aa r X i v : . [ m a t h . L O ] J un Choice principles in local mantles
Farmer Schlutzenberg ∗ [email protected] 2, 2020 Abstract
Assume
ZFC . Let κ be a cardinal. A < κ -ground is a transitive properclass W | = ZFC such that there are P , g such that P ∈ W is a poset, | P | < κ , g is ( W, P )-generic, and the generic extension W [ g ] is equal to thefull set theoretic universe V . The κ -mantle M κ is the intersection of all < κ -grounds. The mantle M is the intersection of all < λ -grounds, overall cardinals λ .We prove here the following instances of choice principles in κ -mantles:If κ is inaccessible then M κ satisfies “for every γ < κ and f : γ → H κ + ,there is a choice function for f ”. If κ is weakly compact then M κ | = κ - DC .We also establish some other related facts, including that if κ is Σ -strongthen V M κ κ +1 = V M κ +1 .Under sufficient large cardinal assumptions, using methods from Woodin’sanalysis of HOD L [ x,G ] , we then analyze M L [ A ] κ , for A a set of ordinals ofsufficient complexity and κ a Silver indiscernible for L [ A ]. We show that M L [ A ] κ is a strategy mouse with a Woodin cardinal, which models ZFC .We also show that the definability of grounds from parameters followsfrom a theory satisfied by H κ , for all strong limit cardinals κ . Let us recall some standard notions from set-theoretic geology. We generallyassume
ZFC , though at times (in particular in §
2) we will also consider a weakertheory T (still with full AC , however).Given a transitive model W of ZFC and a forcing P ∈ W , a ( W, P ) -generic is a filter G ⊆ P which is generic with respect to W . For a cardinal κ , a < κ -ground of V is a transitive proper class W | = ZFC such that there is P ∈ W with P of cardinality < κ (with cardinality as computed in W , or equivalently, in V )and a ( W, P )-generic filter G such that V = W [ G ]. A ground is a < κ -groundfor some cardinal κ . The mantle M is the intersection of all grounds. The κ -mantle M κ is the intersection of all < κ -grounds. ∗ This work supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foun-dation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics M¨unster:Dynamics-Geometry-Structure. Here we work in some sort of second order set theory, so that we can quantify over suchclasses W ; for us a class must have the property that the structure ( V, ∈ , W ) satisfies ZFC inthe language with symbols ˙ ∈ , ˙ W which intepret ∈ and W . Throughout, we consider only set-forcing, no class-forcing.
1y [5], as refined in [1], there is a formula ϕ ( x, y ) in two free variables suchthat (i) for all r , W r = { x (cid:12)(cid:12) ϕ ( r, x ) } is a ground (possibly W r = V ), and (ii)for every ground W there is r such that W = W r . Therefore we can discussgrounds uniformly, and M and M κ are definable transitive classes (so we have ZFC with respect to these classes).In § ZFC : we show that it holds under a certain theory T (see 2.3), which istrue in H κ whenever κ is a strong limit cardinal (assuming ZFC ). The proof isessentially the usual
ZFC proof, however.From now on, we take W r to be defined as in §
2, by which r = ( H γ + ) W forsome γ ≥ ω for which there is a forcing P ∈ r , and there is a ( W, P )-generic G ,such that W [ G ] = V .Now suppose κ is a strong limit cardinal. It was shown by Usuba [12] thatthe grounds are set-directed, and reasonably locally so, such that in particularif R ∈ H κ , then there is s ∈ H κ with W s ⊆ T r ∈ R W r . Using this, he showed M | = ZFC and M κ | = ZF and H M κ = H M κ κ (obviously also M κ | =“ κ is a stronglimit cardinal”). Hence if κ = i κ then κ = i M κ and V κ = H κ and V M κ = H M κ = H M κ κ = V M κ κ | = AC . Usuba then showed in [13], that if κ is an extendible cardinal then M κ = M ,and hence in this case, M κ | = ZFC . Hence Usuba asked in [13] about whether M κ | = ZFC in general. We consider related questions in this paper. Let us firstsketch some further history.By remarks above, if κ is inaccessible then V M κ κ | = ZFC and M κ | =“ κ isinaccessible”, and likewise for Mahloness at κ . However, A. Lietz ([6]) answeredUsuba’s question above negatively (assuming large cardinals), showing that infact it is consistent relative to a Mahlo cardinal that κ is Mahlo but M κ | =“ κ - AC fails”. In fact, Lietz constructs a forcing extension L [ G ] of L in which κ isMahlo and M L [ G ] κ satisfies “there is a function f : κ → H κ + for which there isno choice function”. He also proved other related things.During this time, the theory of Varsovian models was also developed byFuchs, Schindler, Sargsyan and more recently the author. Here, among otherthings, full mantles M of mice (such as M swsw above) are analyzed (assumingthe full iterability of the mice in question; that is, (OR , OR)-iterability), andshown to be strategy mice, satisfying
ZFC . However, an analysis of certain κ -mantles of mice was missing. To state the next result, we need to mention aspecific mouse: Definition 1.1. M swsw denotes the least iterable proper class mouse (fine struc-tural L [ E ]-model) with ordinals δ < κ < δ < κ satisfying “each δ i is Woodinand each κ i is strong”.And M is a mouse just beyond M swsw . ⊣ Using Varsovian model techniques (the general development of which is moreinvolved), the author then analyzed the κ -mantle of M swsw , showing that it is astrategy mouse, modelling ZFC + GCH. An outline is given in §
3; the full proofdepends on the material of [8], and can be seen there. The argument, while foundindependently of Usuba’s extendibility result mentioned above, turned out tohave elements in common with its proof. Schindler then found an argument witha similar structure, showing in general that if κ is measurable then M κ | = AC ,2ence ZFC . We also present the proof of this result in §
3. In this paper, weadapt this mode of argument in a few more ways, deducing further instances ofchoice in M κ from large cardinal properties of κ . Definition 1.2.
Given an ordinal α and set X , let ( α, X ) -Choice be the asser-tion that for every function f : α → X , there is a choice function for f . And( < α, X ) -Choice is the assertion that ( β, X )-Choice holds for all β < α . ⊣ Our first main results, proved in §
3, are variants of Usuba’s extendibilityand Schindler’s measurability results mentioned above. Note that the followingtheorem applies to the kind of function involved in the failure of κ - AC in Lietz’example (but with a smaller domain). Note that we assume ZFC except whereotherwise stated; κ -amenable-closure is defined in 2.19. Theorem (3.15) . Let κ be inaccessible (so M κ | =“ κ is inaccessible”). Then:1. M κ is κ -amenably-closed.2. M κ | =“( κ, H κ )-Choice” iff M κ | =“ V κ is wellordered”.3. M | =“( < κ, H κ + )-Choice holds, and hence, ( H κ + ) <κ ⊆ H κ + ”. Remark 1.3.
In part 3, the “ κ + ” and “ H κ + ” are both in the sense of M κ .However, it can be that κ is Mahlo and M κ | =“( κ, H κ + )-Choice fails, and( H κ + ) κ
6⊆ H κ + ”; indeed, note that this occurs in Lietz’ example L [ G ] men-tioned above.In the following theorem, the initial observation that M κ | =“ V κ is wellorder-able” was due to Lietz: Theorem (3.14) . Let κ be weakly compact. Then:1. M κ | = κ - DC + “ κ is weakly compact”.
2. for each A ∈ M κ ∩ H κ + , M κ | =“ A is wellordered”.
3. if P ( κ ) M κ has cardinality κ then (i) κ is measurable in M κ , and (ii) x exists for every x ∈ P ( κ ) M κ .4. If M κ | =“ µ is a countably complete ultrafilter over γ ≤ κ ”, then theultrapower Ult( M κ , µ ) is wellfounded and the ultrapower embedding i M κ µ : M κ → Ult( M κ , µ )is fully elementary.As a corollary to Schindler’s proof, one easily gets: Corollary (3.3) . Let κ be measurable and µ be a normal measure on κ . Thenfor µ -measure one many γ < κ , M γ | =“ V γ +1 is wellorderable”. Regarding part 2, the author initially showed that M κ | = κ - DC , and later the authorand Lietz independently noticed that one also gets the fact that every set in H κ + ∩ M κ iswellordered in M κ . So also M κ | =“ κ + is regular and H κ + | = ZFC − ”. Note that the “ κ + ” and “ H κ + ” here are computed in V , not M κ .
3s mentioned above, Usuba showed that M = M κ assuming κ is extendible.The next result indicates that there are signs of this in the leadup to an ex-tendible cardinal (for the definition of a Σ -strong cardinal, see 3.5): Theorem (3.9) . Suppose κ is Σ -superstrong. Then V M κ κ +1 = V M κ +1 .Analogously, down lower: Theorem (3.4) . Let A be a set such that A exists. Let κ be an A -indiscernible.Then V M L ( A ) κ κ +1 = V M L ( A ) κ +1 and this set is wellordered in M L ( A ) κ .Finally in §
4, with another variant of the mode of argument above:
Definition 1.4. M denotes the least iterable proper class mouse with a Woodincardinal. And M is the sharp for M . ⊣ Theorem (4.1) . Assume that M exists and is fully iterable; that is, (OR , OR)-iterable. Then M M κ is a fully iterable strategy mouse which models ZFC .From the above theorem we will deduce:
Theorem (4.2) . Assume that M exists and is fully iterable; that is, (OR , OR)-iterable. Let A be a set of ordinals with M ∈ L [ A ]. (Then A exists.) Let κ be an A -indiscernible. Then M L [ A ] κ is a fully iterable strategy mouse whichmodels ZFC . We discuss here some background, starting with the key fact of the definabilityof set-forcing grounds under
ZFC , proved by some combination of Laver, Woodinand Hamkins:
Fact 2.1.
