ITERATING THE COFINALITY- 
$\omega $
 CONSTRUCTIBLE MODEL

Abstract We investigate iterating the construction of 
$C^{*}$
 , the L-like inner model constructed using first order logic augmented with the “cofinality 
$\omega $
 ” quantifier. We first show that 
$\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$
 is equiconsistent with 
$\mathrm {ZFC}$
 , as well as having finite strictly decreasing sequences of iterated 
$C^{*}$
 s. We then show that in models of the form 
$L[U]$
 we get infinite decreasing sequences of length 
$\omega $
 , and that an inner model with a measurable cardinal is required for that.


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The model C * , introduced by Kennedy, Magidor and Väänänen in [5], is the model of sets constructible using the logic L(Q cf ω ) -first order logic augmented with the "cofinality ω" quantifier.As in the case of L -the model of sets constructible using first order logic, this is a model of ZFC, and one can phrase the formula "V = C * ", i.e. ∀x∃α(x ∈ L ′ α ) where L ′ α is the α-th level in the construction of C * .Unlike L, however, it is not always true that C * V = C * , which is equivalent to the question whether (C * ) C * = C * .This is clearly the case if V = L, so the interesting question is whether this can hold with C * = L.In section 2 we show that this is consistent relative to the consistency of ZFC.Next we investigate the possibilities of C * V = C * .In such a case, it makes sense to define recursively the iterated C * s: for any α C * α = β<α C * β for limit α. [8] regarding HOD, where he showed that it is equiconsistent with ZFC that there is a strictly decreasing sequence of iterated HOD of length ω, and the intersection of the sequence can be either a model of ZFC or of ZF + ¬AC.Harrington also showed (in unpublished notes, cf.[11]) that the intersection might not even be a model of ZF.Jech [4] showed that it is possible to have a strictly decreasing sequence of iterated HOD of any arbitrary ordinal length, and later Zadrożny [10] improved this to get an Ord length sequence.In section 3 we show I would like to thank my advisor, Prof. Menachem Magidor, for his guidance and support without which this work would not have been possible.that unlike the case of HOD, without large cardinals we can only have finite decreasing sequences of iterated C * , and that assuming the existence of a measurable cardinal is equivalent to the consistency of a strictly decreasing sequence of length ω.

This type of construction was first investigated by McAloon
In this section we follow the method of Zadrożny [11] to obtain the following result: The idea1 of Zadrożny's proofs, which are based on results of McAloon's [7,8], is to add a generic object (to make V = L), code it using some other generic object, then code the coding, and so on, iterating until we catch our tail.Our coding tool will be the modified Namba forcing of [5, section 6], which adds a countable cofinal sequence to any element of some countable sequence of regular cardinals > ℵ 1 (and only to them).
Revised countable support iterations of this forcing preserves ω 1 .In [5, theroem 6.7], these tools are used to produce a model of ZFC + V = C * = L + 2 ℵ0 = ℵ 2 , but this requires an inaccessible cardinal (as proven there as well).
These two results covers all possibilities, since in [5, corollary to theroem 5.20], it is shown that the statement V = C * implies that 2 ℵ0 ∈ {ℵ 1 , ℵ 2 } and for any κ > ℵ 0 To prove theorem 1, we begin with V = L [A 0 ] where A 0 is a countable set of ordinals such that A 0 does not contain any of its limit points.Set P 0 = {1}.Inductively we assume that P n forces the existence of a countable set of ordinals A n , ξ n = sup A n and we set P n+1 = P n * Qn+1 where Qn+1 is the modified Namba forcing to add a Namba sequence E α to each ℵ L ξn+α+2 such that α ∈ A n .We can require that E α ⊆ ℵ L ξn+α+1 , ℵ L ξn+α+2 , so that A n+1 := E α | α ∈ A n does not contain any of its limit points.P ω is the full support (which is in our case also the revised countable support) iteration.Let G ⊆ P ω generic, and denote A = n<ω A n .By the properties of the modified Namba forcing, for any Remark 2.
(1) ξ n+1 = ℵ L ξn•2 , so inductively depends only A 0 and not on the generics.ξ ω := sup A satisfies ξ ω = ℵ L ξω and is of cofinality ω. (2) From otp(A 0 ) and A we can inductively reconstruct each A n -A 0 is the first otp(A 0 ) elements of A, and if we know Proof.By the properties of the modified Namba forcing, at each stage of the iteration the only cardinals of L [A 0 ] receiving cofinality ω are the ones in A n .The whole iteration will also add new ω sequences to sup A, but this already had cofinality ω as we noted earlier. (where As we noted, for every α ∈ A, E α can be reconstructed from A and α, so so the equality follows.
This finishes the proof of theorem 1 since for any non-empty A 0 we'll get a model of "V = C * = L", and 2 ℵ0 = ℵ 1 still holds.
Before moving to the next section we prove a useful lemma: If there is no inner model with a measurable cardinal, then where K is the Dodd-Jensen core model.
To summarize, we get that for every α, in L [E] we can determine whether cf (α) = ω or not, so 2. The proof is exactly the same, noting that K ⊆ C * by [5, theroem 5.5], and that our assumption implies the covering theorem holds for K.

