aa r X i v : . [ m a t h . L O ] F e b Full normalization for transfinite stacks
Farmer Schlutzenberg ∗ [email protected] 8, 2021 Abstract
We describe the extension of normal iteration strategies with appro-priate condensation properties to strategies for stacks of normal trees,with full normalization. Given a regular uncountable cardinal Ω and an( m, Ω+1)-iteration strategy Σ for a premouse M , such that Σ and M bothhave appropriate condensation properties, we extend Σ to a strategy Σ ∗ for the ( m, Ω , Ω + 1) ∗ -iteration game such that for all λ < Ω and all stacks ~ T = hT α i α<λ via Σ ∗ , consisting of normal trees T α , each of length < Ω,there is a corresponding normal tree X via Σ with M ~ T∞ = M X∞ , along withagreement of iteration maps, when there are no drops in model or degreeon main branches. We also use the methods to analyze the comparison ofmultiple iterates via such a common strategy. The theory of normalization of iteration trees has been developed in the lastfew years by Steel [8], [10], the author [4], Jensen [1] and Siskind. This has builton results of various others in this direction, as discussed more in [8] and [4].In embedding normalization or normal realization , a stack ~ T of normal trees isrealized into a single normal tree X , with a key outcome that we obtain a finalrealization map π : M ~ T∞ → M X∞ , but it is allowed that M ~ T∞ = M X∞ . In full normalization (which we also call just normalization here), it is also demanded that M ~ T∞ = M X∞ and π = id.The main aim of this article is to describe the analogue of the main results of[4] for full normalization. That is, we start with an ( m, Ω + 1)-iteration strategy ∗ Teilweise gef¨ordert durch die Deutsche Forschungsgemeinschaft (DFG) im Rahmen derExzellenzstrategie des Bundes und der L¨ander EXC 2044–390685587, Mathematik M¨unster:Dynamik–Geometrie–Struktur This is a draft of the planned paper. It contains a complete proof of the theorems stated,but some further results are to be added, some related to other results in [4], and some furtherthings. Some parts that are currently here might be changed a little. m -sound premouse M , where Ω is an uncountableregular cardinal, such that Σ has certain condensation properties, and extendit naturally to an ( m, Ω , Ω + 1) ∗ -strategy (for stacks of normal trees) with ap-propriate full normalization properties.The results in this paper hold for fine structural mice M , either of the pure L [ E ] or strategy variety (such as in [8]), with either MS- or λ -indexing, togetherwith iteration strategies Σ for corresponding iteration rules (though see [2]),assuming that M and Σ satisfy the appropriate condensation properties. Sowhenever we say premouse without specifying a restriction on the kind, we meanany of these. Some results are more specific to pure L [ E ]-mice (when the resultinvolves a traditional comparison argument), though in those cases the methodsof [8], and ongoing work of Steel and Siskind, should help with generalization.And some are specific to MS-indexing, though these would presumably adaptto λ -indexing, by adapting the results of [7] and [3] in this manner.The main result to be shown is the following. It is the result of a combinationof work of John Steel and the author. The requirement that Σ have minimalinflation condensation (Definition 3.28) is a variant of very strong hull conden-sation from [8], and of inflation condensation from [4] (the latter adapted tofull normalization, just as very strong hull condensation adapts strong hull con-densation ). The prefix optimal prevents player I from making artificial drops(see [4, § m -standard (for m ≤ ω , Definition 2.1) just requires m -soundness plus some standard condensation facts which for pure L [ E ]-micefollow from ( m, ω , ω + 1) ∗ -iterability (in fact from ( m, ω + 1)-iterability forMS-indexing). Let Ω > ω be regular. Let M be an m -standard premouse. Let Σ be an ( m, Ω + 1) -strategy for M with minimal inflation condensation. Thenthere is an optimal- ( m, Ω , Ω + 1) ∗ -strategy Σ ∗ for M such that:1. Σ ⊆ Σ ∗ ,2. for every stack ~ T = hT α i α<λ via Σ ∗ with a last model and with λ < Ω ,there is an m -maximal successor length tree X on M such that:(a) X ↾ Ω + 1 is via Σ ,(b) if lh( T α ) < Ω for all α < λ then lh( X ) < Ω (and hence X is via Σ inthis case),(c) M ~ T∞ = M X∞ and deg ~ T∞ = deg X∞ ,(d) b ~ T drops in model (degree) iff b X does,(e) if b ~ T does not drop in model or degree then i ~ T ∞ = i X ∞ ,3. Σ ∗ is ∆ ( { Σ } ) , uniformly in Σ ,4. if card( M ) < Ω then Σ ∗ ↾ H Ω is ∆ H Ω (Σ ↾ H Ω ) , uniformly in Σ . So if Ω = ω then Σ ∗ ↾ HC is ∆
HC1 (Σ ↾ HC).2n part 2, note that X ↾ (Ω + 1) is uniquely determined by the conditionsmentioned. If lh( T α ) = Ω + 1 then λ = α + 1 and player II wins, by the rules ofthe ( m, Ω , Ω + 1) ∗ -iteration game. We can’t demand that X be fully “via Σ” inthis case, as it can be that lh( X ) > Ω + ω .A direct consequence of the theorem above, together with Theorem 3.34(which only applies to pure L [ E ]-premice), is for example: Assume that M = M or M = M ω is fully iterable, and M = M or M = M ω respectively. Let δ be the least Woodin of M . Let M ∞ be the direct limit of non-dropping countable iterates of M via trees based on M | δ . Then M ∞ is a normal iterate of M via its unique strategy. Proof.
The strategy Σ for M has Dodd-Jensen. By Theorem 3.34, Σ thereforehas minimal inflation condensation, and hence the theorem applies, as witnessedby Σ ∗ . Moreover, if Σ ′ is any ( ω, ω , ω + 1) ∗ -strategy for M , then Σ ′ agrees withΣ ∗ on non-dropping iterates, by a standard uniqueness argument. This yieldsthe corollary. (cid:3) Of course a similar argument works for many mice M . For the followingcorollary, the author does not know whether DC is necessary, but see [4, § Assume DC . Let Ω > ω be regular. Let M be a countable, m -sound, ( m, Ω , Ω + 1) ∗ -iterable pure L [ E ] -premouse. Then there is an ( m, Ω , Ω +1) ∗ -strategy Σ ∗ for M , with first round Σ , such that Σ , Σ ∗ are related as inTheorem 1.1. Proof.