Let
M, N be proper class transitive inner models modelling
ZFC and γ ∈ OR with P ( γ ) ∩ M = P ( γ ) ∩ N . Let P ∈ M and Q ∈ N , with P , Q ⊆ γ , and let G be ( M, P )-generic and H be ( N, Q )-generic and suppose M [ G ] = N [ H ] = V . Then M = N . Definition 2.2.
Assume
ZFC . Let κ be a cardinal. A < κ -ground is a transitiveproper class W | = ZFC such that for some P ∈ W with card W ( P ) < κ , there is a( W, P )-generic G such that W [ G ] = V . (Note that because κ is a V -cardinal, itwould not change the notion if we said card V ( P ) < κ instead of card W ( P ) < κ .)A ground is a < κ -ground for some κ . ⊣ We will discuss the proof of the result above, for two purposes. First, it iscentral to our concerns, and the proof contains elements which will come upin various places later, so it is natural to all these things together. Second, wewish to prove a version which assumes less background theory (than
ZFC ). Theauthors of [3] make use of an analysis of the complexity of the definability ofgrounds. As shown there, each ground W is, in particular, Σ in a parameter r . However, the Σ definition given there is not particularly local: to compute V Wα , they work in V β , for a significantly larger ordinal β . So for their [3,Theorem 4], they adopt the background theory ZFC δ . We show here that theground definability can be done much more locally (though still requiring Σ complexity), hence requiring significantly less than ZFC δ .4 efinition 2.3. Let T − be the following theory in the language of set theory.The axioms are Extensionality, Foundation, Pairing, Union, Infinity, “Every setis bijectable with an ordinal”, Σ -Separation and Σ -Collection. Now let T = T − + Powerset . ⊣ We will show that models of T can uniformly define their grounds fromparameters. First we give some lemmas. Lemma 2.4.
Assume
ZFC . Then for every cardinal κ , (i) H κ | = T − , and (ii) H κ | = T iff κ is a strong limit cardinal.The usual proofs from ZFC easily adapt to give:
Lemma 2.5.
Assume T . Then (i) for each ordinal ξ , H ξ exists, (ii) V = S ξ ∈ OR H ξ , (iii) H ξ V , (iv) H ξ | = T − , (iv) the Lowenheim-Skolem theoremholds. Lemma 2.6 (Forcing over T − and T ) . Let M | = T − . Let P ∈ M be a posetwith P ⊆ γ ∈ OR M and G be ( M, P )-generic. Then:1. The forcing theorem for Σ -formulas holds for the extension M [ G ].2. The Σ -forcing relation for ( M, P ) is ∆ M ( { P } ). Hence, the restriction ofthe Σ -forcing relation to H Mκ is ∆ H Mκ ( { P } ), uniformly in κ , and henceamenable to H Mκ , for M -cardinals κ > γ .3. M [ G ] | = T − , and if M | = T then M [ G ] | = T .4. M and M [ G ] have the same cardinals κ > γ ,5. for each M -cardinal κ > γ , we have H M [ G ] κ = H Mκ [ G ], andBefore giving the proof, let us remark that such local forcing calcluations arevery common in certain places in the literature, in particular in fine structuretheory, where much more local calculations are often used. Proof.
Parts 1, 2: The usual internal definition of the Σ -forcing relation works locally; in fact, for each ξ ∈ OR M with ξ ≥ γ , the Σ -forcing relation fornames in H ξ , is ∆ H ξ ( { P } ), uniformly in ξ . This gives the Forcing Theorem forΣ formulas in the usual manner.Now define the Σ -strong-forcing relation ∗ over M as follows. Given aΣ formula ϕ and τ , . . . , τ k ∈ M P and p ∈ P , say that p ∗ ∃ y , . . . , y n ϕ ( τ , . . . , τ k , y , . . . , y n )iff there are σ , . . . , σ n ∈ M P such that p ϕ ( τ , . . . , τ k , σ , . . . , σ n ) . Then using the Σ -Forcing Theorem, it is easy to see that M [ G ] | = ∃ ~yϕ ( ~y, τ G )iff there is p ∈ G such that M | = p ∗ ∃ ~yϕ ( ~y, ~τ ).5ow the usual external Σ -forcing relation p ϕ ( ~τ ) (for p ∈ P , ϕ a Σ formula and ~τ ∈ M P ) asserts that for sufficiently large λ ∈ OR, V Col( ω,λ ) | = ∀ H [ p ∈ H is ( M, P )-generic ⇒ M [ H ] | = ϕ ( ~τ H )] . We claim that p ∗ ϕ ( ~τ ) iff p ϕ ( ~τ ). For the non-trivial direction, supposethat p ϕ ( ~τ ).Then we have: ∀ q ≤ p ∃ r ≤ q [ r ∗ ϕ ( ~τ )] . (1)For letting q be otherwise, and letting H be ( M, P )-generic with q ∈ H , thenby the Σ -Forcing Theorem, we must have that M [ H ] | = ¬ ϕ ( ~τ H ), contradictingour assumption.But using line (1), working in M , using Σ -Collection and AC , we can puttogether a name σ ∈ M P showing that p ∗ ϕ ( ~τ ), a contradiction.Part 3: Most of the axioms are routine. Powerset, in the case that M | = T ,comes from the typical nice name calculations. Let us verify that M [ G ] | = Σ -Collection. Fix a Σ formula ϕ and σ, τ ∈ M P . Let t ∈ M be the transitiveclosure of { σ, τ } . Then there is w ∈ M such that for all p ∈ P and ̺ ∈ t , if p ∗ ̺ ∈ σ and ∃ yϕ ( ̺, τ, y ) , then there is y ∈ M P ∩ w such that p ∗ ̺ ∈ σ and ϕ ( ̺, τ, y ) . But then using w , we easily get a bound on witnesses in M [ G ], as desired. Thisand the Σ -Forcing Theorem easily yields Σ -Separation in M [ G ].The remaining parts follow from routine calculations with nice names. Definition 2.7.
Let (
M, E ) | = T − . A ground of M is a W ⊆ M such that:1. ( W, E ↾ W ) is M -transitive ; that is, for all x ∈ W and all y ∈ M , if yEx then y ∈ W ,2. W | = T − ,3. there is P ∈ W and a ( W, P )-generic G ∈ M such that M = W [ G ].4. If ( M, E ) | = T then ( W, E ↾ W ) | = T . ⊣ We now prove that T suffices for the definability of grounds (in the sense ofthe definition above). The proof is essentially that due to some combination ofLaver, Woodin and Hamkins. In the proof we make implicit use of Lemma 2.6,to allow the forcing calculations: Theorem 2.8 (Ground definability under T ) . Assume T . Let γ ∈ OR, H ⊆H γ + and κ ≥ γ + a cardinal. Then there is at most one transitive M ⊆ H κ suchthat M | = T − , ( H γ + ) M = H , and M is a set-ground for H κ via some forcing P ∈ H . 6 roof. We proceed by induction on κ . For κ = γ + it is trivial.Suppose κ is a limit cardinal, and that for each cardinal θ ∈ [ γ + , κ ), there isa (unique) model M θ of ordinal height θ with the stated properties. Then clearly M = S θ<κ M θ is the unique candidate at κ . To see that M works, we just needto verify that M is indeed a set-ground of H κ via some P ∈ H ; i.e. there is P ∈ H and an ( M, P )-generic G ⊆ P such that M [ G ] = H κ . But we can use any ( P , G )which worked at some earlier θ . For let θ ≤ θ < κ , and let ( P , G ) , ( P , G )work for M = M θ and M = M θ . Clearly G is also ( M , P )-generic, andvice versa. And since H M γ + = H = H M γ + , and H [ G ] = H γ + = H [ G ], it followsthat H κ = M [ G ] = M [ G ] and M [ G ] = M [ G ] = H κ , so the specific choiceof ( P , G ) is irrelevant.So consider κ = θ + > γ + . Let M, N be grounds of H κ with the statedproperties. By induction, M ∩ H θ = N ∩ H θ . It just remains to verify that P ( θ ) ∩ M = P ( θ ) ∩ N . The proof is, however, not by contradiction; we will notassume that M = N . Fix ( P , G ) such that P ∈ H and G is ( M, P )-generic and M [ G ] = H κ .Suppose first that cof( θ ) > γ , as this case is easier; however, it is in the endsubsumed into the general case. Let A ⊆ θ . Then: Claim 1. A ∈ M iff A ∩ α ∈ M for all α < θ . Proof.
For the non-trivial direction, suppose A ∩ α ∈ M for every α < θ . Let f : θ → M be f ( α ) = A ∩ α . Then f ∈ H κ . So there is a P -name ˙ f ∈ M with˙ f G = f . Working in M , for p ∈ P , compute D p = { α < θ (cid:12)(cid:12) ∃ x [ p ˙ f (ˇ α ) = ˇ x ] } , and let f p : D p → θ be the function f p ( α ) = unique x such that p ˙ f (ˇ α ) = ˇ x. Then because cof( θ ) > γ , there is p ∈ G such that D p is cofinal in θ . Then f = (cid:16)S α ∈ D p f p ( α ) (cid:17) ∈ M .We now argue in general. Claim 2.
Let A ⊆ θ . Then A ∈ M iff for every X ∈ P ( θ ) ∩ M such thatcard( X ) < ( γ + ) V (as computed in M or V ), we have A ∩ X ∈ M . Proof.
The forward direction is trivial. So let A ⊆ θ with A / ∈ M . Let ˙ A ∈ M be a P -name and p ∈ G such that p ˙ A ⊆ ˇ θ. For each q ≤ p , if there is α < θ such that q ˇ α ∈ ˙ A and q ˇ α / ∈ ˙ A, then let α q be the least such α ; otherwise α q is undefined. Let D be the set ofall q ≤ p such that α q exists. Then G ⊆ D , because otherwise q decides allelements of ˙ A , so A ∈ M .In M , let X = { α q (cid:12)(cid:12) q ∈ D } . Then X ∈ M , card M ( X ) ≤ γ and X ∩ A / ∈ M ,as desired. For given Y ∈ P ( X ) ∩ M , an easy density argument shows that Y = X ∩ A . 7 laim 3. Let X ⊆ θ with card( X ) < ( γ + ) V . Then X ∈ M iff X ∈ N . Proof.