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C * Theorem 5.If ZFC is consistent then so is the existence of a model with a decreasing C *sequence of any finite length.
Proof.Going back to the proof of theorem 1, we note that for any n, (C * ) A k ]: A n can be computed from A n+1 using the cofinality-ω quantifier, which gives ⊇, and on the other hand, from n k=0 A k we know exactly which ordinals will have cofinality ω in L n+1 k=0 A k , which gives ⊆.So by starting e.g. from Without large cardinals this is best possible: Theorem 6.If there is no inner model with a measurable cardinal, then there is k < ω such that C * k = C * (k+1) .Proof.By applying lemma 4.2 inside each C * n , for every n we have where 2 ), and we claim that C * (k+1) = C * (k+2) .To simplify notation we assume w.l.o.g k = 0, i.e ω C * i = ω V i for i = 1, 2 (so we can omit the superscript) and we want to show that (C * ) C * = C * .We have: .
and since ω C *
Our next goal is to show that this is precisely the consistency strength of a decreasing sequence: Theorem 7. If there is an inner model with a measurable cardinal, then it is consistent that the sequence C * n | n < ω is strictly decreasing.
In [5, theroem 5.16] the authors show that if where We improve this by showing that C * is unchanged after adding a Prikry sequence to κ, and then investigate the C * -chain of L µ .First we prove two useful lemmas.
Proof.We use Mathias's characterization of Prikry forcing: Fact 9 (Mathias, cf.[6]).Let M be a transitive model of ZFC, U a normal ultrafilter on κ, then S ⊆ κ of order type ω is generic over M for the Prikry forcing defined from U iff for any X ∈ U , S X is finite.
Lemma 10.For any β < κ and any α, Proof.κ is regular in V , thus it is regular in every M β which is an inner model of V .If M β cf (α) = κ, then there is a cofinal κ-sequence in α (in both M β and V ), and since κ is regular we get V cf (α) = κ.If V cf (α) = κ, then the same argument rules out If β = γ+1, then M β is contained in M γ and closed under κ-sequences in it, so they agree on cofinality κ, and by induction we get that they agree with V as well.So assume β is limit and let α η | η < κ be a cofinal sequence in α. by definition of the limit ultrapower, each α η is of the form j β,β (ᾱ η ) for some β < β.We can also assume that each such β is large enough so that α ∈ Range(j β,β ).Since β < κ, there is some β fitting κ many α η s, so w.l.o.g we can assume β fits all of them.We can assume β > 0 so κ is a fixed point of j β,β .If ᾱ = sup {ᾱ η | η < κ} 2 then, since α = sup j β,β (ᾱ η ) | η < κ and α ∈ Range(j β,β ), we must have that α = j β,β (ᾱ).ᾱ is of cofinality κ in V , so by induction also in M β , hence by elementarity M β cf (α) = κ.
As in the proof of [5, theroem 5.16 As we noted, M ω 2 = L ν where ν is a measure on κ ω 2 and by lemma 8, E is Prikry generic over it, hence, since cofinality κ is unaffected by Prikry forcing on κ ω 2 , we get The other direction of the proof is almost the same as in [5, theroem 5.16]: E is the set of ordinals in the interval (κ, κ ω 2 ) which have cofinality ω in V [G] and are regular in the core model 3 , and from Now we can analyze the C * -chain of V = L µ .To avoid confusion we stick to the notation C * α , starting from C * 0 = L µ = M 0 , with M α being the αth iterate of L µ and κ α the αth image of the measurable cardinal.So by [5, theroem 5.16] we have As we noted earlier M ω 2 is also of the form L µ ′ for the measurable κ ω 2 , and by lemma 8, E 1 is Prikry generic over it, so by proposition 11 which is again the ω 2 -th iterate of M ω 2 , i.e.M ω 2 +ω 2 , plus the corresponding sequence - . We can continue inductively, and get the following: where M α is the αth iterate of V , κ α the αth image of the measurable cardinal and This concludes the proof of theorem 7. To analyze (C * ω ) L µ , we will use the following theorem, due to Bukovský [1,2] and Dehornoy [3]: Fact 13.If κ is measurable, M α is the α-th iterate of V as by a normal ultrafilter on κ and κ α the α-th image of κ, then for any limit ordinal λ exactly one of the following holds: Proof.By definition and our previous calculation, Since ω 3 is of cofinality ω but not of the form α + ω, the conclusion follows from (3) of fact 13.

