δ ⇒ E_{κ′}^{K_{κ}^{M}} = E_{κ′}^{N_{κ}}. So the conclusion of this consequence holds by (5). But E_{κ′}^{K_{κ}^{M}} ≠ E_{κ′}^{N_{κ}} by virtue of the comparison process a contradiction!

Q.E.D.Theorem 1.7

Let K′ be a core model for sequences of measures with o(κ) < (κ)^{++} (for example Mitchell’s Core Model).

Question If ∃j : M \overrightarrow{e} V and j ≠ id, α is a regular cardinal, and there exists stationary many cardinals of cofinality α fixed by j, and M computes correctly cofinalities ≤ α, then:

(a) is this consistent?; or, (b), does K′ ⊧ “∃ unboundedly many measurable cardinals κ with o(κ) = α” (where we set o(κ) = 0 if α = ω)?

2 Some converses

We briefly consider here some partial converses to the above. It might be thought at first that with some extra work, one could obtain from a non-trivial elementary embedding j : M → V, an inner model with a measurable cardinal. The following indicates that this is not possible. Suppose that κ is a strongly inaccessible Jónsson cardinal, and that there is A ⊆ κ with H_{λ} = L_{λ}[A] for cardinals λ < κ, and (by the Jónsson property) that there is a non-trivial j : L_{κ}[\overline{A}] ⟶ L_{κ}[A], then we clearly have candidates for a “j : M ⟶ V ” by taking M = L_{κ}[\overline{A}] and V as L_{κ}[A]. And such a Jónsson cardinal is known to be equiconsistent with a Ramsey cardinal. See [6]. Notice also that this argument shows that if κ, j : L_{κ}[\overline{A}] ⟶ L_{κ}[A] are as above, then we can perform a “lift-up” argument (as in Claim 3 of Theorem 1.1) to extend j to j̃ : L[\overline{A}] ⟶ L[A] (using the regularity of κ). We can also think of this argument as taking an extender ultrapower of L[\overline{A}], Ult(L[\overline{A}], E_{j}) where E_{j} is the long extender derived from the embedding j; by the regularity of κ this is wellfounded. But then V cannot be L[A] since otherwise E_{j} ∈ V and the latter would be an ultrapower of the inner model L[\overline{A}]! Hence:

Lemma 2.1 Suppose V = L[A] for some set A ⊆ κ with κ strongly inaccessible. (i) If j : M ⟶ V, (or even if j ↾ L_{κ}[\overline{A}] ⟶ L_{κ}[A]) then j = id. (ii) κ is not a Jónsson cardinal.

Instead of arguing in terms of the Jónsson property, one can frame a hypothesis of the form “Suppose On is Ramsey ....” (or again reformulation in terms of some V_{κ} where κ is a Ramsey cardinal etc.).

Definition 2.1 (i) On is Ramsey if there is a class I ⊆ On, unbounded, of good indis-

cernibles for ⟨V, ∈⟩, (ii) On is 1-Ramsey, if the class I in (i) is Mahlo.

In (ii) we mean that I has non-empty intersection with any closed and unbounded class C ⊂ On, definable (with parameters allowed) over ⟨V, ∈, I⟩.

Notice that if κ is a Ramsey cardinal, then we have an I ⊆ κ so that in ⟨V_{κ}, ∈ I⟩, On is Ramsey in this sense. Of course, if On is Ramsey we have, definably over ⟨V, ∈, I⟩, a satisfaction relation for ⟨V, ∈⟩, since α < β ∈ I ⇒ V_{α} ≺ V_{β}. Further, using AC, we may define skolem functions, and so skolem hulls for ⟨V_{α}, ∈⟩, so that α < β ∈ I ⇒ Hull_{V_{β}} (γ) = Hull_{V_{α}}(γ) for γ < α. A prototypical situation is again where κ is Ramsey in K and then all the above hold over ⟨K_{κ}, ∈ I⟩.

Theorem 2.2 (i) Suppose I ⊆ On witnesses On is Ramsey. Then, definably over ⟨V, ∈, I⟩, there is a transitive class M, and an elementary embedding j : ⟨M, ∈⟩ ⟶ ⟨V, ∈⟩, with j ≠ id.

(ii) Additionally, if I witnesses On is 1-Ramsey, then j can be taken with unboundedly many regular fixed points.

Proof: The proof of the proposition is straightforward. Let α_{0} = min I. By the remarks above we may define in ⟨V, ∈, I⟩ the hulls H_{β} =_{df} Hull_{V_{β}}((I ∩ β\{α_{0}}) ∪ α_{0}) for β ∈ I, and their transitive collapses with maps π_{β} : M_{β} ⟶ H_{β}. By the usual properties of good indiscernibles, it is easy to verify that crit π_{β} = α_{0}, and that α < β ∈ I ⇒ π_{β} ↾ M_{α} = π_{α}. Hence we may set j = _{⋃β∈I} π_{β}, M = ⋃_{β∈I} M_{β}. This concludes (i). For (ii), simply note that there must be, stationarily often, be inaccessible δ ∈ I with j(δ) = δ. Q.E.D.

Part (i) of the last theorem can be rephrased as a relative consistency result from the existence of a Ramsey cardinal. This shows the answer to the question of Kunen is negative.

A number of questions about the possibilities of finding models which satisfy further closure properties are immediate. For example:

Question 1 Can one arrange for there to be an inner model M and a non-trivial j : M ⟶ V, with stationary many fixed points of cofinality ≥ ω?

Question 2 The same question as for 1, but with M correctly computing the cofinality of ordinals β with cf(β) = ω?

Our original Theorem 1.7 had the hypothesis that ^{ω}M ⊆ M. Matt Foreman pointed out that this was impossible. We thank him for letting us include his proof here.

Theorem 2.3 (Foreman) If ∃j : M ⟶ _{e}V, j ≠ id then ^{ω}M ⊈ M.

Proof: Suppose for a contradiction that there exists such an M and j satisfying also the conclusion. Let α = crit(j). By the supposed closure on M it is easy to see that there are α < λ_{0} … < λ_{n} < … for n < ω with each λ_{n} a singular cardinal of cofinality ω and j(λ_{n}) = λ_{n}. Then ⟨λ_{n}⟩_{n<ω} ∈ M (again by ω-closure). We show that there is a regular fixed point. Consider any ultrapower of ⟨λ_{n}⟩_{n<ω} by a non-principal ultrafilter U ∈ M on ω. Notice that by ω-closure of M it makes no difference whether this ultrapower is considered as taken in M or V. Let κ be its cofinality.

(1) κ is a regular fixed point of j.

Proof: Note that κ is a supremum of fixed points of j. Also κ is definable in M from ⟨λ_{n}⟩_{n<ω} and U. Hence j(κ) = κ. But cf(κ) = κ in M, and hence also in V. Q.E.D.(1)

We now construct in M an ω-Jónsson algebra on κ. Let S = {β < κ | cf(β) = ω}. Split S into κ many disjoint stationary pieces of size κ: S = ∪_{δ<κ}X_{δ} with γ ≠ δ ⟶ X_{γ} ∩ X_{δ} = ∅; and X_{γ} stationary in κ. For any ⟨β_{n}⟩_{n<ω} with β_{n} < β_{n+1} for all n, define F (⟨β_{n}⟩) =_{df} that γ_{0} so that sup β_{n} ∈ X_{γ_{0}}. This defines F : ^{ω}[κ] → κ.

(2) F is an ω-Jónsson algebra in M.

Proof: Let Y ⊆ κ be arbitrary. Let Y^{∗} =_{df} {sup f“ω | f ∈ ^{ω}[Y]}. The Y^{∗} is an ω-club in κ, (i.e. is unbounded and closed under ω sequences) and so Y^{∗} ∩ X_{γ} ≠ ∅ for any γ < κ. Hence F “^{ω}[Y] = κ. Q.E.D.(2)

Clearly F̃ =_{df} j(F) is an ω-Jónsson algebra in V. Now look at Y = j“κ. By the above there is β_{n} = j(γ_{n}) for n < ω with ⟨γ_{n}⟩ ∈ M, and F̃(⟨β_{n}⟩) = α. This is absurd as it implies α = j(F (⟨γ_{n}⟩)) whereas α = crit(j) ∉ ran(j). Q.E.D.

3 Some fine structure

3.1 The Jensen Indiscernibles Lemma

We outline here a proof of the following lemma, due to Jensen, using the fine-structural hierarchy of [3]. The original lemma was proven using the older-style mice. It will establish (Cor. 3.3) not unsurprisingly, the downwards absoluteness of various Ramsey and Erdős properties between V and any universal weasel. We shall also use it to give variant proofs of Theorems 1.1 & 1.3. We assume throughout §3.1 that O^{s} does not exist (otherwise every cardinal is measurable in K and our results become trivially true).

Lemma 3.1 Let κ be a cardinal and suppose A ∈ K ∩ P(κ), and that there is I, an infinite good set of indiscernibles for A = ⟨L_{κ}[E], E, A⟩, (where E = E^{K}) and that cf (otp(I)) > ω. Then there is I′ ∈ K, I′ ⊇ I a set of good indiscernibles for A.

Proof: Standard arguments show that each γ ∈ I is strongly inaccessible in K, and hence A ↾ γ = ⟨L_{γ}[E], E ↾ γ, A ∩ γ⟩ ⊧ ZF C. Closely following the original proof of [2], we set Ã_{γ}^{n} = Hull_{A}(γ + 1 ∪ γ⃗) where γ⃗ is any member of [I\γ + 1]^{n}; A_{γ}^{n} = Ã_{γ}^{n} ∩ K_{γ^{+}} (γ^{+} = γ^{+K} throughout this proof); A_{γ} = ∪_{n}A_{γ}^{n}; δ^{n} = On ∩ A_{γ}^{n}.