Suppose X = X ∈ N . Let ˙ X ∈ M be a P -name for X . Using theforcing relation and ˙ X , there is a set X ∈ P ( θ ) ∩ M with X ⊆ X andcard( X ) < ( γ + ) V . Proceeding back-and-forth, construct (in V ) a continuoussequence of sets h X α i α<γ + such that (i) X = X , (ii) X ωα +2 n +1 ∈ M and X ωα +2 n +2 ∈ N , and (iii) card( X α ) < ( γ + ) V .Now γ + < κ , so h X α i α<γ + ∈ H κ , so M, N have names for this sequence. Soas in the cof( θ ) > γ case, we get a cofinal set D M ⊆ γ + such that D M ∈ M and h X α i α ∈ D M ∈ M . Likewise with a cofinal set D N ∈ N . Let D ′ M be theset of limit points of D M , and D ′ N likewise. So these are club in γ + . Let α ∈ D ′ M ∩ D ′ N . Then note that X α = [ β ∈ D M ∩ α X β = [ β ∈ D N ∩ α X β ∈ M ∩ N. Let π : ξ → X α be the increasing enumeration of X α . Then ξ < γ + and π ∈ M ∩ N . We have X ⊆ rg( π ). Let ¯ X = π − ( X ). Then ¯ X ∈ N . But H Mγ + = H = H Nγ + , so ¯ X ∈ M . So π “ ¯ X = X ∈ M , as desired.This completes the proof of ground definability under T . Definition 2.9.
Assume T . Let ϕ grd ( r, x ) be the formula “ r is a transitiveset, and there are γ, P , G, M, κ such that γ ∈ OR, OR r = ( γ + ), κ is a cardinal, M is transitive, M | = T − , M ⊆ H κ , P ∈ r = ( H γ + ) M , G is ( M, P )-generic, H κ = M [ G ] and x ∈ M ”.We write W ′ r = { x (cid:12)(cid:12) ϕ grd ( r, x ) } . We say r is a true index iff W ′ r is properclass. We write W r = W ′ r for true indices r , and W r = V otherwise. ⊣ Corollary 2.10.
Assume
ZFC + GCH and let λ be a limit cardinal. Then thegrounds of H λ are definable from parameters over H λ . Remark 2.11.
Assume
ZFC + GCH. Then for each limit ordinal ξ , the model V ω + ξ is equivalent in the codes to the model H ℵ ξ . So one can correctly formulate“grounds” of V ω + ξ , and they are definable over that model from parameters.Thus we have the standard uniform definability of grounds, just assuming T : Lemma 2.12.
Let M | = T . Then { W Mr (cid:12)(cid:12) r ∈ M } enumerates exactly thegrounds of M (with repetitions, including M itself). Remark 2.13.
Assume T . Note that ϕ grd is Σ , and the assertion “ r is a trueindex” is Π . (In fact, there are fixed Σ and Π formulas, such that T provesthat these fixed formulas always work.) Moreover, letting ξ = card(trcl( { r, x } )),note that ϕ grd ( r, x ) is absolute between V and H (2 ξ ) + . (It is witnessed by some( H ξ + , M ), a structure of size 2 ξ .) Therefore: Fact 2.14 (Local definability of grounds) . Assume T +“There is a proper classof strong limit cardinals”. Let λ be a strong limit cardinal. Let r ∈ H λ be atrue index. Then H λ | =“ r is a true index” and W H λ r = W r ∩ H λ = H W r λ .8t seems it might be possible, however, that H λ | =“ r is a true index” while r fails to be a true index in V .The remaining facts in this section, and the rest of the paper, have a back-ground theory of ZFC . We have not investigated to what extent things gothrough under T . By [12, Proposition 5.1] and an examination of its proof, wehave: Fact 2.15 (Local set-directedness of grounds (Usuba)) . Assume
ZFC . Let θ bea strong limit cardinal and R ∈ H θ . Then there is t ∈ H θ such that t ∈ T r ∈ R W r and W t ⊆ W r and W t = W W r t for each r ∈ R . In particular, W t ⊆ T r ∈ R W r . Proof.
We refer here to the λ -uniform covering property for V ; see [9, Definition2.1] or [12, Definition 4.2]. Let us set up some of the notation from the proofof [12, Proposition 5.1]. Let X = R (following the notation from [12]). Wemay assume that X is a set of true indices r . For r ∈ X let P r ∈ W r bea forcing witnessing that r is a true index. Let κ be a regular cardinal with κ > card( X ) and κ > card( P r ) for each r (so it suffices if κ > card(trcl( X ))).Then the proof of [12, Proposition 5.1] constructs a ground W ⊆ T r ∈ X W r withthe λ = κ ++ -uniform covering property for V . Therefore by [9, Theorem 3.3],there is P ∈ W such that W | =“card( P ) = 2 <λ ” and W is a ground of V via P . Let γ = card W ( P ) and t = ( H γ +0 ) W . So γ < θ , t is a true index and W = W t . Let B ∈ W be such that W | =“ B is the complete Boolean algebradetermined by P ” (so P is a dense sub-order of B ). So card W ( B ) ≤ (2 γ ) W < θ .Then by [2, Lemma 15.43] (or [12, Fact 3.1]) for each r ∈ X there is some B r ∈ W with B r ⊆ B and there is a ( W, B r )-generic G r such that W [ G r ] = W r .So letting γ = (2 γ ) W , then t = ( H γ + ) W is as desired. Definition 2.16.
Assume
ZFC . The κ -mantle M κ is the intersection of all < κ -grounds. The mantle M is the intersection of all grounds. ⊣ An easy corollary of local directedness is:
Fact 2.17 (Invariance of M κ ) . Assume
ZFC . Let κ be a strong limit cardinaland r ∈ H κ . Then M W r κ = M κ . Lemma 2.18 (Absoluteness of M κ ) . Assume
ZFC . Let κ < λ be strong limitcardinals and suppose H λ = V λ V . Then for each r ∈ H κ , we have:(i) < κ -grounds and M κ are absolute to V λ : W V λ r = W r ∩ V λ = V W r λ and M V λ κ = M κ ∩ V λ = V M κ λ , (ii) V W r λ W r ,(iii) M V Wrλ κ = M W r κ ∩ V W r λ = M κ ∩ V λ = M V λ κ . Proof.
Part (i): The absoluteness of W r is because the class true indices r isΠ , and each W r is Σ ( { r } ). But then clearly M V λ κ = \ r ∈ V κ W V λ r = \ r ∈ V κ V W r λ = V M κ λ . We wrote R in the statement of the fact for consistency with later notation. W r = V then this is trivial. Suppose W r ( V and let ϕ be Σ and x ∈ W r ∩ V λ and suppose that W r | = ϕ ( x ). Then by Fact 2.14, V | = ψ ( x )where ψ asserts “There is a strong limit cardinal ξ such that W H ξ r | = ϕ ( x )”,but this is also Σ , so V λ | = ψ ( x ), so letting ξ < λ witness this, again by Fact2.14, we get W r ∩ H ξ | = ϕ ( x ), so W r ∩ V λ | = ϕ ( x ).Part (iii): This follows from the previous parts and Fact 2.17. Definition 2.19.
Let N be an inner model. Let f : κ → N . Say that f is amenable to N iff f ↾ α ∈ N for every α < κ . Say that N is κ -amenably-closed iff for every f : κ → N , if f is amenable to N then f ∈ N . Say that N is κ -stationarily-computing ( κ -unboundedly-computing ) iff for every f : κ → N ,there is a stationary (unbounded) A ⊆ κ such that f ↾ A ∈ N . ⊣ Lemma 2.20.
Let N be an inner model and κ > ω be regular. If N is κ -stationarily-computing then N is κ -unboundedly-computing. If N is κ -unboundedly-computing then N is κ -amenably-closed. Proof.
The first assertion is immediate. For the second, let g : κ → N beamenable to N , and let f : κ → N be f ( α ) = g ↾ α . (Note that f ( α ) ∈ N foreach α < κ .) Let A ⊆ κ be unbounded, with f ↾ A ∈ N . Then g = S rg( f ), sowe are done. Lemma 2.21.
Let W be a < κ -ground of V , where κ > ω is regular. Then W is κ -stationarily-computing. Proof.
This is a standard forcing argument. Let f : κ → W . Write x α = f ( α ).Fix a forcing P ⊆ θ < κ in W and G a ( W, P )-generic such that W [ G ] = V . Fixa name ˙ f ∈ W such that ˙ f G = f . For each α < κ there is a condition p α ∈ G such that p α “ ˙ f (ˇ α ) = ˇ x α ”. But P ⊆ θ and κ is regular, so there is therefore p ∈ G and a stationary set A ⊆ κ such that p = p α for all α ∈ A . Let q ∈ G besuch that q ≤ p and q “ ˙ f is a function with domain ˇ κ ”. Then letting A ′ = { α < κ (cid:12)(cid:12) ∃ x [ q ˙ f (ˇ α ) = ˇ x ] } , we have A ⊆ A ′ and f ↾ A ′ ∈ W . Lemma 2.22.
The intersection of any family of κ -amenably-closed structuresis κ -amenably-closed.Therefore: Lemma 2.23. If κ is inaccessible then M κ is κ -amenably-closed. Proof.
For each r ∈ V κ , W r is κ -stationarily-computing, hence κ -amenably-closed. So this lemma follows from the previous one. κ -mantle Note that from now on we are working with
ZFC as background theory.10 emark 3.1.