We may take an ( m, Ω , Ω + 1) ∗ -strategy Γ for M with weak DJ (usingDC). By ( m, Ω , Ω + 1) ∗ -iterability (and that M is pure L [ E ]), M is m -standard.By Theorem 3.34, the first round Σ of Γ has minimal inflation condensation,and so the theorem applies to Σ. (cid:3) The key observation beyond the methods of embedding normalization, whichleads from there to full normalization, is due to Steel, and was described by himin preprints of [8] in 2015. In embedding normalization, given an iteration map π : M → N between premice M, M ′ and E in the extender sequence E M + of M , if one wants to copy E using π , then one copies to π ( E ) (or to F N , if E = F M ). But for full normalization, this is not the appropriate copy. Let P E M be such that E = F P . Then the appropriate copy is the active extender E ′ of Ult ( P, F ), where F is the ( δ, π ( δ ))-extender derived from π , and δ is thelargest cardinal of P . For example, if Q is a premouse and E is a Q -extenderand cr( E ) < ρ Qm and R = Ult m ( Q, E ) , and F is the active extender of a premouse, and is an R -extender with cr( E ) < cr( F ) < ν ( E ), then Ult m ( R, F ) = Ult m ( Q, E ′ ), where E ′ = F P ′ is as above.(See e.g. [5, 3.13–3.20] or [7, § E ′ ∈ E M ′ + . Steel showed that the latter is true, assuming that M satisfies some standard condensation facts.3s partially described in an early draft of [8], Steel used this copying processto introduce a full normalization analogue of tree embeddings ( weak tree em-beddings, which we call minimal here, because of the minimality of the degree0 ultrapowers involved), and corresponding adaptation of strong hull condensa-tion, very strong hull condensation , and also a procedure for full normalizationof finite stacks adapting that for embedding normalization. This material wasnot discussed in full detail there, however. Some time later, some details in-volved were ironed out independently by Steel and the author (see Footnote 7,Remark 3.13), and soon after this, in 2016, Steel presented details of his workon the topic in the handwritten notes [10].In fact, most of the ideas needed in the proof of Theorem 1.1 (thus, incorpo-rating infinite stacks) are also present in the papers [8] and [4]; those methodscombined with a bit more analysis is enough. But it does take some work to setthings up, so that the further analysis can be carried out.Conveniently, many complications which arise in normal realization (embed-ding normalization) are eliminated when dealing with (full) normalization.The structure of the proof of Theorem 1.1 we give, and much of the detail,is very similar to that of [4, Theorem 9.1], and where possible, we will omitdetails of proofs which are (essentially) the same. So the reader should have[4] available. If the reader is not familiar with that paper, it seems one mightjust consult it as needed (and as mentioned, certain complications in that paperdo not arise here). The notation employed by the author in [4] is naturallydifferent from that employed by Steel in [8], although many notions match upin meaning. Because this paper is very tightly related to [4], and in order tomake things easier on the reader, we opted to maintain consistency with thenotation of [4] as far as possible. We will give most key definitions in full (butnot all definitions), even though some of these are very similar to those in [4]. We also give an analysis of comparison of normal iterates N , N of a givenmouse M via a strategy Σ with minimal inflation condensation, via the strategiesΣ N , Σ N for N , N given by (the proof of) Theorem 1.1. During the last couple of years, Steel and Siskind have also been developingon a paper with work on full normalization, but focusing on strategy mice andextending Steel’s [8]. ***To do: add some translation between Steel’s notation and the notation here. The work on the material in this paper began in 2015 or early 2016, some time afterSteel and the author had communicated with one another on their respective work in [8]and [4], and Steel suggested considering full normalization for infinite stacks. After this weboth considered the problem, basically independently. By the time of the 2016 UC Irvineconference in inner model theory, the author had sorted out pretty much the proof of fullnormalization presented in this paper. The method for dealing with infinite stacks here reliesheavily on the method presented in [4], and of course our goal here is to extend a given normalstrategy to a strategy for stacks. Steel has considered infinite stacks via a strategy induced fora (strategy) premouse constructed by background construction in a universe for which thereis a sufficiently nice coarse strategy. He has (tentative?) results on normalization for infinitestacks in that context, which may rely somewhat on dealing with the complications discussedin [9] and its successors. This analysis of comparison was observed by the author in early 2016, in conjunction withdiscussions with Steel regarding HOD L [ x ] . .1 Notation See [4, § § E ) for the index of an extender E ∈ E M + , for a premouse M , which is denoted ind( E ) in [4].) We just mention below a few of the moreobscure terminological items that show up in the paper.We deal with both MS-indexed and λ -indexed pure/strategy premice. Weuse MS-fine structure (that is, rΣ n , etc), also for λ -indexed premice. We usesome definitions/facts from [7], which are literally stated there for MS-indexedpure- L [ E ] premice, but as long as there are direct translations to other forms,we assume such a translation. See in particular [7, §
2] for the notion n -liftingembedding .For an active premouse N , lgcd( N ) denotes the largest cardinal of N , and e ν ( F N ) = e ν ( N ) denotes max(lgcd( N ) , ν ( F N )). For a passive premouse N , e ν ( N )denotes OR N . We write M ⊳ card N to say that M ⊳ N and OR M is a cardinalof N .Let T be an m -maximal tree. If α +1 < lh( T ), then ex T α denotes M T α | lh( E T α )( ex for exit ) and e ν T α = e ν ( E T α ) = e ν (ex T α ), so note e ν T α is the exchange ordinalassociated to E T α in T (for either MS-iteration rules or λ -iteration rules). If T has successor length α +1, we say E ∈ E + ( M T α ) is T -normal iff lh( E T β ) ≤ lh( E T α )for all β < α . We say T is terminally non-dropping iff it has successor length α + 1 and [0 , α ] T does not drop in model or degree. A putative m -maximal tree T is like an m -maximal tree, except that if lh( T ) = α +1 then we do not demandthat [0 , α ) T has only finitely many drops (and hence M T α might be ill-defined),and if [0 , α ) T does have only finitely many drops, we do not demand that M T α is wellfounded. ω and degree Let M be an ω -sound premouse. Then there is a natural 1-1 correspondencebetween ω -maximal iteration trees T on M and 0-maximal iteration trees U on J ( M ), such that for all corresponding pairs ( T , U ), we have lh( T ) = lh( U ), < T = < U , E T α = E U α for each α + 1 < lh( T , U ), and for each α < lh( T , U ), wehave:– [0 , α ] T ∩ D T deg = ∅ ⇐⇒ [0 , α ] U ∩ D U = ∅ ,– if [0 , α ] T ∩ D T deg = ∅ then M U α = J ( M T α ) and i T α ⊆ i U α and i U α ( M ) = M T α ,– if [0 , α ] T ∩ D T deg = ∅ then M T α = M U α and there is the natural agreementof iteration maps.This is straightforward to see. Likewise for ω -maximal stacks on M and 0-maximal stacks on J ( M ). In fact, such a correspondence holds not only for5uch iteration trees, but also for abstract iterations via sequences ~E of extendersconsidered in what follows.So throughout the paper, for a little more uniformity, we will ignore iterationtrees and strategies at degree ω for ω -sound premice M , by instead consideringthe corresponding degree 0 trees and strategies for J ( M ), assuming M is a set;if M is proper class, then of course degree n for M is equivalent for all n ≤ ω ,so in this case we just consider degree 0 for M . Just as in [7, Definition 2.10], we abstract out some condensation we need toassume holds of the base premouse M we will be iterating: Let m < ω and let M be an ( m + 1)-sound premouse. We saythat M is ( m + 1) -relevantly-condensing iff for all P, π , if1. P is an ( m + 1)-sound premouse,2. ρ Pm +1 is an M -cardinal,3. π : P → M is a ~p m +1 -preserving m -lifting embedding,4. cr( π ) ≥ ρ Pm +1 and5. π “ ρ Pm is bounded in ρ Mm then P ⊳ M .Say that M is ( m + 1) -sub-condensing iff for all π : P → M as above, exceptthat we replace conditions 3 and 5 respectively with3’. π : P → M is a ~p m +1 -preserving m -embedding,5’. ρ Pm +1 < ρ Mm +1 ,then P ⊳ M .For n < ω , a premouse N is n -standard iff:– N is n -sound and ( m + 1)-relevantly-condensing for every m < n , and– every M ⊳ N is ( m + 1)-relevantly-condensing and ( m + 1)-sub-condensingfor each m < ω . And N is ω -standard iff n -standard for each n < ω . ⊣ If N is an n -sound, ( n, ω + 1)-iterable MS-indexed pure L [ E ]-premouse, then N is n -standard, by [3]. The author expects this should alsowork for λ -indexing, but has not attempted to work through the details. For See 2.2. By § ω -standard in the main calculations; it is only includedas it is used in the statement of some theorems. m -standard explicitly as one of the hypotheses of the main theorems.Like in [7, § -standard is an rΠ property of premice, and ( m + 1) -standard is an rΠ m +1 ( ~p m +1 ) property over ( m + 1)-sound premice N ; therefore,in general, n -standardness is preserved by degree n ultrapower maps.Let π : P → M be as in the definition of ( m + 1) -sub-condensing , exceptthat we drop requirement that ρ Pm +1 < ρ Mm +1 . As also discussed in [7], if alsocr( π ) ≥ ρ Mm +1 , then P = M and π = id; this just follows directly from finestructure. Let M be an m -sound premouse. Let ~E = h E α i α<λ be asequence of short extenders. We say that ~E is ( M, m ) -pre-good iff there is asequence h M α i α ≤ λ such that:– M = M ,– for each α < λ , E α is a weakly amenable M α -extender withcr( E α ) < min( ρ M α m , e ν ( M α )) , – for each α < λ , M α +1 = Ult m ( M α , E α ),– for each limit γ ≤ λ , M γ = dirlim α<γ M α , under the (compositions anddirect limits of) the ultrapower maps,– for each α < λ , M α is wellfounded.We write Ult m ( M, ~E ) = M λ and i M,m~E for the ultrapower map. We say that ~E is ( M, m ) -good iff ~E is ( M, m )-pre-good and M λ is wellfounded. If ~E is ( M, m )-pre-good, given κ ≤ ρ Mm , we say that ~E is < κ -bounded iff cr( E α ) < sup i M,m~E ↾ α “ κ for each α < λ ; and if κ < ρ Mm , say ~E is κ -bounded iff it is < ( κ + 1)-bounded.We say ~E is ( M, m ) -pre-pre-good iff either ~E is ( M, m )-pre-good or there is γ < lh( ~E ) such that ~E ↾ γ is ( M, m )-pre-good but not (
M, m )-good. ⊣ As a corollary to 2.2, we have:
Let N be n -standard and ~E be ( N, n ) -good. Then Ult n ( N, ~E ) is n -standard. The following fact was established in the proof of [7, Lemma 3.17(11)], butthat proof is within a context which makes it a little annoying to isolate, so werepeat the proof here for convenience:
Let N be n -standard and m < n < ω with ρ Nm +1 = ρ Nn . Let ~E be an ( N, n ) -good sequence. Then ~E is ( N, m ) -good. Let U k = Ult k ( N, ~E ) for k ∈ { m, n } . Then U m E U n and ρ U m m +1 = ρ U n n . Proof.
The fact that ~E is ( N, m )-good will follow by induction on lh( ~E ) fromthe rest. So assume this holds.Let i k : M → U k be the ultrapower map and π : U m → U n be the standardfactor map. By 2.4 and calculations as in [6, Corollary 2.24***] and [7], we have7 U n is n -standard and U m is ( m + 1)-sound,– ρ = def ρ U m m +1 = sup i m “ ρ Nm +1 = sup i n “ ρ Nn = ρ U n n ,– if ρ < OR U n then ρ is a U n -cardinal (using that if ρ Mm +1 = ( θ + ) M thenthere is no cofinal rΣ f Mm +1 function f : θ → ρ Mm +1 ),– ~p U k m +1 = i k ( ~p Nm +1 ) for k ∈ { m, n } , and– π is ~p m +1 -preserving m -lifting, with cr( π ) ≥ ρ .So by 2.2 applied to π : U m → U n (noting U n is n -standard) either:– U m ⊳ U n (when sup π “ ρ U m m < ρ U n m ), or– U m = U n (when sup π “ ρ U m m = ρ U n m ),completing the proof. (cid:3) Let N be an n -sound premouse and ( M, m ) E ( N, n ), where m < ω . The extended (( N, n ) , ( M, m )) -dropdown is the sequence h ( M i , m i ) i i ≤ k ,with k as large as possible, where ( M , m ) = ( M, m ), and ( M i +1 , m i +1 ) is theleast ( M ′ , m ′ ) E ( N, n ) such that either– ( M ′ , m ′ ) = ( N, n ), or– ( M i , m i ) ⊳ ( M ′ , m ′ ) and ρ M ′ m ′ +1 < ρ M i m i +1 .The reverse extended (( N, n ) , ( M, m )) -dropdown is h ( M k − i , m k − i ) i i ≤ k .Abbreviate reverse extended with revex and dropdown with dd . ⊣ Steel proved the following dropdown-preservation lemma (for λ -indexing); asmall part of it is independently due to the author: Let N be n -standard and ( M, m ) E ( N, n ) . Let h ( M i , m i ) i i ≤ k be the extended (( N, n ) , ( M, m )) -dropdown. Let ~E = h E α i α<λ be a sequencewhich is ( M i , m i ) -good for each i ≤ k . Let U i = Ult m i ( M i , ~E ) and M ′ = U and N ′ = U k . Then (a) for each i < k , we have ( M ′ , m ) E ( U i , m i ) ⊳ ( U i +1 , m i +1 ) E ( N ′ , n ) , and in fact, (b) the extended (( N ′ , n ) , ( M ′ , m )) -dropdown is h ( U i , m i ) i i ≤ k . Proof. If k = 0 it is trivial so suppose k > i < k . It easily suffices toprove the following: Steel first showed (a) in a 2015 preprint of [8]. He and the author then noticed indepen-dently that it is also important for full normalization to know how the dropdown sequenceis propagated by the ultrapower, and extended (a) to (b). Steel’s formulation and proof ofthis in [10, Lemma 2.4] is somewhat different to the author’s (which is given here, but whichfollows readily from an examination of Steel’s original proof of (a)). The proof for almost the same fact was given in [3, Lemma 10.3], but we include the proofhere also for self-containment, and since the fact is very central to our purposes.