Note δ^{n} makes sense as A_{γ}^{n} is transitive, and δ^{n} is definable in A_{γ}^{n+1}, and (A_{γ}^{n} ≺ A_{γ}^{n+1} ≺ A_{γ}). Hence:

(1) δ^{n} < δ^{n+1}.

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}(ν⃗, γ, γ⃗)) = t_{A}(ν⃗, γ′, γ⃗) for any choice of γ⃗ ∈ [I\γ′ + 1]^{<ω}.

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}⟩ is amenable.

Set γ^{∗} = sup I, and

⟨⟨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 and cf(otp(I)) ≥ ω_{1}:

(3) U_{γ^{∗}} is ω-complete.

Let A_{γ^{∗}} = ⟨L_{β^{∗}}[E^{∗}], E^{∗}, A^{∗}⟩. Clearly A^{∗} ∩ γ^{∗} = A ∩ γ^{∗}, E^{∗} ↾ γ^{∗} = E ↾ γ^{∗}, and, using (2), ⟨L_{β^{∗}}[E^{∗}], U_{γ^{∗}}⟩ is amenable, and, using (3), iterable. Further E ↾ γ, A ∩ γ ∈ A_{γ}^{∗} ⊧ ZF^{−}; hence:

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

For η ≤ β^{∗}, let U_{η} = U_{γ^{∗}} ∩ J_{η}^{E^{∗}} and let N_{η} = ⟨J_{η}^{E^{∗}}, E^{∗} ↾ η, U_{η}⟩. Now let β ≤ β^{∗} be least so that a) A ∩ γ^{∗} ∈ J_{β}^{E^{∗}} b) N = ⟨J_{β}^{E^{∗}}, E^{∗} ↾ β, U_{γ^{∗}}⟩ is amenable. Then recall that N_{η} is a quasi-premouse if it satisfied all conditions for premousehood except possibly the Initial Segment Condition. A quasimouse is an iterable quasi-premouse.

(5) N = N_{β} is a quasimouse. Proof: N ⊧ γ^{∗} is the largest cardinal, and iterability of N comes from the ω-completeness of U_{γ^{∗}} If π : N ⟶ P =_{df} Ult(N, U_{γ^{∗}}) then by amenability γ^{∗}^{+P} = On∩N = ωβ. For coherency notice that E_{ωβ}^{P} must be empty else ⟨J_{β}^{E^{P}} , E^{P} , E_{ωβ}^{P}, U_{γ^{∗}}⟩ is a “sword” mouse. QED (5)

But then N actually is a mouse as its terminating coiteration with K (or more precisely, the ωfull weasel W built over K|κ = N|κ (cf [3] §3.2) shows). By definition of β, ρ_{N}^{1} ≤ crit(U_{γ^{∗}}) = κ. The coiteration just mentioned shows A_{N}^{1}, and so N, are in K. As A ∩ γ^{∗} ∈ |N|, and U_{γ^{∗}} is ω-complete, we now follow the sketch at Claim 5 of Theorem 1.1 above, and we can define over the amenable N a sequence of sets X_{n} ∈ U_{γ^{∗}} whose intersection X ⊇ I and X is a good set of indiscernibles for ⟨L_{γ^{∗}}[E ↾ γ^{∗}], A ∩ γ^{∗}⟩. (For the full details here see the argument of [1] p.131 for example). Then again I′ ∈ K, and we are done.

Q.E.D.Lemma 1.10

Corollary 3.2 Let \overline{K} = L[\overline{E}] be any universal weasel. Then the last lemma remains true with \overline{K}, \overline{E} replacing K, E.

Proof: The previous construction works perfectly well using the predicate \overline{E}. Universality of K was used to argue that the N there constructed was in K. As ρ_{N}^{1} ≤ γ^{∗}, and N|γ^{∗} = \overline{K}|γ^{∗} here, we shall have here N ∈ \overline{K}, and so this suffices.

Q.E.D.Cor.3.2

Corollary 3.3 The following are downward absolute between V and any universal weasel W: “κ is Ramsey”, “κ is almost Ramsey”, “κ is τ-Erdős” where cf^{V}(τ) ≥ ω_{1}.

We now show how Theorem1.1 can be derived using the indiscernibles lemma: in the notation of the proof of Theorem 1.1 we have a mouse N\overline{τ} iterating past the final model \overline{K} on the K^{M}- side. Let τ > \overline{τ} fulfill conditions (i) - (iv). Let I = \{κ_{γ}|\overline{τ} < γ < τ\} be the cub in τ set of iteration points used in the iteration of π_{\overline{τ},τ}^{N} : N\overline{τ} ⟶ N_{τ}. These are then good indiscernibles for ⟨\overline{K}_{τ} , \overline{E}, \overline{A}⟩ where \overline{A} ∈ P(τ)\overline{K}, (= P(τ)^{N_{τ}} ) , \overline{K} = L[\overline{E}]. Continuing with the notation of

Theorem1.1, if \overline{ȷ} : \overline{K} ⟶ K̃, \overline{ȷ}(\overline{A}) = A, \overline{ȷ}“I = I′ ⊆ τ, then I′ is a set of good indiscernibles

for ⟨K̃_{\overline{ȷ}(τ)}, Ẽ, A⟩ (setting \overline{ȷ} = j\overline{τ} , Ẽ = E_{K}) with cf(otp(I′)) > ω.

Applying Corollary 3.2 we have K̃ ⊧ ∀τ′ < τ∃ I′ ⊆ τ (otp(I′) ≥ τ′ ∧ I′ is good for Ã = ⟨K̃_{\overline{ȷ}(τ)}, Ẽ, A⟩).

As \overline{ȷ}“τ ⊆ τ, by elementarity we have \overline{K} ⊧ ∀τ′ < τ∃I′ ⊆ t(otp(I′) ≥ τ′ ∧ I′ is good for A = ⟨\overline{K}, \overline{E}, \overline{A}⟩).

As \overline{A} was arbitrary we thus have \overline{K} ⊧ “τ is almost Ramsey”. There are thus arbitrarily large τ for which this is true. By elementarity, the same is true K^{M} and, via j, hence in K.

To see that Theorem 1.3 is true, pick a sufficiently large regular cardinal τ > \overline{τ}, satisfying (i)-(iv), and so that j(τ) = τ. Now in the above analysis \overline{ȷ}(τ) = τ and so in K̃ we shall have a τ-sequence of good indiscernibles for ⟨K̃_{τ} , Ẽ, A⟩. Hence K̃ ⊧ “τ is Ramsey”. By elementarity of π_{0τ}^{K^{M}} and j, τ is then Ramsey in K.

3.2 Σ^{∗} Fine Structure

In what follows we assume that M = ⟨J_{α}^{A}, ∈, A, B⟩ is an amenable structure with J_{α}^{A} a level of a relativised J-hierarchy, (a “J-model”), that satisfies the axiom of acceptability. Definition 3.1 A J-model M = ⟨J_{α}^{A}, ∈, A, B⟩ as above is acceptable if ∀γ < α P(ωτ) ∩ J_{γ+1}^{A} ⊈ J_{γ}^{A} ⟶ J_{γ+1}^{A} ⊧ card(γ) ≤ τ. Definition 3.2 ρ_{M} (the Σ_{1}-projectum of M) =_{df} the least ρ such that for some p ∈ M there is a Σ_{1}^{M}({p}) definable class A with A ∩ ωρ ∉ M. Remark: ρ_{M} is a Σ_{1}-cardinal of M. Definition 3.3 P_{M} =_{df} {p ∈ M | there is such a class A as above with Σ_{1}^{M}({p}) R_{M} =_{df} p ∈ M | h_{M}“(ω × ([ωρ]^{<ω} × {p}) = |M|}. Remark: We use here the notation that h_{M} is the canonical Σ_{1}-skolem function for such a J-model M, and we abbreviate by h(X) the set h“(ω × [X]^{<ω}). Let ⟨φ_{i} | i ∈ ω⟩ enumerate the Σ_{1}-formulae of the language L_{{ ˙∈, ˙=,Ȧ,Ḃ}} with two free variables.

Definition 3.4 A_{M}^{p} =_{df} {⟨i, x⟩ ∈ ω × J_{ρ_{M}}^{A} | M ⊧ φ_{i}(x, p)} By using Gödel pairing and the remark above one can easily see that p ∈ P_{m} ⟺ A_{M}^{p} ∩ (ω × ωρ) ∉ M. Note also that R_{M} ⊆ P_{M} ≠ ∅. Definition 3.5 M^{p} =_{df} ⟨J_{ρ_{M}}^{A} , A_{M}^{p}⟩ for any p ∈ M. We iterate this idea and define simultaneously by recursion on n < ω: Definition 3.6 The n-th projectum ρ^{n} = ρ_{M}^{n}, the set Δ^{n} = Δ_{M}^{n} of possible parameter sequences, and the n-th reduct M^{n,p} of M for p ∈ Δ^{n}, and H^{n} = H_{M}^{n}.

ρ_{M}^{0} = ωα = On ∩ M; Δ^{0} = {∅}; M^{0,∅} = M; H_{M}^{0} = |M|; A^{0,p} = ∅;

ρ^{n+1} = ρ_{M}^{n+1} = min{ρ_{M^{n,p}} | p ∈ Δ^{n}}; Δ^{n+1} = Π_{i<n+1}H^{i};

For p ∈ Δ^{n+1}:

M^{n+1,p} = (M^{n,p↾n})^{p(n)} | ρ^{n+1} =_{df} ⟨J_{ρ^{n+1}}^{A}, A_{M^{n,p↾n}}^{p(n)} ∩ J_{ρ^{n+1}}^{A}⟩
H^{n+1} = |J_{ρ^{n+1}}^{A}|; A^{n+1,p} =_{df} A_{M^{n,p↾n}}^{p(n)} ∩ J_{ρ^{n+1}}^{A};
ρ_{M}^{ω} = min_{n} ρ_{M}^{n}.