The first positive results along the lines of what we will provehere, are Usuba’s work, including his extendibility result. Some time after this(though independently from it) the author showed that the κ -mantle M Mκ of M = M swsw is a strategy mouse (notation is as above). Here is the outline ofthe argument, including what is relevant to us here, but omitting all specificsto do with Varsovian models. We will also give another related argument, andprovide more details, in § M ∞ , which is the directlimit of (pseudo-)iterates P of M via correct trees T on M , with T ∈ M | κ ,and which are based on M | δ . It also defines a certain fragment Σ of the iterationstrategy for M ∞ , yielding a strategy mouse M ∞ [Σ]. An initial argument, usingthe Varsovian model techniques, shows that M ∞ [Σ] ⊆ M Mκ .The other direction proceeds roughly as follows. Let X ∈ M Mκ be a set ofordinals. We must see that X ∈ M ∞ [Σ]. Now κ is measurable in M . Let E be a normal measure on κ , in the extender sequence of M , and let j : M → U = Ult( M, E )be the ultrapower map. By elementarity, j ( X ) ∈ M Uj ( κ ) . With methods fromthe Varsovian model analysis, one can then construct a specific < j ( κ )-ground W of U , with W ⊆ M ∞ [Σ]. Then j ( X ) ∈ M Uj ( κ ) ⊆ W ⊆ M ∞ [Σ] . Other facts from Varsovian model analysis give j ↾ α ∈ M ∞ [Σ] for each α ∈ OR.But then X ∈ M ∞ [Σ], as desired, since β ∈ X ⇐⇒ j ( β ) ∈ j ( X ) . One can see that the preceding argument has structural similarities to Usuba’sresult (cf. [13]). Schindler then found the following result, using an argumentwith a related structure. We will use an adaptation of the proof for Theorem3.14 later, so we present this one first. We give essentially Schindler’s proof,although the specific details might differ slightly.
Fact 3.2 (Schindler) . Let κ be measurable. Then M κ | = AC , and hence M κ | = ZFC . Proof.
Let A ∈ M κ . We will find a wellorder < A of A with < A ∈ M κ .Let µ be a normal measure on κ and M = Ult( V, µ ) and j = i Vµ : V → M the ultrapower map. So κ = cr( j ) and j ( A ) ∈ M Mj ( κ ) . Claim 4.
We have:1. M Mj ( κ ) ⊆ M Mκ ⊆ M κ , and2. j ↾ M κ is amenable to M κ . This was followed by Lietz’s results in [6], such as that with the Mahlo cardinal. What blocks the more obvious attempt to prove this is that it is not clear that the iterationmaps i PQ between the iterates P, Q of the direct limit system eventually fix X . roof. Part 1: The first ⊆ is immediate. For the second, we have M κ = \ r ∈ V κ W r and M Mκ = \ r ∈ V κ W Mr . Let µ r = µ ∩ W r . Then by standard forcing calculations and elementarity, weget µ r ∈ W r and W Mr = j ( W r ) = Ult( W r , µ ) V = Ult( W r , µ r ) W r , so W Mr ⊆ W r , so M Mκ ⊆ M κ as desired.Part 2: Let r ∈ V κ . Then calculations as above give i W r µ r ↾ W r ⊆ j . But M κ ⊆ W r , and so j ↾ M κ is amenable to W r . Therefore j ↾ M κ is amenable to M κ , as desired.Since κ is a strong limit, Fact 2.15 gives s ∈ V Mj ( κ ) such that M Mj ( κ ) ⊆ W = W Ms ⊆ M Mκ . So j ( A ) ∈ W | = ZFC , so there is a wellorder < ∗ of j ( A ) with < ∗ ∈ W . But W ⊆ M Mκ , so < ∗ ∈ M Mκ ⊆ M κ .Now working in M κ , where we have k = j ↾ A and j ( A ) and < ∗ , we candefine a wellorder < A of A by setting, for x, y ∈ A : x < A y ⇐⇒ k ( x ) < ∗ k ( y ) . This completes the proof.As a corollary to the proof above, we observe:
Corollary 3.3.
Let κ be measurable and µ be a normal measure on κ . Thenfor µ -measure one many γ < κ , M γ | =“ V γ +1 is wellorderable”. Proof.
Continue with the notation from the proof of Fact 3.2. We show that M Mκ | =“ V κ +1 is wellorderable”. Claim 5. V κ +1 ∩ M Mj ( κ ) = V κ +1 ∩ M Mκ = V κ +1 ∩ M κ . Proof.
We have V κ +1 ∩ M κ ⊆ V κ +1 ∩ M Mj ( κ ) since j ↾ M κ : M κ → M Mj ( κ ) is elementary and κ = cr( j ). But by Claim 4 of the proof of Fact 3.2, thissuffices.By Fact 3.2, M κ | = AC , so M Mj ( κ ) | = AC also. Let < ∗ ∈ M Mj ( κ ) be a wellorderof V κ +1 ∩ M Mj ( κ ) . Then by Claim 4 from the proof of Fact 3.2, and also Claim 5above, we have < ∗ ∈ M Mκ and < ∗ is a wellorder of V κ +1 ∩ M Mκ .We next use the simple idea above to prove that certain cardinals are “stable”with respect to the mantle. The first observation is: Theorem 3.4.
Let A be a set such that A exists. Let κ be an A -indiscernible.Then V M L ( A ) κ κ +1 = V M L ( A ) κ +1 and this set is wellordered in M L ( A ) κ .12 roof. Let j : L ( A ) → L ( A ) be elementary with cr( j ) = κ . We write M κ for M L ( A ) κ ; likewise M j ( κ ) . Now j ↾ M κ : M κ → M j ( κ ) is elementary. Clearly M j ( κ ) ⊆ M κ . But also, B = V M κ κ +1 ⊆ V M j ( κ ) κ +1 as in the previous proof. So V M j ( κ ) κ +1 = B . But V M j ( κ ) j ( κ ) | = ZFC , so there is a wellorder of B in M j ( κ ) ⊆ M κ .It now follows that V M κ κ +1 = V M κ +1 , because we can take j ( κ ) as large as welike, hence past any true index. Definition 3.5.
A cardinal κ is Σ -strong iff for every α ∈ OR there is anelementary embedding j : V → M with α < j ( κ ) and V α ⊆ M and Th M Σ ( V α ) =Th V Σ ( V α ). An embedding j : V → M is superstrong iff V j ( κ ) ⊆ M . A cardinal κ is ∞ -superstrong iff for every α ∈ OR there is a superstrong embedding j withcr( j ) = κ and j ( κ ) > α .A superstrong extender is the V β -extender derived from a superstrong em-bedding j : V → M where β = j ( κ ) and κ = cr( j ). ⊣ Lemma 3.6. If E is a superstrong extender and W | = ZFC is a transitive properclass with E ∈ W , then W | =“ E is a superstrong extender”. Proof.
By definition, E is derived from a superstrong embedding j : V → M. Let β = j ( κ ) where κ = cr( j ).Now W can compute Y = Ult( W, E ), and the ultrapower map k : W → Y. Because E ∈ W , we have V Wβ = V β , and it is straightforward to see that W | =“ k is a superstrong embedding”, and moreover, k ( κ ) = β . Remark 3.7.
Say that a cardinal κ is ∞ - -extendible iff for every α ∈ ORthere is β ∈ OR with β ≥ α and and an elementary j : V κ +1 → V β +1 (hence j ( κ ) = β ) with cr( j ) = κ . Recall that κ is extendible iff for every α ∈ ORwith α > κ there is β ∈ OR and an elementary j : V α → V β with cr( j ) = κ and j ( κ ) > α . Theorem 3.8.
We have:1. Every extendible cardinal is ∞ -1-extendible and carries a normal measureconcentrating on ∞ -1-extendible cardinals.2. Every ∞ -1-extendible cardinal is ∞ -superstrong and carries a normal mea-sure concentrating on ∞ -superstrong cardinals. That is, for each Σ formula ϕ and all ~x ∈ ( V α ) <ω , we have M | = ϕ ( ~x ) iff V | = ϕ ( ~x ).
13. Every ∞ -superstrong cardinal is Σ -strong and carries a normal measureconcentrating on Σ -strong cardinals. Proof.
The proof is quite routine, but we provide most of the details for com-pleteness.Part 1: This is routine and left to the reader.Part 2: Let κ be ∞ -1-extendible. Let j : V κ +1 → V β +1 be elementarywith cr( j ) = κ . Let E be the extender derived from j with support V β . Let M = Ult( V, E ) and k : V → M be the ultrapower map. It suffices to show that k is a superstrong embedding with k ( κ ) = β and that M | =“ κ is ∞ -superstrong”. Claim 6. M is wellfounded, V β ⊆ M , M κ ⊆ M and k ( κ ) = β . Proof.
These are standard calculations with ultrapowers via extenders. We have V β ⊆ M and k ( κ ) = β as usual. Let h x α i α<κ ⊆ M . Then there is h f α , a α i α<κ with a α ∈ [ V β ] <ω and f α : [ V κ ] <ω → V such that x α = k ( f α )( a α ). But k ( h f α i α<κ ) ↾ κ = h k ( f α ) i α<κ , and noting that β is inaccessible, we have h a α i α<κ ∈ V β ⊆ M . Therefore we have, as desired, that h x α i α<κ = h k ( f α )( a α ) i α<κ ∈ M. Now consider the structure ( V β , E ). Since β is inaccessible, there is a club C ⊆ β of α such that ( V α , E α ) ( V β , E ) , where E α = E ∩ V α . Let α ∈ C . Let k α : V → M α = Ult( V, E α ) . Then M α is wellfounded, since we have an elementary factor embedding ℓ : M α → M. We have V α ⊆ M α , and by the elementarity, we get k α ( κ ) = α ; so E α is asuperstrong extender.But E α ∈ V β ⊆ M for each α ∈ C . By the lemma, M | =“ E α is a superstrongextender, and κ = cr( k ) and α = k ( κ ) where k is the ultrapower map”. So M | =“ κ is < β -superstrong; that is, for every ξ < β there is a superstrongembedding ℓ : V → M ′ with cr( ℓ ) = κ and ℓ ( κ ) > ξ ”.Now let ξ ∈ OR. Since κ is ∞ -superstrong, M | =“ β = j ( κ ) is ∞ -superstrong”.So M has a superstrong embedding ℓ : M → N with cr( ℓ ) = β and ξ < ℓ ( β ). By the elementarity of ℓ , N | =“ κ is < ℓ ( β )-superstrong”. But V Nℓ ( β ) = V Mℓ ( β ) , so it easily follows that M | =“ κ is < ℓ ( β )-superstrong”. Since ξ was arbitrary, it follows that M | =“ κ is ∞ -superstrong”,as desired.Part 3: Let κ be ∞ -superstrong. We show first that κ is Σ -strong. So let α ∈ OR. We may assume that V α V . Let j : V → M be any superstrongembedding with cr( j ) = κ and α < j ( κ ). It suffices to verify:14 laim 7. Th M Σ ( V α ) = Th V Σ ( V α ). Proof.