C
We summarize what is now known in terms of equiconsistency: (1) ZFC is equiconsistent with (2) Existence of an inaccessible cardinal is equiconsistent with V = C * + 2 ℵ0 = ℵ 2 .
Compared to the results regarding HOD, the following questions remain open: Question 15.
(1) Is it possible, under any large cardinal hypothesis, that ∀n < ω C * n C * (n+1) and C * ω ZFC?More generally, for which ordinals α can we get a decreasing C * sequence of length α?
A natural first attempt towards answering the first question would be to try and work in a model with more measurable cardinals.However, it seems that it would require at least measurably many measurables: in a forthcoming paper [9], we generalize [5, theroem 5.16] and our proposition 11 and show the following: where U = U γ | γ < χ is a sequence of measures on the increasing measurables κ γ | γ < χ where χ < κ 0 .Iterate V according to U where each measurable is iterated ω 2 many times, to obtain , and set M χ as the directed limit of this iteration.Let G be generic over V for the forcing adding a Prikry sequence to every κ γ .Set for every γ < χ So, if V = L [U] as above, C * is of the form L [U 1 ] [G 1 ] for some sequence of measures and a sequence of Prikry sequences on it's measures, and so C * 2 is again of that form, where we iterated the measures in U ω 2 • 2 many times and add Prikry sequences.So again, as we've done here, we'll get that C * ω is the intersection of the models M χ ω 2 •n where we iterated each measure ω 2 • n times .This is due to the facts that changing the order of iteration between the measures doesn't change the final result, and that the Prikry sequences "fall out" during the intersection.Now, we don't have a complete analysis of intersections of iterations by more than one measure, but Dehornoy proves the following more general fact: Fact 17 ([3, section 5.3 proposition 3]).For every α let N α be the αth iteration of V by some measure.Assume λ is such that for every α < λ, N α cf(λ) = ω but there is no ρ such that λ = ρ + ω.Then if M is a transitive inner model of ZFC containing α<λ N α , then there is some α < λ such that N α ⊆ M .
So, if we take N α to be the iteration of V by the first measure in U, we get that C * ω contains α<ω 3 N α , but doesn't contain any N α for α < ω 3 , so C * ω cannot satisfy AC.
Hence a different approach, or larger cardinals, would be required to answer this question.