We then organise the possible parameter sequences as follows: Definition 3.7 R^{0} = P^{0} = {∅};

P^{n+1} = {p ∈ Δ^{n+1} | p ↾ n ∈ P^{n} ∧ ρ^{n+1} = ρ_{M^{n,p↾n}} ∧ p(n) ∈ P_{M^{n,p↾n}}};

R^{n+1} = {p ∈ Δ^{n+1} | p ↾ n ∈ R^{n} ∧ ρ^{n+1} = ρ_{M^{n,p↾n}} ∧ p(n) ∈ R_{M^{n,p↾n}}}.

The sequences p ∈ P_{M}^{n} are those parameter sequences where additional components p(i) are “good” for defining new subsets of ωρ^{i+1} over the ith reduct with respect to the previous components p ↾ i, M^{i,p↾i}. These always exist. Those p ∈ R_{M}^{n} perform a similar function, but in addition, the Σ_{1} Skolem function of this i’th reduct, acting on ωρ^{i+1} ∪ {p(i)} recovers all of that reduct. These are clearly “very good” parameters, and will not exist in general. We do have the following facts:

Fact 3.1 (i) R^{n} ⊆ P^{n} ≠ ∅; (ii) if p ∈ R^{n} and q ∈ Δ^{n} then A_{M}^{n,q} is rudimentary in A_{M}^{n,p} in parameters from H_{M}^{n}. (iii) If p ∈ R^{n} then ρ^{n+1} = ρ_{M^{n,p}}; if p ∈ R^{n+1} then M^{n+1,p} = (M^{n,p})^{p(n)}; (iv) p ∈ R^{n} ⟷ p ∈ Δ^{n} ∧ ∀i < n p(i) ∈ R_{M^{i,p↾i}}. (v) ρ^{n+1} = min{ρ_{M^{n,p}} | p ∈ P^{n}}.

Definition 3.8 If R^{n} = P^{n} then we say M is n-sound. M is sound if it is n-sound for all n. It is sound above μ if

(a) ρ_{M}^{n} ≥ μ ⟶ M is n-sound, and

(b) if ρ_{M}^{n} ≥ μ > ρ_{M}^{n+1} ∧ p ∈ P_{M}^{n+1} then M^{n,p↾n} = h_{M^{n,p↾n}}(μ ∪ {p(n)}).

Notions of “fine-structural” preserving maps (and ultimately ultrapowers) can be smoothly presented in terms of Jensen’s Σ^{∗} hierarchy of formulae. This is a hierarchy of definability over a J-model, with a different order of stratification than the usual Levy hierarchy of formulae. Although the relations Σ^{∗}-definable over M are also the usual Σ_{ω}-definable relations, the intermediate levels of Σ_{n}-definability do not in general correspond to those within Σ^{∗}. More complex initially, the Σ^{∗}-hierarchy possesses nice properties the Levy hierarchy lacks: for example, over L, Jensen’s Σ_{n}-uniformization Theorem states that such relations may be Σ_{n}- uniformized albeit in a parameter. When given the Σ^{∗}-analysis, Σ_{1}^{(n)}(J_{α}) relations (to be defined) enjoy uniform Σ_{1}^{(n)}(J_{α}) Uniformizing functions. (See the two lemmas stated at the end of this subsection.)

We now define this language L_{{ ˙∈, ˙=,Ȧ,Ḃ}}^{∗}. This has variables v_{j}^{i} (i, j < ω) of type i. The

atomic formulae are those of the form v_{j}^{i} ∈ Ȧ(Ḃ), v_{j}^{i} ∈ v_{l}^{k}, v_{j}^{i} = v_{l}^{k}. The formulae are those obtained from the atomic formulae by closing under ¬, ∧ and typed quantification ∃v_{j}^{i}, ∀v_{j}^{i}. The stratification we referred to earlier is as follows:

Definition 3.9 (a) The Σ_{0}^{(n)} formulae, is the smallest class Σ ⊂ L^{∗} such that:

(i) Σ contains the atomic formulae, and the formulae Σ_{1}^{(m)} for any m < n; (ii) Σ is closed under ¬, ∧, and quantification which binds variables of type n by a higher type, that is if φ ∈ Σ, so is ∀x^{n} ∈ y^{m}φ, ∃x^{n} ∈ y^{m}φ, where m ≥ n;

(b) Σ_{k}^{(n)} formulae are obtained by alternating k blocks of quantifiers of type n: ∃x_{1}^{n}∀x_{2}^{n} … Qx_{k}^{n}φ where φ ∈ Σ_{0}^{(n)}. We set Π_{0}^{(n)} = Σ_{0}^{(n)} and Σ^{∗} = ⋃_{n} Σ_{0}^{(n)} ( = ⋃_{n} Σ_{1}^{(n)}).

The variables v_{m}^{n} are intended to range over H_{M}^{n} =_{df} H_{ωρ_{M}^{n}}^{M} of an acceptable J-model M. Note that n ≤ m ⟶ ρ_{M}^{n} ≥ ρ_{M}^{m} and hence that v_{i}^{m} will vary over a smaller domain than v_{j}^{n}. The notion of Σ_{k}^{(n)} relation and function are defined fairly straightforwardly, using these formulae. A full analysis of such relations and functions would take us too far afield at this point, however we shall refer to the good Σ_{1}^{(n)}-functions as those functions that may be substituted into a Σ_{1}^{(n)} relation and still yield a Σ_{1}^{(n)} relation. We shall not use the following definition, but they may be defined as follows:

Definition 3.10 The good Σ_{1}^{(n)}(M)-functions form the smallest class S, such that, setting i, j_{1}, …, j_{k} ≤ n

(i) Each partial Σ_{1}^{(i)}(M) function F (x_{k}^{j_{k}}, …, x_{1}^{j_{1}}) = x^{i} ∈ S; (ii) If F (x_{k}^{j_{k}}, …, x_{1}^{j_{1}}) = x^{i} ∈ S and G_{i}(y⃗) = z^{j_{i}} ∈ S (and the y⃗ all have type ≤ n), then F (G_{k}(y⃗), …, G_{1}(y⃗)) ∈ S.

We may define Σ_{k}^{(n)}-preserving embeddings in a natural way:

Definition 3.11 Let \overline{M}, M be J-models, n, l < ω.

(i) π : \overline{M} ⟶_{Σ_{l}^{(n)}} M iff π : \overline{M} ⟶ M and whenever φ(v_{1}^{j_{1}}, … v_{m}^{j_{m}}) ∈ Σ_{l}^{(n)}, x_{i} ∈ Hj_{i} ; \overline{M} (1 ≤ i ≤

m) then π(x_{i}) ∈ H_{M}^{j_{i}}, and \overline{M} ⊧ φ(x⃗) ⟶ M ⊧ φ(π⃗(x)).

(ii) π : \overline{M} ⟶_{Σ^{∗}} M iff π : \overline{M} ⟶_{Σ_{l}^{(n)}} M for all n < ω.

(iii) π : \overline{M} ⟶_{Σ_{0}^{(n)}} M cofinally iff π : \overline{M} ⟶_{Σ_{0}^{(n)}} M and H_{M}^{n} = ⋃ π“Hn ; \overline{M}.

Fact 3.2 If F is a good Σ_{1}^{(n)}(\overline{M}) definable function, then one may show that F has such a definition that is “functionally absolute”, that is a definition that is robust under Σ_{1}^{(n)} preserving maps. In other words, the definition applied over M where π : \overline{M} ⟶_{Σ_{l}^{(n)}} M also

yields a Σ_{1}^{(n)} function.

Some of the fruits of this analysis are seen in the following two lemmas.

Lemma 3.4 Let R(y^{n}, x_{1}^{j_{1}}, … x_{m}^{j_{m}}) ∈ Σ_{1}^{(n)}(M) with j_{i} ≤ n. Then there is a good Σ_{1}^{(n)}(M) function F (uniformly definable with respect to any such M) into H_{M}^{n} uniformising R. Namely a) domF = {⟨x_{1}^{j_{1}}, … x_{m}^{j_{m}}⟩ ∈ M | ∃y^{n} ∈ MR(y^{n}, x⃗)}. b)∃y^{n}R(y^{n}, x⃗) ⟷ R(F (x⃗), x⃗).

Lemma 3.5 Let n < ω. There is a Σ_{1}^{(n)} formula defining a good Σ_{1}^{(n)}(M) function F_{n}(u, v) into M, (definable uniformly with respect to any J-model M) so that

∀p(p ∈ R_{M}^{n+1} ⟶ |M| = F_{n}“H_{M}^{n+1} × {p}).

3.3 Σ^{∗}-ultrapowers

We give an account of the formation of a fine-structure preserving ultrapower. We shall develop this theory for ultrapowers by extenders that are weakly amenable with respect to the models concerned. This is more than one needs for the models in the previous sections, as these were all dealing with premise with measures of order 0. However, this greater generality now will enable us to quickly dispose of “pseudo-ultrapowers” in the next subsection. In any case it is still basically the account of such ultrapowers from [3]. Suppose that \overline{M} = ⟨J_{α}^{A}, A, B⟩ is a J-model and E is an extender with critical point crit(E) = κ over \overline{M} and E = ⟨E_{a} | α ∈ [ν]^{<ω}. We assume that E is weakly amenable with respect to \overline{M}. This is defined as for any a ∈ [ν]^{<ω}, for any ⟨X_{α} | α < κ⟩ ∈ \overline{M}, \{α | X_{α} ∈ E_{a}\} ∈ \overline{M}. This is equivalent to saying that P(κ) ∩ \overline{M} = P(κ) ∩ Ult(\overline{M}, E) (where the latter is the usual “Σ_{0}-ultrapower” of \overline{M} using functions belonging to \overline{M}).