Let ϕ be Σ and ~x ∈ ( V α ) <ω . If V | = ϕ ( ~x ) then V α | = ϕ ( ~x ), which implies M | = ϕ ( ~x ). Conversely, suppose M | = ϕ ( ~x ). Because κ is ∞ -superstrong, it isclearly strong, which implies that V κ V . Therefore V Mj ( κ ) M . Therefore V Mj ( κ ) | = ϕ ( ~x ). But V Mj ( κ ) = V j ( κ ) , so V j ( κ ) | = ϕ ( ~x ), so V | = ϕ ( ~x ), as desired.Now let j : V → M be any superstrong embedding. We will show that M | =“ κ is Σ -strong”, which completes the proof. Claim 8. M | =“ κ is < β -Σ -strong”, where β = j ( κ ). That is, for each α < β , M has an elementary k : M → N with cr( k ) = κ and V α ⊆ N and Th N Σ ( V α ) =Th M Σ ( V α ). Proof.
Since M | =“ β is strong”, V Mβ M and there are club many α < β such that V Mα = V α M . Fix some such α . Let E α be the V α -extender derivedfrom j . Then E α ∈ V β ⊆ M , and M | =“ E α is an extender”. Moreover, letting N α = Ult( M, E α ), we have V α ⊆ N α andTh N α Σ ( V α ) = Th M Σ ( V α ) . For let t = Th V Σ ( V κ ) = Th M Σ ( V κ ). Then letting k α : M → N α be the ultrapowermap, j ( t ) = Th M Σ ( V β ) and k α ( t ) = Th N Σ ( V Nk α ( κ ) ) . So Th M Σ ( V α ) = j ( t ) ∩ V α = k α ( t ) ∩ V α = Th N Σ ( V α ).Now since κ is Σ -strong, M | =“ β = j ( κ ) is Σ -strong”. So let α ∈ OR bea strong limit cardinal. Then M has an embedding ℓ : M → N with cr( ℓ ) = β and V Mα = V Nα and Th M Σ ( V Mα ) = Th N Σ ( V Mα ). By the claim and elementarity, N | =“ κ is < ℓ ( β )-Σ -strong”. But then extenders in N which witness < α -Σ -strength in N also witness this in M . Since α was arbitrary, we are done.We now prove an analogue of Usuba’s extendibility result down lower: Theorem 3.9.
Suppose κ is Σ -superstrong. Then V M κ κ +1 = V M κ +1 . Proof.
Suppose not and let r be such that V W r κ +1 ( V M κ κ +1 . Let λ ∈ OR be suchthat i λ = λ and r ∈ V λ . Let j : V → M witness Σ -strength with respect to λ .Since the class of true indices is Π , M | =“ r is a true index”. Also, by thelocal definability of grounds, W Mr ∩ V λ = W V Mλ r = W V λ r = W r ∩ V λ . In particular, V W Mr κ +1 = V W r κ +1 ( V M κ κ +1 .Since r ∈ V λ ⊆ V Mj ( κ ) , therefore M Mj ( κ ) ∩ V κ +1 ( M κ ∩ V κ +1 . But sincecr( j ) = κ , as in the proof of Theorem 3.2, we have M κ ∩ V κ +1 ⊆ M Mj ( κ ) ∩ V κ +1 , a contradiction. Question 3.10.
Suppose κ is strong. Is V M κ +1 = V M κ κ +1 ?15e now move toward the positive results in the cases that κ is inaccessibleand/or weakly compact. Toward these we first prove a couple of lemmas. Lemma 3.11 ( κ -uniform hulls) . Let κ be inaccessible. For true indices r ∈ V κ ,let ( P r , G r ) witness this, and otherwise let P r = G r = ∅ . Let λ = i λ withcof( λ ) > κ and V λ V . Let S ∈ V λ . Then there is X such that, letting X r = X ∩ V W r λ for r ∈ V κ , we have:1. V κ ∪ { S, κ } ⊆ X V λ and X <κ ⊆ X and | X | = κ ,2. X r ∈ W r and X r V W r λ W r ,and letting ¯ X be the transitive collapse of X and σ : ¯ X → X the uncollapseand ¯ X r , σ r likewise, then:3. ¯ X r ⊆ ¯ X and in fact, ¯ X r = W ¯ Xr ,4. σ : ¯ X → V λ is fully elementary with cr( σ ) > κ ,5. σ r : ¯ X r → V W r λ is fully elementary with cr( σ r ) > κ ,6. σ r ⊆ σ ,7. G r is ( ¯ X r , P r )-generic and ¯ X = ¯ X r [ G r ],8. M ¯ Xκ = M ¯ X r κ = T s ∈ V κ ¯ X s ; hence M ¯ Xκ ∈ M κ ,9. ¯ X <κ ⊆ ¯ X and ¯ X <κr ∩ W r ⊆ ¯ X r and ( M ¯ Xκ ) <κ ∩ M κ ⊆ M ¯ Xκ ,10. σ ↾ M ¯ Xκ = σ r ↾ M ¯ X r κ ; hence σ ↾ M ¯ Xκ ∈ M κ ,11. σ ↾ M ¯ Xκ : M ¯ Xκ → M V λ κ is fully elementary.12. V λ , X, ¯ X, X r , ¯ X r each satisfy T and the following statements:(a) “There are unboundedly many η such that η = i η ”,(b) “Fact 2.15”,(c) “There is ξ = i ξ such that for each r ∈ V κ and s ∈ V W r κ , we have W r | =“ s is true” iff V W r ξ | =“ s is true”. Proof.
The fact that V W r λ W r is by Lemma 2.18.Construct an increasing sequence h X α i α<κ such that X α V λ and V κ ∪{ x } ⊆ X α and X <κα ⊆ X α and | X α | = κ , and such that for each r ∈ V κ there arecofinally many α < κ such that X α ∩ W r ∈ W r .To construct this sequence, suppose we have constructed X α , and let r ∈ V κ .Let ¯ X = X α ∩ W r . By elementarity, X α = ¯ X [ G r ] = { τ G r (cid:12)(cid:12) τ ∈ ¯ X } . Since | ¯ X | = κ , there is some X ′ ∈ W r with | X ′ | = κ (hence W r | =“ | X ′ | = κ ”),and ¯ X ⊆ X ′ , so there is also X ′′ ∈ W r with X ′′ V W r λ and X ′ ⊆ X ′′ and | X ′′ | = κ (in V and W r ) and such that W r | =“( X ′′ ) <κ ⊆ X ′′ ”. It follows that X α ⊆ X ′′ [ G r ] = { τ G r (cid:12)(cid:12) τ ∈ X ′′ } V λ , X ′′ [ G r ] ∩ W r = X ′′ . We set X α +1 = X ′′ [ G r ]. Then everything is clear exept for the requirementthat X <κα +1 ⊆ X α +1 . So let f : γ → X α +1 where γ < κ (with f ∈ V ); we claimthat f ∈ X α +1 . Let g : γ → X ′′ be such that g ( α ) G r = f ( α ) for each α < γ .So g ∈ V , but we don’t know that g ∈ W r . But there is a P r -name ˙ g ∈ V W r λ such that ˙ g G r = g . And X ′′ ∈ W r , so there is p ∈ G r forcing that rg( ˙ g ) ⊆ X ′′ .Working in W r then, we may fix for each α < γ an antichain A α ⊆ P r maximalbelow p and for each p ∈ A α some τ αp ∈ X ′′ such that p forces that ˙ g ( α ) = τ αp .Then the sequence h τ αp i ( α,p ) ∈ I , where I = { ( α, p ) (cid:12)(cid:12) α < γ and p ∈ A α } , is ⊆ X ′′ , and hence in X ′′ . But clearly this gives a name ˙ g ′′ ∈ X ′′ such that p forces ˙ g ′′ = ˙ g , and therefore g = ˙ g G r = ˙ g ′′ G r ∈ X ′′ [ G r ] = X α +1 . But since G r ∈ X α +1 , therefore f ∈ X α +1 , so X <κα +1 ⊆ X α +1 as desired. Withthe obvious bookkeeping then, we get an appropriate sequence.Let now X = S α<κ X α . We claim that X is as desired. The only thing weneed to verify is that for each r ∈ V κ , we have X r = X ∩ W r ∈ W r . Fix r . There is a P r -name τ ∈ W r such that τ G r = h X α i α<κ , and forcofinally many α < κ there is p α ∈ G r and X rα ∈ W r such that p α τ α ∩ W r = ˇ X rα (hence X rα = X α ∩ W r ). But since P r ∈ V κ , there is therefore a fixed p ∈ P r such that p α = p for cofinally many α . But then X r = [ α ∈ I X rα where I = { α < κ (cid:12)(cid:12) ∃ x [ p τ α = ˇ x ] } . So W r has some sequence h x rα i α ∈ I suchthat p τ α = ˇ x rα for each α ∈ I . Therefore X r = [ α ∈ I X rα ! = [ α ∈ I x rα ! ∈ W r . This completes the construction. The verification of the properties listed inthe statement of the lemma is now straightforward. We omit discussing them,other than two remarks. In part 9, the third statement follows directly from thefirst two together with part 8; the first two follow readily from the construction.And in part 12, note that ξ exists because cof( λ ) > κ = | V κ | . Fact 3.12.