8. ( U i , m i ) ⊳ ( U i +1 , m i +1 ).2. The extended (( U i +1 , m i +1 ) , ( U i , m i ))-dropdown is h ( U i , m i ) , ( U i +1 , m i +1 ) i ,3. If i + 1 < k (so ρ M i +1 m i +1 +1 < ρ M i m i +1 ) then ρ U i +1 m i +1 +1 < ρ U i m i +1 .4. If k > M k − = N and n > m k − and ρ Nn = ρ Nm k − +1 and ρ N ′ n = ρ U k − m k − +1 ,or(b) M k − ⊳ N and ρ Nn = ρ M k − m k − +1 = ρ M k − ω and ρ N ′ n = ρ U k − m k − +1 = ρ U k − ω ,or(c) M k − ⊳ N and ρ Nn > ρ M k − m k − +1 = ρ M k − ω and ρ N ′ n > ρ U k − m k − +1 = ρ U k − ω .We just give the proof assuming that λ = lh( ~E ) = 1; the general case isthen a straightforward induction on lh( ~E ). Let E = E and κ = cr( E ). Notethat for each i < k , we have κ < ρ M i m i +1 and ( κ + ) N = ( κ + ) M i , and also κ < ρ Nn .Write R = M i , r = m i , S = M i +1 , s = m i +1 , R ′ = U i , and S ′ = U i +1 . So( R, r ) ⊳ ( S, s ).To start with we prove parts 1–3 assuming that i + 1 < k . Case . R ⊳ S and if we are using MS-indexing then
R ⊳ S sq .Let ρ = ρ Rr +1 . So ρ = ρ Rω is a cardinal of S and ρ Ss +1 < ρ ≤ ρ Ss . Therefore thefunctions [ κ ] <ω → α , for α < ρ , which are used in forming R ′ = Ult r ( R, E ), areexactly those used in forming S ′ = Ult s ( S, E ). Let i R : R → R ′ and i S : S → S ′ be the ultrapower maps and π : R ′ → i S ( R ) ⊳ S ′ the natural factor map. Thenlike in the proof of Lemma 2.5, ρ R ′ r +1 = sup i R “ ρ = sup i S “ ρ ≤ cr( π )and ρ R ′ r +1 < sup i S “ ρ Ss = ρ S ′ s . Moreover, ρ R ′ r +1 is a cardinal of S ′ , because ρ isa cardinal of S , and if ρ = ( γ + ) S then ρ is regular in S . Note that either π satisfies the requirements for ( r + 1)-relevant (if π “ ρ Rr is bounded in ρ i S ( R ) r ), orfor ( r + 1)-sub-condensing (if π “ ρ Rr is unbounded in ρ i S ( R ) r but ρ Rr +1 < ρ i S ( R ) r +1 ),or R ′ = i S ( R ) and π = id (if π “ ρ Rr is unbounded in ρ i S ( R ) r and ρ Rr +1 = ρ i S ( R ) r +1 ).But S ′ is 0-standard by 2.4, so R ′ E i S ( R ) ⊳ S ′ . Since also ρ Ss +1 < ρ and ρ S ′ s +1 = sup i S “ ρ Ss +1 , we have ρ S ′ s +1 < ρ R ′ r +1 ≤ ρ S ′ s . So parts 1–3 for this casefollow. Case . R ⊳ S but we are using MS-indexing and R ⋪ S sq . Argue as in the previous case, replacing i S ( R ) (which is not defined asdom( i S ) = S sq ) with b i S ( R ), noting that ρ R ′ r +1 ≤ ν ( F S ′ ), so R ′ ⊳ S ′ . Here b i S : Ult( S, F S ) → Ult( S ′ , F S ′ )is the map induced by i S via the Shift Lemma. This case is a variant of an observation due to the author from a separate context. ase . R = S .So r < s , and note that ρ Ss +1 < ρ Ss = ρ Sr +1 . Lemmas 2.5 gives that R ′ E S ′ ,and note that ρ U s s +1 < ρ U s s = ρ U r r +1 = def ρ ′ . (1)Note that h ( U r , r ) , ( U s , s ) i is the extended dropdown of (( U s , s ) , ( U r , r )): For if U r = U s this follows from line (1) above; if U r ⊳ U s it is by line (1) and because ρ ′ = ρ U r r +1 = ρ U r ω is a cardinal of U s (if ρ Ss = ( γ + ) S then ρ Ss is rΣ f Ss -regular).This completes the proof of parts 1–3 assuming that i + 1 < k .Now suppose that k >
0. Suppose M k − ⊳ N . Then ρ = def ρ M k − m k − +1 = ρ M k − ω is an N -cardinal. We have ρ Nn ≥ ρ , because if ρ Nn < ρ then n >
0, and letting n ′ be least such that ρ Nn ′ +1 < ρ , then ( N, n ′ ) should have been in the dropdownsequence, a contradiction.Suppose instead that M k − = N . Then by definition, m k − < n , so ρ Nn ≤ ρ Nm k − +1 . But if ρ Nn < ρ Nm k − +1 then again, there should have been anotherelement in the dropdown sequence. So ρ Nn = ρ Nm k − +1 .Using these observations, one proceeds as before to establish parts 1, 2 and4 for the case that i + 1 = k . (cid:3) Note that in the context above, if M i ⊳ M i +1 then U i ⊳ U i +1 , butit is possible that M i = M i +1 and U i ⊳ U i +1 . Essentially the following lemma was shown in [7, § § (Extender commutativity) . See Figure 1. Let M be m -sound and P be active. Let G = F P and κ = cr( G ) . Suppose M || ( κ + ) M = P | ( κ + ) P and κ < ρ Mm . Suppose either ( κ + ) M < OR M or M is active and κ < e ν ( M ) . Let U = Ult m ( M, G ) and suppose U is wellfounded.Let ~E be ( M, m ) -good, ( P, -good and κ -bounded. Let M ⊛ = Ult m ( M, ~E ) and P ⊛ = Ult ( P, ~E ) and G ⊛ = F P ⊛ , so κ ⊛ = def i M,m~E ( κ ) = i P, ~E ( κ ) = cr( G ⊛ ) < min( ρ M ⊛ m , ρ P ⊛ ) . If ~E is ( U, m ) -pre-good, also let U ⊛ = Ult m ( U, ~E ) .Let ~F be ( P ⊛ , -good with κ ⊛ < cr( ~F ) . Let ~D = ~E b ~F . Let P ⊛ = Ult ( P, ~E b ~F ) and G ⊛ = F P ⊛ . Let κ ⊛ = cr( G ⊛ ) = cr( G ⊛ ) = κ ⊛ .If ~E b ~F is ( U, m ) -pre-good, also let U ⊛ = Ult m ( U, ~E b ~F ) = Ult m ( U ⊛ , ~F ) . This assumption is not so important, but will always hold where we use the lemma. ⊛ = e U ⊛ P ⊛ U ⊛ = e U ⊛ M ⊛ P ⊛ U M P ~E, m~F , m G, m ~E, mG ⊛ , mG ⊛ , m ~E, ~F , ~D, m ~D, Figure 1: Extender commutativity. The diagrams commute, where ~D = ~E b ~F ,and a label ~C, k denotes a degree k abstract iteration map given by ~C . Let e U ⊛ = Ult m ( M ⊛ , G ⊛ ) and e U ⊛ = Ult m ( M ⊛ , G ⊛ )( see part 2 below ) , and suppose e U ⊛ is wellfounded. Then:1. ~E b ~F is ( U, m ) -good.2. M ⊛ || κ + M ⊛ ⊛ = P ⊛ | κ + P ⊛ ⊛ = P ⊛ | ( κ ⊛ ) + P ⊛ (so e U ⊛ and e U ⊛ are well-defined),3. U ⊛ = e U ⊛ and U ⊛ = e U ⊛ .4. The various ultrapower maps commute, as indicated in Figure 1; that is, i U,m~E b ~F ◦ i M,mG = i e U ⊛ ,m~F ◦ i M ⊛ ,mG ⊛ ◦ i M,m~E = i M ⊛ ,mG ⊛ ◦ i M,m~E .
5. The hypotheses for the Shift Lemma hold with respect to ( M, P ) , ( M ⊛ , P ⊛ ) ,and the maps i M,m~E : M → M ⊛ and i P, ~E b ~F : P → P ⊛ . Moreover, i U,m~E b ~F is just the Shift Lemma map. Proof.
Let ~D = ~E b ~F .Since G = F P , we have OR P ≤ ρ Um and either– P pv ⊳ card U , or– P pv = U pv , F P is of superstrong type and M is MS-indexed type 2 withlargest cardinal κ . 11rite P α = Ult ( P, ~D ↾ α ) and U α = Ult m ( U, ~D ↾ α ), where α is is large aspossible that ~D ↾ α is ( U, m )-pre-good. Since ~D is ( P, β ≤ α that OR P β ≤ ρ U β m and either– ( P β ) pv ⊳ card U β , with OR P β in the wellfounded part of U β , or– ( P β ) pv = ( U β ) pv , which is wellfounded,and the ultrapower maps agree over P , or over P sq if P is MS-indexed type3. So either ~D is ( U, m )-good, ~D ↾ α is ( U, m )-pre-good but U α is illfounded.So renaming, we may assume that α = lh( ~D ), so ~D is ( U, m )-pre-good and P α = P ⊛ and U α = U ⊛ .Let H be the (long) M -extender measuring P ( κ ) ∩ M , derived from j = i M ⊛ ,mG ⊛ ◦ i M,m~E : M → e U ⊛ , of length e ν ( G ⊛ ). Clearly e U ⊛ = Ult m ( M, H ) and j = i M,mH .Let H ′ be the (long) M -extender measuring P ( κ ) ∩ M , derived from j ′ = def i U,m~D ◦ i M,mG : M → U ⊛ , of length e ν ( G ⊛ ). Let σ : Ult m ( M, H ′ ) → U ⊛ be the standard factor map.Then Ult m ( M, H ′ ) = U ⊛ and σ = id, because G is generated by e ν ( G ) and ~D is < e ν ( G )-bounded and H ′ has length e ν ( G ⊛ ), and e ν ( G ⊛ ) = sup i P, ~D “ e ν ( G ) = sup i U,m~D “ e ν ( G ) . Therefore j ′ = i M,mH ′ . Claim . H ′ = H . Proof.