As is well known there is a Σ_{0} and cofinal embedding π : \overline{M} ⟶_{E} M = Ult(\overline{M}, E). In general this is as much definable preservation that one could hope for. The notion of Σ^{∗}-ultrapower involves using functions that lie outside of \overline{M} but that are definable classes using formulae of the language L^{∗}. Let \overline{M} and E be as above.

Definition 3.12 The relation π : \overline{M} ⟶_{E}^{∗} M holds iff the following conditions are met:

(i) π : \overline{M} ⟶_{Σ_{0}^{(m)}} M; where M is transitive, for any m so that ωρ_{M}^{m} > κ.

(ii) Let ρn ; \overline{M} = min\{ρ_{M}^{m} | ρ_{M}^{m} > κ\}. Let H = ⋃_{x∈Hn ; \overline{M}} π(x). Then π ↾ Hn ; \overline{M} : Hn ; \overline{M} ⟶_{E} H.

(iii) Additionally if there is k with κ < ωρk+1 ; \overline{M}, then M is the closure of H ∪ ran(π) under

good Σ_{1}^{(k)}(M) functions.

In most situations we shall want M to be wellfounded (hence the definition), but sometimes it is convenient to ask only that the wellfounded core of M (which will be assumed transitive) contains ν. Just by the notation one has straightaway that

Fact 3.3 (i) If ωρ1 ; \overline{M} < κ then π : \overline{M} ⟶_{E} M iff π : \overline{M} ⟶_{E}^{∗} M; (ii) If ωρω ; \overline{M} > κ then π : \overline{M} ⟶_{E}^{∗} M implies π : \overline{M} ⟶_{Σ^{∗}} M.

We shall see that the existence of such a π, and an M with π : \overline{M} ⟶_{E}^{∗} M satisfying the above, implies that such π and M are unique.

Definition 3.13 Γ = Γ(κ, \overline{M}) is the set of functions f with dom (f) = [κ]^{n} (some n < ω) and either f is a good Σ_{1}^{(n)}(\overline{M}) function for some n with ωρn+1 ; \overline{M} ≥ κ, or f ∈ \overline{M}.

The extension of π to elements of Γ is effected as follows: if f ∈ Γ and dom(f) = κ, and f is good Σ_{1}^{(m)}(\overline{M}), where ωρm+1 ; \overline{M} > κ then the statement that any two functionally absolute

definitions define the same function is Π_{0}^{(m+1)}. Under the definition above, then, this will carry over to M, and define a unique function f′ over M. We then may set π(f) = f′. This will then imply that

(1) M = {π(f)(a) | f ∈ Γ, a ∈ [ν]^{<ω}}.

From this (using a Łos theorem for Σ_{0}^{(m+1)}-formulae) one then derives immediately the uniqueness mentioned above.

We define a suitable domain D with relations e, I, Ȧ, whose wellfoundedness of e as usual will give us a transitive ultrapower, and the existence of the π : \overline{M} ⟶_{E}^{∗} M.

Definition 3.14 D = \{⟨a, f⟩|f ∈ Γ\}. Set: ⟨a, f⟩{e \atop I}⟨b, g⟩ iff \{u|f(u){∈ \atop =}g(v)\} ∈ E_{a∪b}); Ġ(⟨a, f⟩) iff {u|f(u) ∈ G}) ∈ E_{a} (for G = A, B); D = ⟨D, I, e, Ȧ, Ḃ⟩.

One can then prove a Łos theorem for Σ_{0} formulae:

(2) Let ⟨a_{1}, f_{1}⟩, ⋯, ⟨a_{n}, f_{n}⟩ ∈ D, and let b ⊇ ∪_{i}a_{i}, and let φ be a Σ_{0} formula. Then D ⊧ φ[⟨\overrightarrow{⟨a_{i}, f_{i}⟩}⟩] ⟺ \{u | \overline{M} ⊧ φ[\overrightarrow{f_{i}^{a_{i}b}(u)}]\} ∈ E_{b}

(We use the notation that if a ⊆ b ∈ [ν]^{<ω}, with card(a) = n, then if f : [κ]^{n} ⟶ V then f^{ab}(u) = f(u^{ab}) where u^{ab} = ⟨u_{i_{1}}, … u_{i_{n}}⟩ for u with card(u) = card(b), and a = ⟨b_{i_{1}}, … b_{i_{n}}⟩.) Then I satisfies the axioms for equality, and D satisfies extensionality. If e is wellfounded then there is an isomorphism [ ] : D ˜⟷M where M is transitive and [x]{= \atop ∈}[y] iff x{I \atop e}y. We can define π : \overline{M} ⟶_{Σ_{0}} M by π(x) = [⟨∅, c_{x}⟩] (the latter c_{x} the constant function x). M will be of the form ⟨J_{β}^{A′}, A′⟩. and we shall show that π satisfies the properties of the definition. We assume the wellfoundedness of e.

We first note:

(3) Let n be chosen least to satisfy the hypothesis of 3.12 (ii). Then condition (ii) holds.

Hence κ = crit(π) and [⟨a, f⟩] = π(f)(a) for f ∈ \overline{H} = Hn ; \overline{M}, a ∈ [ν]^{<ω}.

Proof: By the definition of Γ, whenever f ∈ Γ and ran(f) ⊆ v ∈ \overline{H}, then in fact f ∈ \overline{M}.

Q.E.D.

We now prove

Lemma 3.6 Ultrapower Lemma (i) M is an acceptable J-model.

(ii) π : \overline{M} ⟶_{Σ_{0}^{(n)}} M if ωρn ; \overline{M} > κ. (iii) π : \overline{M} ⟶_{Σ_{2}^{(n)}} M if ωρn+1 ; \overline{M} > κ.

(iv) Let φ be a Σ_{0}^{(n)} (or in Σ_{1}^{(n)}, if κ < ωρn+1 ; \overline{M}). Let b ⊇ ∪_{i}a_{i}.

M ⊧ φ(π(f_{1})(a_{1}), …, π(f_{n})(a_{n})) ⟺ \{u | \overline{M} ⊧ φ(\overrightarrow{f_{i}^{a_{i}b}(u)}\} ∈ E_{b}

The proof of the lemma is in two stages. The main difficulty is in showing that the maps are between the relevant reducts. What this amounts to is showing that the elements in the natural “strata” that one defines from the functions in Γ are actually those obtained by using the iterated definition of projectum over M. The “strata” referred to arise as the H_{n} in the following definition: Set

Γ_{n} = \{f ∈ Γ | ran(f) ⊆ Hn ; \overline{M}\} if ωρn+1 ; \overline{M} ≥ \overline{κ}
= \{f ∈ Γ | ran(f) ∈ Hn ; \overline{M}\} if ωρn+1 ; \overline{M} < \overline{κ} ≤ ωρn ; \overline{M}.
H_{n} = {[⟨a, f⟩] | f ∈ Γ_{n} ∧ ⟨a, f⟩ ∈ D} if Γ_{n} is defined. ωρ_{n} = On ∩ H_{n}.

Note that H_{n} is transitive. For any i with ωρi ; \overline{M} > κ , we shall interpret on the M-side the

variables v_{j}^{i} as varying over H_{i}, For such an i we can take φ ∈ Σ_{k}^{(i)} and say that M ⊧ φ(r⃗) in the pseudo-interpretation if the variables v_{j}^{i} are so interpreted as r^{i} ∈ H_{i}. We show first the remark:

Lemma 3.7 (iv) in the Ultrapower Lemma holds.

Proof: This is an induction on n and then for each n by complexity of the formula φ. Q.E.D.

Corollary 3.8 (ii) and (iii) of the Ultrapower Lemma hold.

Proof: (ii) is immediate. (iii): Suppose M ⊧ ∃x^{n}φ(x^{n}, π⃗(z)) where φ is Π_{1}^{(n)}. Let ⟨a, g⟩ ∈ D, g ∈ Γ_{n} such that M ⊧ φ([⟨a, g⟩], π⃗(z)). By the last lemma \{u ∈ [κ]card(a) | \overline{M} ⊧

φ(g(u), z⃗) ∈ E_{a}. Hence ∃u_{0}\overline{M} ⊧ φ(g(u_{0}), z⃗). As ran g ⊆ Hn ; \overline{M}, \overline{M} ⊧ ∃^{n}φ(x^{n}, z⃗). Q.E.D.

Corollary 3.9 Q-properties are preserved: π : \overline{M} ⟶_{Q} M. In particular M is an acceptable J-model.

Proof: (Recall that a Q-formula was one expressible as ∀x∃y(x ∈ y ∧ Trans(y) ∧ ψ(y)) where x does not occur free in ψ ∈ Σ_{1}. The axiom of acceptability is expressible by a Q-condition.) If ωρ1 ; \overline{M} ≤ κ then π : \overline{M} ⟶_{Σ_{0}} M cofinally, so this suffices. If ωρ1 ; \overline{M} > κ then π : \overline{M} ⟶_{Σ_{2}} M

since π is Σ_{2}^{(0)} preserving, whilst H_{M}^{0} = M. Q.E.D.

Corollary 3.10 If M = ⟨J_{α′}^{A′}, B′⟩, then H_{n} = |J_{ρ_{n}}^{A′}|.

Proof: π ↾ Hn ; \overline{M} ⟶_{Q} H_{n} for all n < ω. If ωρn+1 ; \overline{M} ≤ κ < ωρn ; \overline{M}, then π : Hn ; \overline{M} ⟶ ⟨H_{n}, A′⟩

is Σ_{0}-cofinal, and again if ωρn+1 ; \overline{M} > κ, then π is Σ_{2}-preserving. For ρ^{n} < κ π = id ↾ HN ; \overline{M}. Q.E.D.