Let κ be weakly compact. Then X be transitive with κ ∈ X and X <κ ⊆ X . Then there is a non-principal X - κ -complete X -normal ultrafilter µ over κ such that letting Y = Ult( X, µ ) and i Xµ the ultrapower embedding, then Y is wellfounded. Moreover, i Xµ is Σ -elementary and cofinal and cr( i Xµ ) = κ . That is, κ -completeness and normality with respect to sequences in X . roof. Let π : X → Z be any elementary embedding with Z transitive andcr( π ) = κ . Let µ be the normal measure derived from π . Note that µ works.We now extend the situation above, adding the assumption that κ is weaklycompact. Lemma 3.13 ( κ -uniform weak compactness embedding) . Adopt the assump-tions and notation from the statement and proof of Lemma 3.11. Assume furtherthat κ is weakly compact. Let π : X → Y witness the weak compactness of κ in V , with Y = Ult( X, µ ) for an X - κ -complete X -normal ultrafilter µ over κ ,and π = i Xµ . For r ∈ V κ , let µ r = µ ∩ X r . Then:1. µ r ∈ W r and µ r is an X r - κ -complete ultrafilter over κ ; let Y r = Ult( X r , µ r ) and π r : X r → Y r the ultrapower map; so Y r , π r ∈ W r ,2. µ is the X -ultrafilter generated by µ r (the upward closure).3. Functions in X are represented in X r : For each f ∈ X with f : κ → X there is f r ∈ X r with f r : κ → X r and f r ( α ) = f ( α ) for µ -measure onemany α < κ .4. The ultrapowers satisfy Los’ theorem for Σ formulas, and π r , π are Σ -elementary.5. Y, Y r | = T and Y r is transitive, Y r = W Yr , and Y = Y r [ G r ].6. π r ⊆ π .7. M Yπ ( κ ) = M Y r π r ( κ ) ∈ W r ; hence this belongs to M κ .8. π ↾ M Xκ : M Xκ → M Yπ ( κ ) is cofinal Σ -elementary; this map belongs to M κ .9. M Yκ = T s ∈ V κ Y s = M Y r κ ∈ W r ; hence this belongs to M κ .10. Y, Y r each satisfy T and the following statements:(a) “There are unboundedly many η such that η = i η ”,(b) “Fact 2.15 holds at θ = π ( κ ) = i π ( κ ) ”,(c) “There is ξ = i ξ such that for each r ∈ V π ( κ ) and s ∈ V W r π ( κ ) , we have W r | =“ s is true” iff V W r ξ | =“ s is true”.Therefore there is t ∈ V Yπ ( κ ) with W Yt ⊆ M Yκ . Proof.
Part 1: Let ˙ µ be a P r -name with ˙ µ G r = µ . For each A ∈ P ( κ ) ∩ X r , thereis p A ∈ G r deciding whether A ∈ µ . We show that there is p ∈ G r decidingthis for all A ∈ P ( κ ) ∩ X r simultaneously, giving the claim. So suppose not.Working in W r , for each p ∈ P r , if there is A ∈ P ( κ ) ∩ X r such that p does notdecide whether A ∈ ˙ µ , then let A p be some such A , and otherwise set A p = κ .Let ˙ A be the name for the intersection of { A p , κ \ A p (cid:12)(cid:12) p ∈ P r } ∩ ˙ µ. P ˙ A ∈ ˙ µ . Note that h A p i p ∈ P ∈ X r .In V , let B p = A p if A p ∈ µ , and let B p = κ \ A p otherwise. Then B = h B p i p ∈ P is a < κ -sequence ⊆ X , so belongs to X . So C = \ B = ˙ A G r ∈ µ. Let ˙ C ∈ X r be a P -name for C . Working in X r , for p ∈ P let C p = { α < κ (cid:12)(cid:12) p ˇ α ∈ ˙ C p } . So C p , h C p i p ∈ P ∈ X r ⊆ X , and C = S p ∈ G r C p , so there is p ∈ G r such that C p ∈ µ . Since C p ⊆ C , note that for p ∈ P ,either C p ∩ A p = ∅ or C p ∩ ( κ \ A p ) = ∅ . Let p ∈ G r with p ≤ p be such that p ˇ C p ∈ ˙ µ . Then either:– C p ∩ A p = ∅ and p A p / ∈ ˙ µ , or– C p ∩ ( κ \ A p ) = ∅ and p κ \ A p / ∈ ˙ µ ,so p decides whether A p ∈ ˙ µ , a contradiction.Part 2: Work in W r . Let τ ∈ X r be such that P r τ r ∈ ˙ µ . Working in X r ,for p ∈ P , let C p = { α < κ (cid:12)(cid:12) p ˇ α ∈ τ } . Then since τ ∈ X r , we have C p , h C p i p ∈ P ∈ X r , and using κ -completeness likebefore, we get some p ∈ G r such that C p ∈ µ , as desired.Part 3: Work in W r . Let ˙ f ∈ X r be such that P r ˙ f : ˇ κ → ˇ X r . Workingin X r , for p ∈ P let C p = { α < κ (cid:12)(cid:12) ∃ x [ p ˙ f (ˇ α ) = ˇ x ] } . As before, there is p ∈ G r such that C p ∈ µ . But then f ↾ C p ∈ X r , whichsuffices.Part 4: Note that V λ satisfies Σ -Collection and “For all α ∈ OR, V α existsand i α ∈ OR exists, and OR = i OR ”, so X r , X do also. Therefore if ϕ is Σ and x ∈ X and X | = ∀ α < κ ∃ y ϕ ( x, y, α )then some V Xξ ∈ X satisfies the same statement, and hence there is f ∈ X pick-ing witnesses y . This gives Los’ theorem for Σ formulas. The Σ -elementarityof π : X → Y follows. Likewise for X r , π r .Parts 5, 6: The fact that Y, Y r | = T follows from Σ -elementarity andcofinality of π, π r , and (for Σ -Collection) that for each ξ ∈ OR X , we have H Xξ X and H X r ξ X r . The rest follows as usual from the fact thatfunctions in X are represented in X r (part 3), and again the Σ -elementarity of π, π r .Parts 7, 8: By uniformity of mantles, we have M Xκ = M X r κ , and by part12 of 3.11, there is ξ < OR X such that for each r ∈ V κ and s ∈ V W r κ , we have X r | =“ s is true” iff V X r ξ | =“ s is true”. Let T r = { s ∈ V W r κ (cid:12)(cid:12) W r | = “ s is true” } . T r ∈ X r and has the same definition there; likewise for T r ∈ Y r , since π r isΣ -elementary. And because of the existence of ξ , π ( T r ) = { s ∈ V Y r π r ( κ ) (cid:12)(cid:12) Y r | = “ s is true” , } and it follows (in the case of r = ∅ , but similarly in general), M Yπ ( κ ) = \ s ∈ V Yπ ( κ ) W Ys = [ ζ ∈ [ ξ, OR X ) π ( M V ζ κ ) . But M Xκ = M X r κ and π r ⊆ π , so M Yπ ( κ ) = M Y r π r ( κ ) . The calculations above alsoshow that π ↾ M Xκ : M Xκ → M Yπ ( κ ) is cofinal Σ -elementary, and likewise for π r ⊆ π .Part 9: By part 5, W Ys = Y s , so M Yκ = T s ∈ V κ Y s . And note that the densityof the grounds of X r in the grounds of X is lifted to that for those of Y r in thoseof Y . (That is, for example, if r, s are such that X r ⊆ X s , then Y r ⊆ Y s , as thisis preserved pointwise by the maps.) So M Y r κ = M Yκ , as desired.Part 10a: For each ξ ∈ X with ξ = ( i ξ ) X , we have π ( ξ ) = i Yπ ( ξ ) .Part 10c: If ξ witnesses the corresponding statement in X , note that π ( ξ )works in Y .Part 10b: We consider literally Y , but the same proof works for Y r . Notethat there is a function f : V κ → V κ with f ∈ X , such that for each R ∈ V κ , X | =“ t = f ( R ) is a true index and t witnesses Fact 2.15 for R ” ( f exists by theelementarity of σ ). We claim that π ( f ) has the same property for Y . For byΠ -elementarity, Y | =“Every t ∈ rg( π ( f )) is a true index”. Moreover, let ξ beas above (witnessing the previous statement in X ). Then for each ζ such that ξ < ζ < OR X and ζ = i Xζ , V Xζ satisfies “ W f ( R ) ⊆ W r for each r ∈ R ”. Thislifts to Y under π , and since π is cofinal, this suffices.Part 10: Apply part 10b in Y to π ( κ ) and R = V κ , giving t ∈ V Yπ ( κ ) with W Yt ⊆ M Yκ .We are now ready to prove the main theorem for weakly compact κ . Thefirst proof that, under this assumption, M κ | =“ V κ is wellordered” is due toLietz: Theorem 3.14.
Let κ be weakly compact. Then:1. M κ | = κ - DC + “ κ is weakly compact”.
2. for each A ∈ M κ ∩ H κ + , M κ | =“ A is wellordered”.