Let A ∈ P ([ κ ] <ω ) ∩ M . For ease of reading we assume that A ∈ P ( κ )and that P is not MS-indexed type 3 (hence e ν ( G ⊛ ) = i P, ( e ν ( G ))), but the othercases are simple variants. We want to see j ′ ( A ) ∩ e ν ( G ⊛ ) = j ( A ) ∩ e ν ( G ⊛ ) . But j ′ ( A ) ∩ e ν ( G ⊛ ) = i U,m~D ( i G ( A ) ∩ e ν ( G )) = i P, ~D ( i G ( A ) ∩ e ν ( G ))(the second equality as i G ( A ) ∩ e ν ( G ) ∈ P , over which i U,m~D agrees with i P, ~D )= i G ⊛ ( i P, ~D ( A )) ∩ e ν ( G ⊛ )(by definition of how F P shifts to F P ⊛ under ultrapowers)= i G ⊛ ( i P, ~E ( A )) ∩ e ν ( G ⊛ )(since i P, ~D ( A ) = i P, ~E ( A ), since cr( ~F ) > κ ⊛ )= i M ⊛ ,mG ⊛ ( i M,m~E ( A )) ∩ e ν ( G ⊛ ) = j ( A ) ∩ e ν ( G ⊛ )(by agreement of ultrapower maps), as desired. (cid:3)
12y the claim, U ⊛ = e U ⊛ and the corresponding ultrapower maps commute.The rest of parts 2–4 follow from this, by considering the special cases thateither ~F = ∅ or ~E = ∅ .Part 5 follows from the commutativity and agreement between i U,m~D and i P, ~D ,and by the elementarity of the maps (it is also like in [4, Lemma 4.20]). (cid:3) We next want to generalize the preceding lemma to deal with the case of a(normal) sequence ~G of extenders, instead of just a single extender G . Let P be an active premouse and ~F be a sequence of extenderswhich is ( P, P η = Ult ( P, ~F ↾ η ). We say that ~F is:– ( P, -strictly- e ν -bounded iff < e ν ( P )-bounded,– ( P, -critical-bounded iff cr( F P )-bounded.Let ~P = h P α i α<λ be a sequence of active premice. Say ~P and (cid:10) F P α (cid:11) α<λ are normal iff e ν ( P α ) ≤ cr( F P β ) and ( P α ) pv ⊳ card P β for α < β < λ . ⊣ Let h Q α i α<λ be a normal sequence of active premice. Let G α = F Q α and ~G = h G α i α<λ . Let h P α , F α i α<θ and ~F be likewise.Let α < λ . Let η α be the largest η ≤ θ such that ~F ↾ η is ( Q α , < e ν ( Q α )-bounded. Suppose that ~F ↾ η α is ( Q α , ξ α be thelargest ξ ≤ θ such that ~F ↾ ξ is ( Q α , F Q α )-bounded. Notethat ξ α ≤ η α , so ~F ↾ ξ α is also ( Q α , η β ≤ ξ α for β < α .Write Q ⊛ α = Ult ( Q α , ~F ↾ η α ) and G ⊛ α = F Q ⊛ α . Given β < θ , say that F β is nested (with respect to this ⊛ -product) iff ξ α ≤ β < η α for some α < θ ; and unnested otherwise. Then the ⊛ -product ~G ⊛ ~F denotes the enumeration of X = { G ⊛ α } α<λ ∪ { F α | α < θ and F α is unnested } in order of increasing critical point. And ~Q ⊛ ~P denotes the correspondingenumeration of { Q ⊛ α } α<λ ∪ { P α | α < θ and F α is unnested } . In this context, we also write Q α ⊛ = Ult ( Q α , ~F ↾ ξ α ) and G α ⊛ = F Q α ⊛ . ⊣ Adopt the hypotheses and notation of 2.11. Let M be m -sound. Suppose that ~G is ( M, m ) -pre-good and ~G ⊛ ~F is ( M, m ) -good. Let U = Ult m ( M, ~G ) . Then:1. If E, F ∈ X then cr( E ) = cr( F ) , so the ordering of ~G ⊛ ~F is well-defined.2. ~G ⊛ ~F and ~Q ⊛ ~P are normal sequences.3. ~G ⊛ ~F is equivalent to ~G b ~F ; that is, ~G is ( M, m ) -good, ~F is ( U, m ) -good, Ult m ( M, ~G b ~F ) = Ult m ( M, ~G ⊛ ~F ) and the associated ultrapower maps ( and hence derived extenders ) agree. roof. Parts 1 and 2 are routine. Consider part 3. Its proof is basically averification that the diagram in Figure 2 commutes. Note that in the diagram,all arrows labelled with extenders or sequences thereof correspond to degree m ultrapowers by those extenders. In the diagram and in what follows, given asequence ~E of extenders, we write ~E [ α,β ) for ~E ↾ [ α, β ). The maps σ αβ displayedin the diagram are the natural factor maps between degree m ultrapowers of M by certain natural segments of ~G ⊛ ~F , and will be specified below. The readerwill then happily verify that U , in the top right corner of the diagram, is justUlt m ( M, ~G ⊛ ~F ), with its ultrapower map derived along the main diagonal ofthe diagram (passing from M to U ); and also that U is just Ult m ( M, ~G b ~F ),with its ultrapower map derived from the composition of the maps along thebottom and right side. So this will complete the proof.Now for each ε ≤ λ let η <ε = sup α<ε η α . Let M ε = Ult m ( M, ~G ↾ ε )(so M = M and M λ = U = Ult m ( M, ~G )) and U ε = Ult m ( M ε , ~F [0 ,η <ε ) )(so U = M ).Let ε < λ . Let κ = cr( G ε ). Note that 2.9 applies to the sub-diagram ofFigure 2 with corners M ε , M ε +1 , ¯ U ε +1 and U ε +1 , and where (by induction)¯ U ε +1 = Ult m ( U ε , ~F [ η <ε ,ξ ε ) ) = Ult m ( M ε , ~F [0 ,ξ ε ) ) , e U ε +1 = Ult m ( ¯ U ε +1 , G ε ⊛ ) = Ult m ( M ε +1 , ~F [0 ,ξ ε ) ) , and in particular, U ε +1 = Ult m ( ¯ U ε +1 , G ⊛ ε )and the sub-diagram commutes. So let σ ε,ε +1 : U ε → U ε +1 be the resultingmap, that is, σ ε,ε +1 = i ¯ U ε +1 ,mG ⊛ ε ◦ i U ε ,m~F [ η<ε,ξε ) . Now for α ≤ β ≤ λ let σ αβ : U α → U β be the commuting map now inducedby composition and direct limits. We claim this makes sense, in that for eachlimit δ ≤ λ , U δ = dirlim α ≤ β<δ ( U α , U β ; σ αβ ) , (2)and that moreover, for all δ ≤ λ and α ≤ δ , we have k = def σ αδ ◦ i M α ,m~F [0 ,η<α ) = i M δ ,m~F [0 ,η<δ ) ◦ i M α ,m~G [ α,δ ) = def k ′ . This is verified by a straightforward induction on δ . For successor δ it is asdiscussed above, and for limit δ , assuming for simplicity that m = 0, letting σ ′ αδ : U α → U δ The notation in the proof does not match well with the previous lemma; this will beremedied in a future version. U λ U δ U ε +1 ¯ U ε +1 e U ε +1 U ε U ¯ U e U M M M ε M ε +1 M δ M λG ~G [1 ,ε ) G ε ~G [ ε +1 ,δ ) ~G [ δ,λ ) ~F [0 ,ξ ~F [0 ,ξ ~F [ ξ ,η ~F [0 ,η<ε ) ~F [ η<ε,ξε ) ~F [0 ,ξε ) ~F [ ξε,ηε ) ~F [0 ,η<δ ) ~F [0 ,η<λ ) ~F [ η<λ,θ ) G ⊛ G ⊛ σ ε G ε ⊛ G ⊛ ε σ ε +1 ,δ σ δλ Figure 2: The diagram commutes. Arrows labelled with (sequences of) exten-der(s) indicate the degree m ultrapower map determined by that (sequence of)extender(s), and that the structure at the tip of the arrow is the degree m ultrapower of the structure at its base. The unlabelled arrows correspond toultrapowers by the appropriate middle segment of ~G ⊛ ~F . Given a sequence ~E , ~E [ α,β ) denotes ~E ↾ [ α, β ). 