In the second stage we show that this interpretation is the correct one: i.e. H_{i} does indeed equal H_{ωρ_{M}^{n}}^{M}. The point is now to show that the ρ_{n}, which were defined from the H_{n} are the “right” ordinals. This is stated in the next lemma. Actually for hierarchies involving measures, as here, stronger statements are provable (which are below). We have divided the preservations properties into two at this, as for hierarchies involving extenders further amenability assumptions are required to gain this extra strength. For measures we have in any case full amenability which we can exploit.

An embedding π : \overline{M} ⟶_{Σ_{0}^{(n)}} M is cofinal if π is Σ_{0}^{(n)} preserving and H_{M}^{n} = ∪π“Hn ; \overline{M}. This

implies that π ↾ Hn ; \overline{M} : Hn ; \overline{M} ⟶_{Σ_{0}} H_{M}^{n} cofinally in the usual sense, and that π : \overline{M} ⟶_{Σ_{0}^{(n)}} M.

It only remains to show that the projecta above κ correspond to the ordinal ρ_{n}.

Lemma 3.11 ρ_{m} = ρ_{M}^{m} if κ ≤ ωρm+1 ; \overline{M}; ρ_{m} ≤ ρ_{M}^{m} if κ ≤ ωρm ; \overline{M}.

Proof: By induction on m. This is trivial for m = 0. We show ρ_{m} ≤ ρ_{M}^{m} by showing that

(4) ⟨H_{m}, A⟩ is amenable for A ∈ P (ωρ_{m})∩ Σ_{1}^{(m−1)}(M). Suppose A ⊆ ωρ_{m} and is Σ_{1}^{(m−1),M}([⟨a, f⟩]) , say A(x) ⟺ M ⊧ φ(x, [⟨a, f⟩]) with φ ∈ Σ_{1}^{(m−1)}. Let w = [⟨b, g⟩] ∈ H_{m}. We need A ∩ w ∈ H_{m}. Let card(a) = n. Define h : [κ]^{n} ⟶ H_{\overline{M}}^{m} by h(u) = \{t ∈ g(u) | \overline{M} ⊧ φ(t, f(u))\}. Then h ∈ Γ_{m}. So [⟨a, h⟩] ∈ H_{m}, and using 3.6(iv) [⟨a, h⟩] = [⟨b, g⟩] ∩ A.

We now show ρ_{m} ≥ ρ_{M}^{m} if κ ≤ ωρm+1 ; \overline{M}. We show:

(5) There is A ⊆ ωρ_{m}, A ∈ Σ_{1}^{(m−1)}(M) with A ∉ M.

Suppose \overline{A} is Σ_{1}^{(m−1)} (\overline{M}) definable in a parameter \overline{p}.

Let A be Σ_{1}^{(m−1)}(M) in π(\overline{p}) = p by the same definition. We show that A ∩ ωρ_{m} ∉ M. Suppose otherwise. Let A ∩ ωρ_{m} = [⟨a, f⟩] ∈ M. Then

A ∩ ωρ_{m} = x ⟺ ∀z^{m}(z_{m} ∈ A ⟺ z^{m} ∈ x).

The right hand side here is a Π_{1}^{(m)}, in p, formula. M ⊧ φ[[⟨a, f⟩], π(\overline{p})] so \{u|\overline{M} ⊧ φ[f(u), \overline{p}]\} ⊆ \{u|f(u) = \overline{A}\} ∈ E_{a}, by 3.6 (iv). So ∃uf(u) = \overline{A} ∈ \overline{M}. A contradiction.

Q.E.D.

Corollary 3.12 (i) Let ωρn+1 ; \overline{M} > κ. Then π“P n ; \overline{M} ⊆ P_{M}^{n}. (ii) π : \overline{M} ⟶_{E}^{∗} M; in particular for f ∈ D, [f] = π(f)(κ).

Proof: (i) This was shown in the last part of the last proof. For (ii) all that we have left to show is (iv) in Def 3.12, that M is the closure of ranπ under good Σ_{1}^{(n)}-functions, for ωρn+1 ; \overline{M} < κ. We’ve extended π to such functions so it is enough to show M = {π(f)(a) | f ∈ D, a ∈ [ν]^{n}, dom(f) = [κ]^{n} \}. By (3) above [c_{α}] = α for α < κ. As \{u | \overline{M} ⊧ f(u) = f(id(u))\} ∈ E_{a} by the Łos theorem for f ∈ Γ, we have D ⊧ f = c_{f}(id) and so M ⊧ [⟨a, f⟩] = π(f)(a).

Q.E.D.

If we are using measures to take ultrapowers, we can build mouse hierarchies with the measures fully amenable with respect to the structures concerned. This enables us to improve the last inequality above, and indeed, continue the analysis down below the critical point. However we shall continue to assume that we have an extender ultrapower and impose some additional requirements on E in order to do this.

We now use weak amenability to enable us to improve the last lemma and corollary at the point where the projectum crosses the critical point.

Lemma 3.13 Suppose ωρn ; \overline{M} > κ ≥ ωρn+1 ; \overline{M}, and that P(ωρn+1 ; \overline{M}) ∩ \overline{M} = ωρn+1 ; \overline{M} ∩ M. Then:

(i) π : \overline{M} ⟶_{Σ_{0}^{(n)}} M cofinally; (ii) ωρn ; \overline{M} = ωρ_{M}^{n};

(iii) ωρ_{M}^{n+1} ≤ ωρn+1 ; \overline{M}.

Proof: Assume n > 0.

Claim Let p ∈ P n+1 ; \overline{M}. Then there is a set A ∈ Σ_{1}^{(n−1)M}(\{π(p)\}) such that A ⊆ ωρ_{n} with A ∉ M.

Proof: Let \overline{A} = A_{\overline{M}^{n−1,p↾n−1}}^{p(n−1)}. Then \overline{A} ∈ Σ_{1}^{(n−1)}(\overline{M}) and \overline{M}^{n,p↾n} = ⟨H\overline{M} ; ωρ^{n}, \overline{A}⟩ is amenable.

Let B ⊆ H_{n} be defined by the same Σ_{1}^{(n−1)} formula φ in parameter π(p). Let \overline{H} = H\overline{M} ; ωρn ; \overline{M}.

π ↾ \overline{H} : \overline{H} ⟶_{Σ_{0}} H_{n} cofinally.

(1) B = ⋃_{x∈\overline{H}} π(x ∩ \overline{A}).

Proof: Pick x^{n} ∈ \overline{H}. Let y^{n} = x^{n} ∩ \overline{A} ∈ \overline{H}. Then

\overline{M} ⊧ ∀z^{n} ∈ x^{n}(z^{n} ∈ y^{n} ⟷ φ(z^{n}, p)) ⟺ M ⊧ ∀z^{n} ∈ π(x^{n})(z^{n} ∈ π(y^{n}) ⟷ φ(z^{n}, π(p))

(As π is Σ_{0} preserving in the pseudo-interpretation.) This then simply says π(x^{n} ∩ \overline{A}) = π(x^{n}) ∩ B. As π ↾ \overline{H} is cofinal (1) follows. Q.E.D.(1).

(2) B ∉ M

Proof: Assume B ∈ M. Then ⟨H_{n}, B⟩ ∈ M (by Corollary 3.10.) As p ∈ P n+1 ; \overline{M}, there is D ∈

Σ_{1}(\overline{M}^{n,p↾n}) ∩ P(ωρn+1 ; \overline{M}) with D ∉ \overline{M}. As π ↾ ⟨\overline{H}, \overline{A}⟩ ⟶_{Σ_{0},cof} ⟨H_{n}, B⟩, and π ↾ ωρn+1 ; \overline{M} = id,

this D is Σ_{1}(⟨H_{n}, B⟩) ⊆ M. But by assumption P(ωρn+1 ; \overline{M}) is the same in both \overline{M} and M. Contradiction! Q.E.D.(2)

Now just code B onto ωρ_{n} to get a suitable A. As ρ_{n} is p.r.closed, and H_{n} = J_{ρ_{n}}^{A′}, let f be Σ_{1}(M) with f : ωρ_{n} ⟶ H_{n} be onto; let A = {ξ | f(ξ) ∈ B}. Then A is as required.

Q.E.D.Claim

From the Claim we see that ωρ_{M}^{n} ≤ ωρ_{n}. The reverse inequality we already established in the last lemma. This gives us (ii) here; and we can now assert (i).

The D of (2) is Σ_{1}^{(n)\overline{M}}(p), and the same definition over M in π(p) yields a D′ with D′∩ωρn+1 ; \overline{M} = D ∉ M. Hence ωρ_{M}^{n+1} ≤ ωρn+1 ; \overline{M} which is (iii). Q.E.D.

Remark: It is possible to replace the assumption on weak amenability with just that Rn ; \overline{M} ≠ ∅ and get the same conclusions.

Definition 3.15 E is Σ_{1}-amenable with respect to \overline{M}, if E_{a} is Σ_{1}(\overline{M}) for any a ∈ [ν]^{<ω}.

Note that the parameters involved in the definition are not assumed to be given uniformly. The reader may like to check that this notion yields that Σ_{1}(M) ∩ P(κ) ⊆ Σ_{1}(\overline{M}) ∩ P(κ).

In addition to weak amenability we now make the following assumption on the extender E, that it is Σ_{1}-amenable. Now we obtain:

Lemma 3.14 Suppose ωρn ; \overline{M} > κ ≥ ωρn+1 ; \overline{M}, and that E is Σ_{1}-amenable. Then

P(κ) ∩ Σ_{1}^{(n)}(M) ⊆ Σ_{1}^{(n)}\overline{M}.