3. if P ( κ ) M κ has cardinality κ then (i) κ is measurable in M κ , and (ii) x exists for every x ∈ P ( κ ) M κ .4. If M κ | =“ µ is a countably complete ultrafilter over γ ≤ κ ”, then theultrapower Ult( M κ , µ ) is wellfounded and the ultrapower embedding i M κ µ : M κ → Ult( M κ , µ )is fully elementary. So also M κ | =“ κ + is regular and H κ + | = ZFC − ”. Note that the “ κ + ” and “ H κ + ” here are computed in V , not M κ . roof. Part 4 follows directly from part 1, as the wellfoundedness of Ult( M κ , µ )requires only ω -DC, and the proof of Los’ theorem here only uses κ -choice.The conclusion that x exists in part 3 follows easily from the rest, using theelementarity of i µ and that Ult( M κ , µ ) is wellfounded. To see that M κ | =“ κ isweakly compact”, let T ⊆ <κ M κ . Then T has a cofinal branch b in V , by weak compactness in V . But b ∩ V α ∈ M κ for each α < κ . Thereforeby 2.21, b ∈ M κ .The initial observation that M κ | =“ V κ is wellordered” was due to Lietz; hereis his direct argument. Working in M κ , let T be the tree of all attempts tobuild a wellorder of V κ . (For example, let T ⊆ <κ V κ be the set of all functions f : α → V κ where α < κ , such that for each β < α , f ( β ) is a wellorder of V β ,and for all β < β < α , f ( β ) is an end extension of f ( β ).) Since V M κ κ | = ZFC , T is unbounded in V κ , and clearly T ↾ α ∈ V κ for each α < κ . Therefore byweakly compactness in M κ , M κ has a T -cofinal branch, and clearly this givesa wellorder of V κ ∩ M κ .We proceed now to the proof that M κ | = κ - DC , and that every set A ∈ M κ ∩H κ + is wellordered in M κ . Let T ∈ M be a κ - DC -tree, and let A ∈ M κ ∩H κ + .Let S = ( T , A ) ∈ V λ and X be a κ -uniform hull, etc, with everything as inLemma 3.11, and let π : X → Y , etc, be as in Lemma 3.13. So we also have σ : X → V λ , which is fully elementary, with γ < cr( σ ). We have σ ( ¯ T ) = T and σ ( A ) = A .By 3.13, π ′ = π ↾ M Xκ : M Xκ → M Yπ ( κ ) is cofinal Σ -elementary, and thesemodels and map belong to M κ . We have A, ¯ T ∈ M Xκ .We first find a wellorder of A in M κ , by arguing as in Schindler’s proof ofFact 3.2, but using the weak compactness embedding. We have π ′ ( A ) ∈ M Yπ ( κ ) .By 3.13, there is a ground W of M Yπ ( κ ) such that M Yπ ( κ ) ⊆ W ⊆ M Yκ ∈ M κ . So W | = AC and π ′ ( A ) ∈ W . Let < ∗ ∈ W be a wellorder of π ′ ( A ). So < ∗ ∈ M κ .Working in M κ , we can therefore wellorder A by setting, for x, y ∈ A : x < A y ⇐⇒ π ′ ( x ) < ∗ π ′ ( y ) . We now find a branch through ¯ T in M κ , with length κ . Let B ∈ M Xκ bethe field of ¯ T . As above, there is a B is wellorder < ∗ of B in M κ . Working in M κ , we recursively construct a sequence h x α i α<κ constituting a branch through¯ T , using < ∗ to pick next elements, and noting that at limit stages η < κ , weget h x α i α<η ∈ M Xκ , because by 3.13 part 9 we have ( M Xκ ) <κ ∩ M κ ⊆ M Xκ . By3.11, σ ′ = σ ↾ M Xκ ∈ M κ , and note that h σ ′ ( x α ) i α<κ is a cofinal branch through T , as desired.Part 3: Now suppose P ( κ ) ∩ M κ ∈ H κ + . Then we may assume that A = P ( κ ) ∩ M κ above. Therefore π ′ : M Xκ → M Yπ ( κ ) is M κ -total. Therefore κ ismeasurable in M κ . Since M κ | = κ - DC , the rest now follows, as discussed in thefirst paragraph of the proof. The author first mistakenly thought that a similar argument worked with κ only inacces-sible, but Lietz noted that one seems to need weak compactness for this. That is, a set F of functions f such that dom( f ) < κ , with F closed under initial segment,and no maximal elements; that is, for every f ∈ F there is g ∈ F with dom( f ) < dom( g ) and f = g ↾ dom( f ). Note that κ - DC is just the assertion that for every κ - DC tree T , there is a T -maximal branch; that is, a function f / ∈ T such that f ↾ α ∈ T for all α < dom( f ). α, X )-Choice from 1.2: Theorem 3.15.
Let κ be inaccessible (so M κ | =“ κ is inaccessible”). Then:1. M κ is κ -amenably-closed.2. M κ | =“( κ, H κ )-Choice” iff M κ | =“ V κ is wellordered”.3. M | =“( < κ, H κ + )-Choice holds, and hence, ( H κ + ) <κ ⊆ H κ + ”. Remark 3.16.
Note that in part 3, the “ κ + ” and “ H κ + ” are both in the sense of M κ . Note that also, as κ is inaccessible, V M κ κ | = ZFC , M κ | =“ κ is inaccessible”,and M κ is κ -amenable closed, by Lemma 2.23. Proof.
Part 2: Since V M κ κ | = ZFC , easily V M κ κ = H M κ κ . So if M κ | =“ V κ iswellordered” then clearly M κ | =“( κ, H κ )-Choice”. For the converse, suppose M κ | =“( κ, H κ )-Choice” and in M κ , let f : κ → V M κ κ be such that f ( α ) = theset of wellorders of V α . Then any choice function for f is easily converted intoa wellorder of V κ , so we are done.Part 3: Let γ < κ and f ∈ M κ be such that f : γ → ( H κ + ) M κ . We find a choice function for f in M κ .For each x ∈ rg( f ), fix a surjection g x : κ → x with g x ∈ M κ ∩ X , and let c x ⊆ κ be the induced code for g x (so c x ∈ M κ ∩ X also).Fix λ ∈ OR and X ′ V λ a κ -uniform hull with f, h c x , g x i x ∈ rg( f ) ∈ X ′ and everything else as in 3.11. Let X be the transitive collapse of X ′ , and X r the version for r ∈ V κ , so X r is the transitive collapse of X ′ r = X ′ ∩ W r . Let σ : X → X ′ be the uncollapse map. So cr( σ ) > κ and σ ( f ) = f . Fix a club C of ¯ κ < κ such that γ < ¯ κ and V ¯ κ V κ and such that we get a correspondingsystem of structures X r ¯ κ and elementary embeddings π r ¯ κ : X r ¯ κ → X r , for r ∈ V ¯ κ , with X r ¯ κ , π r ¯ κ ∈ W r , X r, ¯ κ of cardinality ¯ κ in W r , cr( π r ¯ κ ) = ¯ κ and π r ¯ κ (¯ κ ) = κ , and each X r ¯ κ [ G r ] = X ∅ ¯ κ and π r ¯ κ ⊆ π ∅ ¯ κ . Then f, g x ∈ rg( π r ¯ κ ).Write π ∅ ¯ κ ( c ¯ κ,x , g ¯ κ,x ) = ( c x , g x ). So c ¯ κ,x = c x ∩ ¯ κ , so c ¯ κ,x , g ¯ κ,x ∈ ( H ¯ κ + ) M κ .In V (where we have AC ), pick a sequence h < ¯ κ i ¯ κ ∈ C of wellorders < ¯ κ of( H ¯ κ + ) M κ with < ¯ κ in M κ . Let z x, ¯ κ be the < ¯ κ -least element of g x, ¯ κ , and let α x, ¯ κ < ¯ κ be the least code for z x, ¯ κ with respect to the coding given by c x, ¯ κ .Let S be the stationary set of all strong limit cardinals ¯ κ ∈ C of cofinality γ + . Enumerate κ γ as { s β } β<κ , with ¯ κ γ = { s β } β< ¯ κ for each ¯ κ ∈ S . For ¯ κ ∈ S ,let β ¯ κ be the β such that s β = (cid:10) α f ( ξ ) , ¯ κ (cid:11) ξ<γ . Let S ′ ⊆ S be stationary and suchthat the ordinal β ¯ κ is constant for ¯ κ ∈ S ′ .Now let c : γ → M κ be the choice function for f given by lifting the choicesat ¯ κ ∈ S ′ pointwise with π ∅ ¯ κ . That is, c ( ξ ) = π ∅ ¯ κ ( z f ( ξ ) , ¯ κ ). Note that c isindependent of the choice of ¯ κ ∈ S ′ . For if ¯ κ , ¯ κ ∈ S ′ with ¯ κ < ¯ κ , then foreach ξ < γ and x = f ( ξ ), we have α = α x, ¯ κ = α x, ¯ κ , so π ∅ ¯ κ ( c x, ¯ κ , α ) = ( c x , α ) = π ∅ ¯ κ ( c x, ¯ κ , α ) , π ∅ ¯ κ ( z x, ¯ κ ) = π ∅ ¯ κ ( z x, ¯ κ ).But c ∈ M κ . For given r ∈ V κ , let ¯ κ ∈ S ′ with r ∈ V ¯ κ . We have f ∈ rg( π ∅ ¯ κ ) ∩ M κ ; say π ∅ ¯ κ ( ¯ f ) = f . Then ¯ f ∈ X r ¯ κ and π r ¯ κ ( ¯ f ) = f , since π r ¯ κ ⊆ π ∅ ¯ κ .But X r ¯ κ ∈ W r , so ¯ f ∈ W r . But < ∗ ¯ κ is also in W r , and so ¯ c = (cid:10) z f ( ξ )¯ κ (cid:11) ξ<γ ∈ W r .And since π r ¯ κ ⊆ π ∅ ¯ κ , π r ¯ κ also lifts ¯ c pointwise to c . Since π r ¯ κ ∈ W r , therefore c ∈ W r .So c ∈ M κ | =“ c is a choice function for f ”, so we are done. L [ A ] , M and κ -mantles In this section, we assume M exists and is fully iterable (that is, (OR , OR)-iterable), and analyze the following two related κ -mantles:– the κ -mantle of M , where κ is an M -indiscernible, and– the κ -mantle of L [ A ], where κ is an A -indscernible, for a set A of ordinalswith M ∈ L [ A ].The analysis will be a straightforward corollary of Woodin’s analysis ofHOD L [ x,G ] . For details on this, the reader should refer to [4]. We must adaptthat analysis slightly. Write I M = (cid:10) κ M α (cid:11) α ∈ OR for the increasing enumerationof the (Silver) indiscernibles of M , and similarly I P for normal non-droppingiterates P of M . Write I x = h κ xα i α ∈ OR for those of L [ x ]. If N is M -like, write δ N for the unique Woodin cardinal of N . Given two normal, non-dropping it-erates P, Q of M such that Q is a normal iterate of P , let i P Q : P → Q be theiteration map. Then:– P = Hull P ( δ P ∪ I P ),– i P Q is elementary, so i P Q ( δ P ) = δ Q ,– I Q = i P Q “ I Q Let κ = κ M ξ . Then F = F M κ denotes the “set” of maximal pseudo-iteratesof M via trees in M | κ . Given P, Q ∈ F , we write P ≤ Q iff Q is a normaliterate of P . Then ≤ is a directed partial order (literally using [11]), and h P, Q, i
P Q i P ≤ Q ∈ F forms a directed system. Let M ∞ be its direct limit and i P ∞ : P → M ∞ the direct limit map, for P ∈ F . Then M ∞ is an iterate of M , and in fact by[11], a normal iterate. For α ∈ OR define α ∗ = min P ∈ F i P ∞ ( α ) . Then M ∞ and the map α α ∗ are definable over M , uniformly in the param-eter κ .Let N = M ∞ . We now define M N ∞ , etc, analogously, using pseudo-iteratesof N via maximal trees in N | κ ∗ . Then M N ∞ is a normal iterate of N . Let k : N → M N ∞ be the iteration map. 23or α ∈ OR, we say that P is α -stable iff i P Q ( α ) = α for all Q ∈ F κ with Q ≥ P . We write c F κ for the set of those P ∈ F κ which are < κ -grounds of M .We have:1. k ( α ) = α ∗ for all α ∈ OR.2. c F κ is dense in F κ , and also in the < κ -grounds of M .3. For each α ∈ OR, there is P ∈ c F κ which is α -stable (hence all Q ∈ F κ with Q ≥ P are α -stable).4. M is κ M α -stable for each α ≥ ξ (basically via the proof in [10] or [7]), so I P = I for each P ∈ F κ .5. M ∞ [ ∗ ] = L [ A ] for a set of ordinals A , hence models ZFC .6. Woodin’s analysis of HOD L [ x,G ] (see [4]) adapts to show thatHOD M [ G ] = M ∞ [ ∗ ] = M ∞ [ b ] , where G is ( M , Col( ω, < κ ))-generic and b is the wellfounded cofinalbranch through the normal tree T on N = M ∞ with last model M N ∞ .7. M ∞ [ b ] is a fully iterable strategy mouse modelling ZFC .We will now establish a new characterization of M ∞ [ ∗ ]: Theorem 4.1.