15e σ ′ αδ (cid:0) i M α ~F [0 ,η<α ) ( f )( a ) (cid:1) = i M δ ~F [0 ,η<δ ) (cid:0) i M α ~G [ α,δ ) ( f ) (cid:1) ( a )for a ∈ [cr( ~F [ η <α , ∞ ) )] <ω , then for every x ∈ U δ , there is α < δ with x ∈ rg( σ ′ αδ ),and note that this then yields the inductive hypotheses and that σ αδ = σ ′ αδ .(One can also argue like in part of the proof of 2.9: by commutativity, onederives the same extender from k as from k ′ (the maps above), and the directlimit is in fact the ultrapower by this extender, since it is a direct limit of smallerultrapowers by sub-extenders thereof (note here that on both sides, the derivedextenders do have the same generators).)So the diagram commutes, and the lemma easily follows. (cid:3) We now proceed to adapt much of [4], with the most fundamental change beingin how extenders are copied from a tree T into a (now minimal ) inflation X . (Tree dropdown) . Let M be a m -sound premouse and let T bea putative m -maximal tree on M .For β + 1 < lh( T ) let ( λ β , d β ) = (lh( E T β ) , β + 1 = lh( T ) (if lh( T )is a successor and M T β well-defined) let ( λ β , d β ) = (OR( M T β ) , deg T ( β )). Let β < lh( T ). Let h M βi , m βi i i ≤ k β be the reversed extended dropdown of(( M T β , deg T ( β )) , ( M T β | λ β , d β ))(note this defines k β ). Then k T β = def k β and M T βi = def M βi and m T βi = m βi .Let θ ≤ lh( T ). We define the dropdown domain ddd ( T ,θ ) of ( T , θ ) by∆ = ddd ( T ,θ ) = def { ( β, i ) | β < θ & i ≤ k β } , and define the dropdown sequence dds ( T ,θ ) of ( T , θ ) bydds ( T ,θ ) = def h ( M βi , m βi ) i ( β,i ) ∈ ∆ . The dropdown sequence dds T of T is dds ( T , lh( T )) , and the dropdown domain ddd T of T is ddd ( T , lh( T )) .Given κ < e ν T α for some α + 1 < lh( T ), α T κ denotes the least such α , and n T κ denotes the largest n ≤ k T α such that n = 0 or ρ m αn +1 ( M αi ) ≤ κ . If insteadlh( T ) = α + 1 and κ ≤ OR( M T α ) but e ν T β ≤ κ for all β + 1 < lh( T ), then α T κ = α and n T κ = 0. ⊣ So if κ = cr( E T β ) then pred T ( β + 1) = α T κ and M ∗T β +1 = M T α T κ n T κ .16 .2 Remark. Recall that for an iteration tree X , clint X denotes the set ofclosed < X -intervals.We now define the notion of an minimal tree embedding Π : T ֒ → min X between normal trees T , X (actually we allow T to be a putative tree). Thedefinition is just that of tree embedding from [4], except that we modify howlift extenders E T α of T into X , and therefore must also modify how we lift theassociated dropdown sequence. In [4] the lift of E T α is just its image π ( E T α )under a copy map π . Here, associated with our copy maps π we will alsohave a sequence ~E of extenders, and π will just be the ultrapower map associ-ated to Ult n ( N, ~E ), for some (
N, n ) in the dropdown sequence of T , such that(ex T α , E ( N, n ) E ( M T α , deg T ( α )), and ~E will be (ex T α , E T α to Ult ( E T α , ~E ); that is, the active extender of Ult (ex T α , ~E ). The dropdownsequence is lifted analogously. The rest is just a straightforward modification ofthe notion of tree embedding. (Tree pre-embedding) . (Cf. [4, Figure 1].) Let M be an m -sound premouse, let T , X be putative m -maximal trees on M , with X a truetree, and θ ≤ lh( T ). A tree pre-embedding from ( T , θ ) to X , denotedΠ : ( T , θ ) ֒ → pre X , is a sequence Π = h I α i α<θ such that (cf. [4, Figure 1]):1. I β ∈ clint X for each β < θ . Let [ γ β , δ β ] X = def I β and Γ : θ → lh( X ) beΓ( β ) = γ β .2. γ = 0.3. Γ preserves < , is continuous, and sends successors to successors.4. β < T β ⇐⇒ γ β < X γ β .5. deg X ( γ β ) = deg T ( β ).6. For β + 1 < θ , we have γ β +1 = δ β + 1.7. For β + 1 < θ , letting ξ = pred T ( β + 1), we have pred X ( γ β +1 ) ∈ I ξ (in [4,Figure 1], η β +1 = pred X ( γ β +1 )) and D X ∩ ( γ ξ , γ β +1 ] X = ∅ ⇐⇒ β + 1 / ∈ D T . We say Π has degree m . ⊣ It follows that:(i) the < -intervals [ γ β , δ β ] partition sup β<θ δ β ,(ii) for ξ, ζ < θ , we have ( γ ξ , γ ζ ] X ∩ D X = ∅ iff ( ξ, ζ ] T ∩ D T = ∅ ,17iii) for each limit β < θ , we have Γ“[0 , β ) T ⊆ cof [0 , γ β ) X ,(iv) if lh( T ) = α + 1 then M T α is well-defined, and(v) as in [4], if α ∈ I ξ and δ ≤ X α then δ ∈ I ζ for some ζ ≤ T ξ . Let X be an iteration tree and α ≤ X β . Let D = { γ | γ + 1 ∈ ( α, β ] X } . Then ~E X αβ denotes (cid:10) E X γ (cid:11) γ ∈ D (note that when ( α, β ] X does not drop, this exten-der sequence corresponds to i X αβ ). ⊣ Let Π : ( T , θ ) ֒ → pre X and write I α = I Π α etc. For ξ ∈ S α<θ I α ,define the inflationary extender sequence ~F ξ = ~F Π ξ by:– ~F γ = ~F = ∅ – for ξ ∈ ( γ α , δ α ] X , ~F ξ = ~F γ α b ~E X γ α ξ ,– for α + 1 < θ , ~F γ α +1 = ~F δ α ,– for limit α < θ , ~F γ α = S ξ< X γ α ~F ξ . ⊣ The kinds of tree embeddings relevant to full normalization are the minimal ones, defined next, essentially the weak hull embeddings defined by Steel. Thedefinition is actually much shorter than the analogous definition in [4]; we willonly keep track of embeddings from ex T α into segments of models of X , not fromthe full models M T α . Thus, the definition will not immediately yield that T haswellfounded models. We will soon see, however, that if Π is minimal and M is m -standard where T is m -maximal then there is an embedding M T α → M X γ α , so T will have wellfounded models. (Minimal tree embedding) . Let Π : ( T , θ ) ֒ → pre X be a treepre-embedding. We say Π is minimal , denotedΠ : ( T , θ ) ֒ → min X , provided writing γ α = γ Π α , etc, for each α < θ we have:1. If α +1 < lh( T ) then ~F δ α is (ex T α , Q αξ = Ult (ex T α , ~F ξ ) E M X ξ for each ξ ∈ I α .2. if α + 1 < θ then E X δ α = F Q αδα , and3. if α + 1 = lh( T ) then ( γ α , δ α ] X does not drop in model or degree.If Π is a minimal tree embedding, we say Π is bounding iff lh( E X ξ ) ≤ OR Q αξ foreach α < θ and ξ ∈ I α such that α + 1 < lh( T ) and ξ + 1 < lh( X ), and exactlybounding iff lh( E X ξ ) < OR Q αξ for each such α, ξ with ξ ∈ [ γ α , δ α ) X .We write Π : T ֒ → min X iff Π : ( T , lh( T )) ֒ → min X . ⊣
18e will mostly be interested in exactly bounding minimal tree embeddings.