Proof: Let B ∈ P(κ) ∩ Σ_{1}^{(n)}(M). Suppose B(z) ⟷ ∃x^{n}B′(x^{n}, z, π(f)(a)) with B′ ∈ Σ_{0}^{(n)}(M). Using Σ_{0}^{(n)} cofinality, we can write

B(z) ⟷ ∃u ∈ Hn ; \overline{M}∃x ∈ π(u)B′(x, z, π(f)(a)) ⟷ ∃u ∈ Hn ; \overline{M}D(π(u), z, π(f)(a))

(where D ∈ Σ_{0}^{(n)}(M)) Let \overline{D} ∈ Σ_{0}^{(n)}(\overline{M}) have the same definition.

B(z) ⟷ ∃u ∈ Hn ; \overline{M} \{v | \overline{D}(u, z, f(v))\} ∈ E_{a} ⟷ ∃u^{n}∃x^{n}(x^{n} = \{v | \overline{D}(u^{n}, z, f(v))\} ∧ x^{n} ∈ E_{a})

for any z < κ. The matrix here is Σ_{0}^{(n)} ∩ Σ_{1}^{(0)}. Q.E.D.

Remark: There is a non-uniformity in the above due to the final “x^{n} ∈ E_{a}”, but that is the only one. We now go down below κ.

Lemma 3.15 Let E be Σ_{1}-amenable, and assume ωρm ; \overline{M} ≤ κ. Then

(i) H_{M}^{m} = Hm ; \overline{M}; (ii) π : \overline{M} ⟶_{Σ_{1}^{(m)}} M;

(iii) Σ_{1}^{(m)}(M) ∩ P(H_{M}^{m}) = Σ_{1}^{(m)}(\overline{M}) ∩ P(Hm ; \overline{M});

(iv) π“P m ; \overline{M} ⊆ P_{M}^{m}.

Proof: By induction on m. We first show (i). Note that H_{M}^{m} by definition is J_{ωρ_{M}^{m}}^{A′}, and ωρ_{M}^{m} is a cardinal of M = ⟨J_{α′}^{A′}, B′⟩. Similarly for \overline{M} = ⟨J_{α}^{A}, B⟩. We also know that J_{κ}^{A} = J_{κ}^{A′}. So it suffices for (i) to show that ωρm ; \overline{M} = ωρ_{M}^{m}. This will also show (ii), since π ↾ Hm ; \overline{M} = id. Suppose that m = k + 1. ρm ; \overline{M} ≥ ρ_{M}^{m}: since there is B ⊆ ωρm ; \overline{M}, B ∈ Σ_{1}^{(k)}(\overline{M})\\overline{M}. By the inductive hypothesis, as

π : \overline{M} ⟶_{Σ_{1}^{(k)}} M and π ↾ Hm ; \overline{M} = id, B is thus also Σ_{1}^{(k)}(M). But also B ∉ M, since P(ωρ_{M}^{m}) is the same in both models.

ρ_{M}^{m} ≥ ρm ; \overline{M}: suppose not. Then there is B ⊆ ωρ_{M}^{m} < ωρm ; \overline{M} with B ∈ Σ_{1}^{(k)}(M)\M. But then

B ∈ Σ_{1}^{(k)}(\overline{M}) by the previous lemma, if ωρ^{k} > κ, and by the inductive hypothesis (iii) for k otherwise. But B cannot be in \overline{M}, since P(κ) ∩ \overline{M} ⊆ M. So ωρ_{M}^{m} ≮ ωρm ; \overline{M}. This proves (i) and as remarked, (ii).

For (iii): let H = Hm ; \overline{M} = H_{M}^{m} and suppose B ⊆ H.

B ∈ Σ_{1}^{(m)}(M) ⟷ ∃D ∈ Σ_{1}^{(k)}(M) B ∈ Σ_{1}(⟨H, D⟩)
⟷ ∃D ∈ Σ_{1}^{(k)}(\overline{M}) B ∈ Σ_{1}(⟨H, D⟩) ⟷ B ∈ Σ_{1}^{(m)}(\overline{M}).

where the second equivalence uses the I.H. (iii) or the last lemma. This is (iii) for m. We consider (iv). Let \overline{p} ∈ P m ; \overline{M}, p = π(\overline{p}). We need to show:

Claim A_{M}^{k,p↾k} ∉ M for 1 ≤ k ≤ m.

We prove this by induction on m − k. If m = k, then as π ↾ H = id, and π : \overline{M} ⟶_{Σ_{1}^{(m)}}

M, so Ak,\overline{p}↾k ; \overline{M} = A_{M}^{k,p↾k}, and the latter is not in \overline{M}, and hence not in M either, because P(H) ∩ M = P(H) ∩ \overline{M}. Suppose it holds for k + 1 ≤ m. Then A_{M}^{k+1,p↾k+1} = B ∩ H_{M}^{k+1} where B ∈ Σ_{1}(⟨H_{M}^{k}, A_{M}^{k,p↾k}⟩) in parameter p(k). But A_{M}^{k,p↾k} ∈ M ⇒ ⟨H_{M}^{k}, A_{M}^{k,p↾k}⟩ ∈ M which contradicts the inductive hypothesis for k. Q.E.D.

Collecting together the pieces of the above one obtains (with a small amount swept under the carpet in (ii) for ωρ_{M}^{m} < κ, as the argument requires some more analysis of the Σ_{1}^{(n)} relations than we have catered for here):

Corollary 3.16 Let E be Σ_{1}-amenable. Then

(i) π : \overline{M} ⟶_{Σ^{∗}} M (ii) P(κ) ∩ Σ_{1}^{(n)}(\overline{M}) = P(κ) ∩ Σ_{1}^{(n)}(M) for all n < ω.

3.4 An Upwards Extension of Embeddings Lemma

Lemma 3.17 Suppose \overline{M} = ⟨J E\overline{M} ; \overline{β} , E\overline{M}, F \overline{M}⟩ is a mouse, and \overline{Q} = ⟨J E\overline{M} ; \overline{ν} , E\overline{M} ↾ \overline{ν}⟩ and there is a cofinal embedding σ : \overline{Q}⟶_{Σ_{0}}Q. We suppose there is \overline{κ} < \overline{ν} with \overline{Q} ⊧ “\overline{κ} is the largest cardinal”, and either \overline{β} = \overline{ν} or \overline{M} ⊧ \overline{ν} = \overline{κ}^{+}. Suppose that cf(\overline{ν}) > ω. Then there is σ̃ ⊇ σ

and M = ⟨J_{β}^{E^{M}} , E^{M}, F^{M}⟩ with M a mouse and σ̃ : \overline{M} ⟶_{Σ_{1}^{(n)}} M, for any n with ωρn+1 ; \overline{M} ≥ κ,

where κ = σ(\overline{κ}). Further, if \overline{M} is sound above \overline{κ}, and for some m < ω, ωρm+1 ; \overline{M} ≤ \overline{κ} < ωρm ; \overline{M},

then ωρ_{M}^{m+1} ≤ κ < ωρ_{M}^{m}, M is the closure of κ ∪ {p} under good Σ_{1}^{(m)}-functions for a p ∈ M, and σ̃ : \overline{M} ⟶_{Σ_{1}^{(m)}} M.

We first remark that in many situations one may use the lemma with κ a cardinal (or at least the cardinal of a model) which we are not assuming here. One can often improve the conclusion to require that M is sound above κ (and that σ̃(p\overline{M}) = p_{M}\κ where p\overline{M} is the standard parameter of \overline{M}) - but this requires more work on parameters, which we have eschewn. The proof of this lemma involves forming a “pseudo-ultrapower” M of \overline{M} that has also a fine-structure preserving extension of σ mapping \overline{M} into M. The construction is very similar to that for forming an ultrapower by an extender (more precisely it is that for the “long” extender or “derived” extender from the map σ). So naturally it bares many similarities to what we have just done in the previous subsection. Much of what we shall do is redefinition of the previous notions, and many of the changes in the proofs require only a change of notation. We omit some detail therefore. We shall first redefine the class of functions Γ, the stratified functions Γ_{n}, and form the term model D, and the “strata” H_{n}. There is little that can be said about the H_{n} for n > m + 1 and indeed we shall not define these. The “lift-up” structure M is wellfounded (and iterable) by the crucial use of the uncountable cofinality of ν.

Definition 3.16 Γ is the set of functions f with dom (f) ⊆ u for some u ∈ \overline{Q} and either f is a good Σ_{1}^{(n)}(\overline{M}) function for some n with ωρn+1 ; \overline{M} ≥ \overline{ν}, or f ∈ Hn ; \overline{M} where ωρn ; \overline{M} ≥ \overline{ν}.

We define e, I, Ė, and D as indicated §1.

Definition 3.17 D = {⟨a, f⟩|f ∈ Γ ∧ a ∈ σ(f)}; set:

⟨a, b⟩{e \atop I}⟨b, g⟩ iff ⟨a, f⟩ ∈ σ(\{⟨u, v⟩|f(u){∈ \atop =}g(v)\})

Ġ(⟨a, f⟩) iff a ∈ σ({u|f(u) ∈ G}) for G = E, F; set: D = ⟨D, I, e, Ė˙(F)⟩.