Assume that M exists and is fully iterable; that is, (OR , OR)-iterable. Then M M κ is a fully iterable strategy mouse which models ZFC . Infact, in the notation above, M M κ = M ∞ [ b ]. Proof.
We may assume that κ = κ M , by indiscernibility (and the statement isfirst-order about κ , since M ∞ [ ∗ ] is defined uniformly over M from κ ).We first show that M ∞ [ ∗ ] ⊆ M M κ , a fact which is not new.We know the points P ∈ c F κ are dense in the < κ -grounds of M . Moreover,each such P computes M ∞ [ ∗ ] in the same manner as does M . So M ∞ [ ∗ ] ⊆ \ P ∈ b F κ = M M κ , as desired.We now proceed to the converse, that M M κ ⊆ M ∞ [ ∗ ].We first show that P ( < OR) ∩ M κ ⊆ M ∞ [ ∗ ]. So let X ⊆ α ∈ OR with X ∈ M M κ . Let j : M → M be an embedding with cr( j ) = κ . Then j “ I ⊆ I .Let G be ( M , Col( ω, < κ )))-generic. Then j ( X ) ∈ M M j ( κ ) ⊆ HOD M [ G ] ;the “ ∈ ” is by elementarity, and the “ ⊆ ” is because HOD M [ G ] is a ground for M via Vopenka, a forcing of size < j ( κ ) (one can compute a bound on the sizedirectly, or just observe that it has size < j ( κ ) because j ( κ ) ∈ I and HOD M [ G ] is defined over M from the parameter κ ).24o we can fix a formula ϕ and η ∈ OR such that for α ∈ OR, we have α ∈ j ( X ) ⇐⇒ M | = Col( ω, < κ ) ϕ ( η, α ) , so for all P ∈ c F κ , α ∈ j ( X ) ⇐⇒ P | = Col( ω, < κ ) ϕ ( η, α ) . (2)Fix P ∈ c F κ which is η -stable ( P is also κ -stable by Fact 4) above. Claim 9. i P Q ( j ( X )) = j ( X ) for all Q ∈ c F κ with Q ≥ P . Proof.
Since i P Q ( κ, η ) = ( κ, η ), this follows from line (2) applied to each of P and Q . Claim 10. j ◦ i P Q = i P Q ◦ j . Proof.
We have δ P ≤ δ Q < κ = cr( j ). Also, P = L [ P | δ P ] and Q = L [ Q | δ Q ]. So j ↾ P : P → P and j ↾ Q : Q → Q are elementary.Now P = Hull P ( δ P ∪ I ). So it suffices to see that the claimed commutativityholds for all elements of δ P ∪ I .Given ξ < δ P , since δ P ≤ δ Q = i P Q ( δ P ) < κ = cr( j ), we have j ( i P Q ( ξ )) = i P Q ( ξ ) = i P Q ( j ( ξ )) , as desired. Now let µ ∈ I . Since j “ I ⊆ I and by Fact 4 above, i P Q ↾ I = id,so j ( i P Q ( µ )) = j ( µ ) = i P Q ( j ( µ )) , completing the proof. Claim 11. i P Q ( X ) = X for all Q ∈ c F κ with Q ≥ P . Proof.
Let Y = i P Q ( X ). By Claims 9 and 10, we have j ( Y ) = j ( i P Q ( X )) = i P Q ( j ( X )) = j ( X ) , but j is injective, so Y = X as desired.The fact that X ∈ M ∞ [ ∗ ] follows from the previous claim via the followingstandard calculation. Let X ∗ = i P ∞ ( X ) ∈ M ∞ . Then X ∗ = i Q ∞ ( X ) for all Q ∈ c F κ with Q ≥ P , since i P Q ( X ) = X . Let α ∈ OR. By taking Q as aboveand also α -stable, it follows that α ∈ X ⇐⇒ Q | = “ α ∈ X ” ⇐⇒ M ∞ | = “ α ∗ ∈ X ∗ , since i Q ∞ is elementary and i Q ∞ ( α, X ) = ( α ∗ , X ∗ ). Note that the last statementis independent of Q . And since X ∗ and ∗ ↾ sup( X ) are both in M ∞ [ ∗ ], therefore X ∈ M ∞ [ ∗ ].Now we know M ∞ [ ∗ ] | = ZFC , and have shown M ∞ [ ∗ ] ⊆ M κ and P ( < OR) ∩ M κ ⊆ M ∞ [ ∗ ] .
25t follows that M κ ⊆ M ∞ [ ∗ ]. For suppose not, and let η ∈ OR be largest suchthat V M κ η = V M ∞ [ ∗ ] η . Therefore V M κ η is coded by a set X of ordinals in M ∞ [ ∗ ].But M ∞ [ ∗ ] ⊆ M κ , so X ∈ M κ . It follows that every Y ∈ V M κ η +1 is coded by aset X Y ∈ M κ of ordinals. Hence X Y ∈ M ∞ [ ∗ ]. But then V M κ η +1 = V M ∞ [ ∗ ] η +1 , acontradiction, completing the proof.We can now deduce: Theorem 4.2.
Assume that M exists and is fully iterable; that is, (OR , OR)-iterable. Let A be a set of ordinals with M ∈ L [ A ]. (Then A exists.) Let κ be an A -indiscernible. Then M L [ A ] κ is a fully iterable strategy mouse whichmodels ZFC . In fact, M L [ A ] κ = M ∞ [ b ], where M ∞ is a certain iterate of M , and b is the cofinal branch through a certain normal tree T on M ∞ , with T ∈ M ∞ . Proof.
Here M ∞ is the direct limit of all pseudo-iterates of M via maximaltrees in L κ [ A ]. Let κ ∗ = i M M ∞ ( κ ), let N = M ∞ and then define M N ∞ asbefore, via the directed system generated by maximal trees in M ∞ | κ ∗ . We set b to be the correct branch through the normal tree leading from N = M ∞ to M N ∞ .So we claim that M L [ A ] κ = M ∞ [ b ]. This is a direct corollary of Theorem4.1. For we have M ∈ L [ A ], so M ∈ L γ [ A ] for some γ < κ . Let γ be leastsuch. Then working in L [ A ], we can form a genericity iteration of M , making A generic, leading to a pseudo-iterate P of M with δ P = ( γ + ) L [ A ] and P aground of L [ A ] via a forcing of size δ P < κ .Now calculations as in the proof of Fact 4 above give that I P = I A . Inparticular, κ ∈ I P . Let i M P (¯ κ ) = κ . Then M L [ A ] κ = M Pκ = i M P ( M M ¯ κ ) , which by Theorem 4.1 gives M L [ A ] κ = M P ∞ [ b P ] = i M P ( M M ∞ [ b M ]) , where M M ∞ , b M are the model/branch determined in M at ¯ κ ∈ I M .But standard calculations give that M P ∞ = M ∞ and hence b P = b is theunique wellfounded branch through the specified tree.Finally, the iterability of M P ∞ [ b P ] is a standard fact. This proves the theorem. References [1] Joel David Hamkins Gunter Fuchs and Jonas Reitz. Set theoretic geology.
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