A tree pre-embedding Π : ( T , θ ) ֒ → pre X is puta-minimal ,written Π : ( T , θ ) ֒ → putamin X , iff the requirements of minimality hold, exceptthat we replace condition 3.7(1) with the following, defining Q αξ as before:1’. Let α + 1 < lh( T ). Then:(a) If α +1 < θ then ~F δ α is (ex T α , Q αξ E M X ξ for each ξ ∈ I α .(b) If α + 1 = θ and ~F γ α is (ex T α , Q αγ α E M X γ α then ~F δ α is(ex T α , Q αξ E M X ξ for each ξ ∈ I α \{ δ α } . ⊣ We will need to define various bookkeeping devices (mice and maps) anal-ogous to those used in [4], in order to see that minimal tree embeddings makesense, for m -standard M . The definition will list a lot of properties of thevarious objects, and we will verify that they exist later in Lemma 3.12. (Dropdown lifts) . Let Π : ( T , θ ) ֒ → putamin X of degree m onan m -standard M . Write I α = I Π α , etc. Let ∆ = ddd( T , θ ). For x = ( β, i ) ∈ ∆let k β = k T β and ( M βi , m βi ) = ( M x , m x ) = ( M T x , m T x ) . For ( β, i ) ∈ ∆ let δ βi be the largest δ ∈ S α<θ I α such that ~F δ is ( M βi , m βi )-pre-good. Say that Π is pre-standard iff for each ( β, i ) ∈ ∆, we have δ βk β = δ β , δ βi ∈ I β and if ( β, i + 1) ∈ ∆ then δ βi ≤ δ β,i +1 .Suppose Π is pre-standard. For ξ ∈ [ γ β , δ βi ] X define– U βiξ = Ult m βi ( M βi , ~F ξ ), and– π βiξ : M βiξ → U βiξ is the associated ultrapower map.Let U βi = U βiγ β and π βi = π βiγ β and π β = π β . Let γ β = γ β and γ β,i +1 = δ βi for i + 1 ≤ k β . Let I βi = [ γ βi , δ βi ] X and J βi = [ γ β , δ βi ] X . For ξ ∈ I β let i βξ =the least i ′ such that ξ ∈ I βi ′ . If β ∈ lh( T ) − then for ξ ∈ I β let Q βξ = U βk β ξ and ω βξ = π βk β ξ . ⊣ Note that by pre-standardness, for ( β, i ) ∈ ∆ we have have I βi = [ γ βi , δ βi ] X ⊆ I β . (Cf. [4, Figures 2, 3, 4].) Let M, m, T , X , Π , ∆ be as in 3.9.We say Π is standard iffT1. Π : ( T , θ ) ֒ → min X and Π is pre-standard.T2. (Dropdowns lift) For ( α, i ) ∈ ∆ and ξ ∈ J αi we have: So i ≤ k β , M β = M β , m β = deg T ( β ), and if β + 1 < lh( T ) then M βk β = ex T β and m βk β = 0. ***To do: change this use of variable “ i ” due to conflict with iteration maps. U α , m α ) = ( M X γ α , deg X ( γ α )),(b) ( U αiξ , m αiξ ) E ( M X ξ , deg X ( ξ )),(c) ( U αiξ , m αiξ ) = ( M X ξ , deg X ( ξ )) if γ αi < ξ ≤ δ αi ,(d) D X deg ∩ ( γ α , δ α ] X = ∅ .(e) Suppose i > γ αi < δ αi . Let ε αi = succ X ( γ αi , δ αi ). Then:i. ( γ αi , δ αi ] X ∩ D X deg = { ε αi } ,ii. ( M ∗X ε αi , deg X ( ε αi )) = ( U αiγ αi , m αi ).(f) Suppose α ∈ lh( T ) − . Then:i. h ( U αj , m αj ) i j ≤ k α is the revex (( M X γ α , deg X ( γ α )) , ( Q αγ α , γ αi < ξ ≤ δ αi then h ( U αjξ , m αj ) i i ≤ j ≤ k α is the revex (( M X ξ , deg X ( ξ )) , ( Q αξ , α < θ with α +1 < lh( T ) and κ < e ν ( E T α ) with α = α T κ and i = n T κ . Let ξ ∈ J αi be least such that, letting µ = π αiξ ( κ ),either ξ = δ αi or µ < cr( E X η ) where η + 1 = succ X ( ξ, δ αi ); note that ξ ≥ γ αi . Write ξ κ = ξ . Let U = U αiξ and π = π αiξ .Whenever ( α ′ , i ′ , ξ ′ ) ≥ ( α, i, ξ ), U ′ = U α ′ i ′ ξ ′ and π ′ = π α ′ i ′ ξ ′ , we have:– U || ( µ + ) U = U ′ || ( µ + ) U ′ , – π ↾ P ( κ ) ⊆ π ′ and– if α < α ′ and ( i, ξ ) = ( k T α , δ α ) then:– π ↾ ι T α ⊆ π ′ and b π ( δ ) ≤ π ′ ( δ ) for every δ < lh( E T α ), – if lh( E T α ) < OR( M α ′ i ′ ) then lh( E X δ α ) ≤ b π ′ (lh( E T α )),– if lh( E T α ) = OR( M α ′ i ′ ) then α ′ = α + 1, i ′ = 0, lh( E X δ α ) =OR( M X γ α +1 ) , π α +1 , = b π ↾ M α +1 , , and M ∗X γ α +1 = Q εδ ε where ε = pred T ( α + 1). T4. (Commutativity) Let ( χ, i ) , ( β + 1 , , ( ε, ∈ ∆ with χ < T β + 1 ≤ T ε and χ = pred T ( β + 1) and( M χi , m χi ) = ( M ∗T β +1 , deg T ( β + 1)) . (So i > β + 1 ∈ D T deg .) Let ξ = pred X ( γ β +1 ). Then:(a) ξ ∈ I χi , and if i > ξ = γ χi iff γ β +1 ∈ D X deg ]. We might have ( µ + ) U = OR U , but then i = k T α , ξ = δ α , and we are using MS-indexing, M T α is active type 2 and κ = lgcd( M T α ). ι T α = lgcd(ex T α ) unless ex T α is MS-indexed type 2, in which case ι T α = lh( E T α ). So if Π is exactly bounding then γ α +1 = δ α +1 . It follows that we are using MS-indexing, E T α is superstrong and M T α +1 is active type 2, E X δ α is superstrong and M X γ α +1 is active type 2. π β +1 , ◦ i ∗T β +1 = i ∗X γ β +1 ◦ π χiξ .(c) If ( χ, ε ] T ∩ D T deg = ∅ (so i = 0 and ( γ χ , γ ε ] X ∩ D X deg = ∅ ) then π ε ◦ i T χε = i X γ χ γ ε ◦ π χ . (d) (Shift Lemma) Let κ = cr( E T β ), so κ < e ν ( E T χ ) and i = n T κ , so T3applies. Then (i) ξ (defined above) is also as defined in T3 and( M ∗X γ β +1 , deg X γ β +1 ) = ( U χiξ , m χi ) . So by T3, the Shift Lemma applies to the embeddings π χiξ and ω βδ β : ex T β → Q βδ β . Moreover, (ii) π β +1 , is just the map given by the Shift Lemma (thismakes sense as M X γ β +1 = U β +1 , by T2). ⊣ Π : ( T , θ ) ֒ → putamin X have degree m , on an m -standard M .Then Π : ( T , θ ) ֒ → min X , Π is standard and T ↾ θ has (well-defined and)wellfounded models. Proof.