One can then again prove a Łos theorem for Σ_{0} formulae:

(1) Let ⟨a_{1}, f_{1}⟩, ⋯, ⟨a_{n}, f_{n}⟩ ∈ D and let φ be a Σ_{0} formula. Then D ⊧ φ[⟨\overrightarrow{a_{i}, f_{i}}⟩] ⟺ \overrightarrow{a_{i}} ∈ σ(\{\overrightarrow{u}|\overline{M} ⊧ φ[\overrightarrow{f_{i}}(\overrightarrow{u})]\})

Then I satisfies the axioms for equality, and D satisfies extensionality. If e is wellfounded then there is an isomorphism [ ] : D ˜⟷M where M is transitive and [x]{= \atop ∈}[y] iff x{I \atop e}y. We can define σ̃ : \overline{M} ⟶_{Σ_{0}} M by σ̃(x) = [⟨0, \{⟨0, x⟩\}⟩]. M will be of the form ⟨J_{β}^{E}, E, F ⟩, and we shall show that it is a mouse and σ̃ satisfies the properties of the Lemma. Since wellfoundedness of e can be considered a special case of the proposed iterability of M, we omit the proof of the wellfoundedness of e, and simply assume it for the moment.

We first note:

(2) σ̃ ↾ \overline{Q} = σ

Proof: Note that for f, g ∈ \overline{Q} [⟨a, f⟩] ∈ [⟨b, g⟩] ⇔ σ(f)(a) ∈ σ(g)(b). Hence for such f we may define a map [⟨a, f⟩] ↦ σ(f)(a). This map is onto and so by the above is the identity. Hence [⟨a, f⟩] = σ(f)(a) = σ̃(f)(a) and so σ̃(x) = [⟨0, {⟨0, x⟩}⟩] = σ(x).

QED

Define

Γ_{n} = \{f ∈ Γ | ran(f) ⊆ Hn ; \overline{M}\} if ωρn+1 ; \overline{M} ≥ \overline{ν}
= \{f ∈ Γ | ran(f) ∈ Hn ; \overline{M}\} if ωρn+1 ; \overline{M} < \overline{ν} ≤ ωρn ; \overline{M}.
H_{n} = {[⟨a, f⟩] | f ∈ Γ_{n} ∧ ⟨a, f⟩ ∈ D} if Γ_{n} is defined. ωρ_{n} = On ∩ H_{n}.

Note that H_{n} is transitive as before.

(3) σ̃ : \overline{M} ⟶_{Σ_{0}^{(n)}} M cofinally for any n with ωρn ; \overline{M} ≥ \overline{ν}. Proof: We shall prove (3) in three stages. The first two are just the manoeuvres of Lemma 3.11. The third stage will use the extra assumption on the cofinality of ν to check that (3) holds for the m of crossing projectum of Lemma 3.17. In the first step we interpret the variables of type k with ωρk ; \overline{M} ≥ \overline{ν} on the right hand side as varying over H_{k}. We prove (3) using this pseudo-interpretation. Again we then shall show in the second step that this interpretation is “correct” in that H_{k} in fact is H_{M}^{k} for any k ≤ m

The proofs of (4),(5) & (6) are, with minor alterations, those of the last subsection.

(4) Let ⟨\overrightarrow{a_{i}, f_{i}}⟩ ∈ D and let φ ∈ Σ_{0}^{(n)} where ωρn ; \overline{M} ≥ \overline{ν}. Then M ⊧ φ[[⟨\overrightarrow{a_{i}, f_{i}}⟩]] ⟺ \overrightarrow{a_{i}} ∈ σ(\{\overrightarrow{u} | M ⊧ φ[\overrightarrow{f}(\overrightarrow{u})]\}).

(5) σ̃ : \overline{M} ⟶_{Σ_{0}^{(n)}} M cofinally, for ωρn ; \overline{M} ≥ \overline{ν}. Proof: We mean here in the pseudo-interpretation. By cofinally, we mean σ̃ ↾ Hm ; \overline{M} is cofinal into H_{M}^{m}, in case m is defined. But this is straightforward. Q.E.D.(5)

As σ̃ is Σ_{0} and cofinal into M, acceptability of \overline{M} translates to acceptability of M, and so H_{n} = J_{ρ_{n}}^{M} for ωρn ; \overline{M} ≥ \overline{ν}. It remains to show that ρ_{n} = ρ_{M}^{n} for ωρn ; \overline{M} ≥ \overline{ν}.

(6) ρ_{n} = ρ_{M}^{n} if \overline{ν} ≤ ωρn+1 ; \overline{M}; ρ_{n} ≤ ρ_{M}^{n} if ν ≤ ωρn ; \overline{M}.

Proof: The reader is referred to (4) and (5) of Lemma 3.11. The notational changes are simply that in (4) the function h now has domain dom(g). The map between the two structures is now σ̃ (not π); and in (5) we need to conclude with: a ∈ σ(\{u|\overline{M} ⊧ φ[f(u), \overline{p}]\}) = σ(\{u|f(u) = \overline{A}\}) rather than claim that the set is in E_{a}. Q.E.D.(6)

We now use the extra condition on \overline{Q}, \overline{ν}.

(7) If ωρm+1 ; \overline{M} < \overline{ν} ≤ ωρm ; \overline{M} then

(i) ωρ_{M}^{m+1} < ν ≤ ωρ_{M}^{m}; (ii) ρ_{m} = ρ_{M}^{m}. (iii) M is the closure of κ ∪ {r} under good Σ_{1}^{(m)}(M)-functions, for some r ∈ R_{M}^{m+1}.

Proof: We first show σ̃(\overline{ν}) = ν = sup σ̃“\overline{ν} (setting σ̃(On ∩ M) = On ∩ M). Let ξ = σ̃(f)(a) < σ̃(\overline{ν}) with [⟨a, f⟩] ∈ D. Without loss of generality we can assume ran(f) ⊆ \overline{ν}. By the definition of Γ, and using \overline{ν} is a cardinal of \overline{M}, ran(f) cannot be cofinal in \overline{ν}. Hence there is η with ran(f) ⊆ η < \overline{ν} and so σ̃(f)(a) < σ(η) < ν.

If σ(\overline{κ}) = κ, then one may simply check that κ^{+M} = ν. By (6) ν ≤ ωρ_{M}^{m}. Suppose ωρm+1 ; \overline{M} <

\overline{ν} ≤ ωρm ; \overline{M}. Let σ̃(\overline{p}) = p and \overline{p} ∈ P m+1 ; \overline{M}.

For clause (ii) we only have left to show ρ_{M}^{m} ≤ ρ_{m}. We do this now. Let A be defined by

the cofinal map σ̃ ↾ Hm ; \overline{M} : ⟨M^{m,\overline{p}}, Am,\overline{p} ; \overline{M}⟩ ⟶_{Σ_{0}} ⟨H_{m}, A⟩. That is, let A = ⋃_{x∈Hm ; \overline{M}} σ̃(x ∩ Am,\overline{p} ; \overline{M}).

Then ⟨H_{m}, A⟩ ≺_{Σ_{0}} M^{m,p}. By the Downwards Extensions of Embeddings Lemma, there are M̃, p̃ such that M̃^{m,_{p}} = ⟨H_{m}, A⟩ and p̃ ∈ Rm ; _{M}. We have the further properties that there is

π : M̃ ⟶_{Σ_{0}^{(n)}} M with π ↾ H_{m} = id and π(p̃) = p. Since \overline{p} ∈ Rm ; \overline{M} we can factor through,

and see that there is σ̂ : \overline{M} ⟶_{Σ_{1}^{(n)}} M̃ such that σ̂ ↾ Hm ; \overline{M} = σ̃ ↾ HM ; \overline{M} and σ̂(\overline{p}) = p̃. Then π(σ̂(f)(a)) = σ̃(f)(a) for ⟨a, f⟩ ∈ D. Hence πσ̂ = σ̃ and π is onto; hence π = id, σ̂ = σ̃, and M̃ = M. Then A cannot be in M whilst it is Σ_{1}^{(m)}(M). Hence ρ_{M}^{m} ≤ ρ_{m}.

Let \overline{p} ∈ P m+1 ; \overline{M} and let p = σ̃(\overline{p}). Any x ∈ H_{M}^{m} has the form σ̃(f)(a) where f ∈ Γ_{m}(\overline{M}) and a ∈ σ(dom(f)) If f ∈ h_{\overline{M}_{m,\overline{p}}}(i, ⟨\overline{ξ}, \overline{p}(m)⟩) some \overline{ξ} < \overline{κ}, then σ̃(f)(a) = h_{M^{m,p}}(j, ⟨⟨ξ, a⟩, p(m)⟩) for some j < ω, ξ = σ(\overline{ξ}); hence M^{m,p} ⊆ h_{M^{m,p}}(ν ∪ \{p(m)\}). But as at the beginning of this proof, h_{M^{m,p}}(κ ∪ {p(m)}) is cofinal in ν. As ν = (κ^{+})^{M^{m,p}}, (or equals On ∩ M^{m,p}), this suffices to show h_{M^{m,p}}(κ ∪ {p(m)}) ⊇ M^{m,p}. This shows that both that M is the closure of κ ∪ {p(m)} under good Σ_{1}^{(m)}−functions, and that ωρ_{M}^{m+1} ≤ κ, which are clauses (i) and (iii).

Q.E.D.(7) & (3)

Claim M is wellfounded and iterable above κ.

Proof: Suppose not; let I = ⟨M_{α}|α < θ⟩ be an iteration of M_{0} = M above κ that cannot be continued. Then there is a set {x_{i}}_{i∈ω} ⊆ M of supports of functions coding an illfounded ∈_{θ}-chain through the “ordinals” of a model M_{θ}. We treat wellfoundness of M as a special case of iterability. In this case set x_{i+1} ∈ x_{i} as a descending e-chain through M = M_{0}. Let H be some sufficiently large ZF C^{−}-model that is transitive and contains I, {x_{i}}. Let I ∪ \{x_{i}\} ⊆ Y ≺ H where Y is countable. Let k : \overline{H} ˜⟷Y be the inverse of the transitive collapse. Let k(⟨N, κ′, \overline{x}_{i}, \overline{I}⟩) = ⟨M, κ, x_{i}, I⟩. Then in \overline{H}, \{\overline{x}_{i}\} is a set of supports for functions coding illfoundedness of the last model \overline{M}\overline{θ} of \overline{I} = ⟨\overline{M}_{α}|α < \overline{θ}⟩. Let \{y_{i}\}_{i∈ω} enumerate N. Then each k(y_{i}) is of the form σ̃(f_{i})(a_{i}) for some f_{i} ∈ Hn ; \overline{M}, or f_{i} a good Σ_{1}^{(n)}(\overline{M}) function with domf_{i} ∈ \overline{Q}.

We define by cases a model P and a map π : P ⟶ \overline{M}.

Case 1 n ≥ 1 and \overline{ν} < ρn ; \overline{M}.

Let μ < \overline{ν} be chosen so that

(i) \overline{P} = _{df} ⟨\overline{M}^{n,\overline{p}}|μ, A^{n,\overline{p}}|μ⟩ is amenable, ρ1 ; \overline{P} ≤ \overline{κ}.

(ii) sup_{i}domf_{i} < μ

(iii) id : \overline{P} ⟶_{Σ_{1}} ⟨\overline{M}^{n,\overline{p}}, An,\overline{p} ; \overline{M}⟩

(i) & (ii) are clearly possible since cf(\overline{ν}) > ω. As \overline{ν} < ρn ; \overline{M}, \overline{ν} is a cardinal of \overline{M}^{n,\overline{p}}, so

⟨\overline{M}^{n,\overline{p}}|\overline{ν}, A^{n,\overline{p}}|\overline{ν}⟩ ≺_{Σ_{1}} ⟨\overline{M}^{n,\overline{p}}, A^{n,\overline{p}}⟩. But there are arbitrarily large μ < \overline{ν} with such a \overline{P} ≺_{Σ_{1}} ⟨\overline{M}^{n,\overline{p}} | \overline{ν}, An,\overline{p} ; \overline{M}|\overline{ν}⟩. Hence \overline{P} exists. Now any f_{i} is either in Hn ; \overline{M} = \overline{M}^{n,\overline{p}} or has a good Σ_{1}^{(k)}(\overline{M}) definition involving parameters ξ⃗_{i} < ωρk ; \overline{M}. By the soundness of \overline{M} above \overline{κ}, either way such

an f_{i} has a good Σ_{1}^{(n)}(\overline{M}) definition involving parameters ξ⃗_{i} below \overline{κ}. By the Extensions of Embeddings Lemma there is a premouse P, iterable above \overline{κ}, and q ∈ R_{P}^{n}, and a π : P ⟶_{Σ_{0}^{(n)}} \overline{M}, with \overline{P} = P^{n,q} and π(q) = \overline{p}, and A_{P}^{n,q} = A^{n,\overline{p}}|μ. As

A_{P}^{n,q} ∈ Hn ; \overline{M}, P ∈ Hn ; \overline{M}. Let \overline{f}_{i} have the same good Σ_{1}^{(n)} definition over P from ξ⃗_{i}, q as f_{i} did

over \overline{M} in ξ⃗_{i}, \overline{p}. Note that π is in fact Σ_{1}^{(n)} preserving. Let σ(p) = p̃. But as domf_{i} = dom\overline{f}_{i} and π is Σ_{1}^{(n)} -elementary π(\overline{f}_{i}(u)) = f_{i}(u) for any u ∈ dom\overline{f}_{i}.

Case 2 n ≥ 1 and \overline{ν} = ρn ; \overline{M}.

\overline{μ} may be chosen as in Case 1, however in (iii) we may only have that \overline{P} is Σ_{0}-embedded into ⟨\overline{M}^{n,\overline{p}}, An,\overline{p} ; \overline{M}⟩ by id. However this suffices to still extend id to a π : P ⟶_{Σ_{0}^{(n)}} \overline{M}, but

which is nevertheless Σ_{1}^{(n−1)} preserving. (This is just part of the Extensions of Embeddings Lemmas.) However we may now at no additional cost impose on μ, using cf(\overline{ν}) > ω, and the Case hypothesis:

(iv) Any f_{i} that has a good Σ_{1}^{(n−1)}(M) definition in parameters ξ⃗ < ρn ; \overline{M}, then such a ξ⃗ is below μ.

Again we have π : P ⟶_{Σ_{0}^{(n)}} \overline{M}, with P a premouse iterable above \overline{κ}, P ∈ H\overline{M} ; \overline{ν}. As π is

Σ_{1}^{(n−1)} preserving, we can let \overline{f}_{i} be the functions defined over P with the same definitions as that of f_{i} over M, with sup_{i}domf_{i} < On ∩ \overline{P}. In both cases we define l = N ⟶ P̃, where P̃ = σ(P), by l(y_{i})σ̃(\overline{f}_{i})(a_{i}). l will embed, in particular, the supports for a descending sequence of ordinals of an iterate of P̃ above κ. This will contradict the latter’s iterability above κ.

(9) l is Σ_{0}^{(n)} preserving.

Proof:

P̃ ⊧ φ(l(y_{i})) ⟺ P̃ ⊧ φ(σ̃(\overline{f}_{i})(a_{i}))
⟺ a_{i} ∈ \{u | P̃ ⊧ φ(σ(\overline{f}_{i})(u))\}
⟺ a_{i} ∈ σ({u | P ⊧ φ(f̃_{i}(u))}) (as σ ↾ P is Σ_{ω})
⟺ a_{i} ∈ σ(\{u | \overline{M} ⊧ φ(\overline{f}_{i}(u))\}) (as π is Σ_{0}^{(n)} or Σ_{1}^{(n)} depending on the case)
⟺ a_{i} ∈ \{u | \overline{M} ⊧ φ(σ̃(f_{i})(u))\} (as σ̃ is Σ_{0}^{(n)})
⟺ M ⊧ φ(σ̃(f_{i})(a_{i})) ⟺ N ⊧ φ(y_{i}) (as k is Σ_{ω}).

QED(9)

Note that P̃ is truly iterable above κ, since “P is iterable above κ” holds in some admissible set A with P ∈ A ∈ \overline{M}, and hence by elementarity holds in Ã = σ(A) of P̃. But ω_{1} ⊆ Ã so this is absolute. But this is now a contradiction since \{l(\overline{x}_{i})\}_{i∈ω} contains supports coding an illfounded iteration of P̃ above κ.

QED Cases 1 & 2

Case 3 n = 0. A standard argument shows that cf(\overline{ν}) > ω ⟹ cf(On ∩ \overline{M}) > ω. As σ̃ is Σ_{0} -cofinal the same is true of M. There is thus τ̃ = σ(τ) with

(i) ∀y k(y) ∈ J M ; τ̃

(ii) ∃ f_{n} ∈ J\overline{M} ; τ ∃a_{n} ∈ \overline{Q} σ̃(f_{n})(a_{n}) = k(y_{n}).

We can assume that τ̃ is chosen sufficiently large so that each f_{n} can be taken in h_{\overline{M}|τ} (\overline{κ}∪p\overline{M}).

Let X ≺_{Σ_{1}} J\overline{M} ; τ be such that \overline{ν} ∪ \{p\overline{M}\} ⊆ X, let π : P ˜ ⟷X. Let π(q) = p\overline{M}, and let \overline{f}_{n} have the same Σ_{1}^{P} (\{q, ξ⃗_{n}\}) definition as f_{n} had in Σ^{\overline{M}|τ} (\{p\overline{M}, ξ⃗\}). Note that sup_{i} domf_{i} ≤ \overline{ν} ∩ X. Also note that P ∈ J\overline{M} ; \overline{ν} (since A_{P}^{1} ⊂ J_{κ}^{P}, as ρ_{P}^{1} ≤ \overline{κ}, and then a simple comparison argument shows A_{P}^{1} ∈ \overline{M}).

Again define l : N ⟶ P̃ =_{df} σ(P) by l(y_{n}) = σ̃(\overline{f}_{n})(a_{n}).

(10) l is Σ_{0}-preserving. Proof The equivalences are exactly those of (9), noting in this case that π : P ⟶_{Σ_{1}} \overline{M}|τ, and so π : P ⟶_{Σ_{0}} \overline{M}. QED (10)

But P̃ is iterable above κ as before, so l embeds a supposedly non-iterable premouse above κ′ into one that is iterable above l(κ′) = κ! This is a contradiction. QED.

References

[1] A.J.Dodd, The Core Model, Lond. Math. Soc. Lecture Notes in Maths, 61, CUP, 1982.

[2] H-D Donder, R.B. Jensen, B.J.Koppelberg, Some applications of the Core Model, in Set Theory and Model Theory, Springer Lecture Notes in Maths., 872, 1981, 55-97.

[3] R.B. Jensen, The Core Model for Measures of Order Zero, R. Jensen, circulated manuscript, 1989.

[4] R.B. Jensen & J. Vickers, Absoluteness of Jónsson Cardinals, accepted for the J. Symbolic Logic, 1997.

[5] , A covering lemma part 2.

[6] A. Kanamori, The Higher Infinite, Perspectives in Mathematical Logic, 1994, Springer Verlag, Berlin.

[7] A. Miller, Some Interesting Problems, in Set Theory of the Reals, Ed. H. Judah, Israeli Mathematical Conference Proceedings, 6 , 1993, 645-654, Amer. Math. Soc. or http://www.math.wisc.edu/∼ miller

[8] J. R. Steel, The Core Model Iterability Problem, Springer Lecture Notes in Logic, 8 Springer Verlag, 1996.

[9] J. Vickers, A covering lemma part 1. manuscript. 1990(?)

[10] P.D.Welch, Countable Unions of Simple Sets in the Core Model, in J.Symbolic Logic, 61 No. 1, 1996, pp 293-312.

[11] W.H.Woodin, Σ_{1}^{2}-absoluteness and supercompact cardinals, handwritten notes, May 15, 1985.