We adopt the notation of 3.9. The proof is by induction on θ .Suppose θ = 1. If lh( T ) = 1 then everything is easy. So suppose lh( T ) > E T exists. Property T4 is trivial. By puta-minimality, ~F δ is (ex T , Q ξ E M X ξ for every ξ < X δ . Now δ is the largest ξ ∈ I such that ~F ξ is ( M, m )-pre-good. Property T2 for ( α, i ) = (0 , δ ≤ δ i for each i ≤ k T , then both follow from Lemma 2.7, by induction on ξ ∈ I .It doesn’t matter here whether Π is bounding. In fact, note that if β + 1 ≤ X δ and κ = cr( E X β ) and h ( N ′ n , m ′ n ) i n ≤ k ′ is the revex (( M X ξ , deg X ξ ) , (ex X ξ , h ( N n , m n ) i n ≤ k the revex (( M X ξ , deg X ξ ) , ( Q ξ , N n , m n ) = ( N ′ n , m ′ n )whenever n = 0 or [ n ≤ k and ρ N n m n +1 ≤ κ ] or [ n ≤ k ′ and ρ N ′ n m ′ n +1 ≤ κ ]. Hence,if δ < δ and β + 1 = succ X ( δ , δ ), then ( M ∗X β +1 , deg X β +1 ) = ( N n , m n ) forsome n >
0, which is one of the ( U iξ , m iξ ). This gives that δ = γ i ′ = δ i ′ = γ i < δ i and property T2 for ( α, i ′ ) for all i ′ < i . For ( α, i ), property T2 andthat δ i ≤ δ i ′ for all i ′ > i now follows similarly to before. Preceding in thisway, we get the full properties T1 and T2 (recalling θ = 1). Property T3 is nowstraightforward.Now suppose that Π ↾ ( β + 1) : ( T , β + 1) ֒ → min X is standard and β + 1 < lh( T ). We prove Π ↾ ( β + 2) : ( T , β + 2) ֒ → min and is standard.Property T4: We must just verify this for β + 1, with ε = β + 1. Adoptnotation as there (this defines ξ, i, κ etc). Parts (a) and (d)(i) follow routinelyfrom the inductive hypotheses. Given these, we verify the rest. We have χ =pred T ( β + 1) and ( M ∗T β +1 , deg T β +1 ) = ( M χi , m χi ). Note that Lemma 2.9 appliesto ( M χi , m χi ) and P = ex T β , with extender sequences ~E = ~F ξ and ~F where ~F δ β = ~F ξ b ~F . Moreover, 21 χ = pred T ( β + 1) and ξ = pred X ( γ β +1 ),– n = def deg T ( β + 1) = m β +1 , = m χi = deg X ( γ β +1 ),– M χi = M ∗T β +1 and U χiξ = M ∗X γ β +1 ,– M T β +1 = Ult m ( M χi , E T β ) and M X γ β +1 = Ult m ( U χiξ , E X δ β ),– ~F γ β +1 = ~F δ β = ~E b ~F where ~E is κ -bounded and µ = def i ex T β , ~E ( κ ) = i ex T β , ~F δβ ( κ ) = cr( E X δ β ) < cr( ~F ) , – U χiξ = Ult m ( M χi , ~E ) and π χiξ is the ultrapower map,– Q βδ β = Ult (ex T β , ~F δ β ) and ω βδ β is the ultrapower map,– U β +1 , = Ult m ( M T β +1 , ~F γ β +1 ) and π β +1 , is the ultrapower map.So by Lemma 2.9, U β +1 , = M X γ β +1 and π β +1 , is the Shift Lemma map, givingpart (d)(ii), and commutativity holds, giving (b) and (c).Property T2 for α = β +1 and ξ = γ β +1 , and the fact that γ β +1 ≤ δ βi for each i ≤ k T β +1 , follow from the observations above (such as that U β +1 , = M X γ β +1 and m β +1 , = deg X ( γ β +1 )), together with the m -standardness of M and Lemma2.7, and using that lh( E T β ) ≤ lh( E T β +1 ) when verifying that ~F γ β +1 = ~F δ β is(ex T β +1 , α = β + 1are then like in the case that θ = 1.Property T3: In the main instance of interest, α = β and α ′ = β + 1. In thisinstance, the property follows as usual from the Shift Lemma, using the factthat π β +1 , is in fact the Shift Lemma map. The rest is routine.This completes the proof that Π ↾ ( β + 2) is standard.Now let β be a limit and suppose that Π ↾ β is standard. We verify thatΠ ↾ ( β + 1) is standard. The main issue is to see that ~F γ β is ( M T β , m β )-good, M X γ β = U β and π β ◦ i T αβ = i X γ α γ β ◦ π α for sufficiently large α < T β ; the restis as before. Let π ∗ : M T β → M X γ β be the map commuting in this way (byinduction, π ∗ exists and is a deg T β = deg X γ β -embedding). Let δ = δ ( T ↾ β ) and δ ′ = δ ( X ↾ γ β ). By commutativity, ~F γ β is equivalent to the ( δ, δ ′ )-extenderderived from π ∗ (and δ ′ = sup π ∗ “ δ ≤ π ∗ ( δ )). It easily follows that U β = M X γ β and π ∗ = π β +1 , , as desired. (cid:3) The basic observation which made the lemma above possible– the fact that extenders in E + ( M T α ) lift to extenders in E + ( M X β ) for the ap-propriate α, β , under degree 0 ultrapower maps – and the ensuing idea for fullnormalization (as opposed to embedding normalization), weak hull embeddingsand very strong hull condensation – was due to Steel. The fact that one mustalso keep track of how the dropdown sequence is shifted all along the entireinterval I α (in order to see that the T and X drop to corresponding segments)was noticed somewhat later, independently by both the author and Steel.22 .14 Definition. Let Π be a minimal tree embedding and adopt notation asbefore. We use notation analogous to that of [4]; the subscript “Π” indicatesobjects associated to Π. That is, I Π α = I α , γ Π α = γ α , etc. ⊣ (Cf. [4, Figure 4].) Let Π : ( T , θ ) ֒ → min X and γ β = γ Π β ,etc. Let β < θ and κ ≤ OR( M T β ) with β = α T κ , and let n = n T κ (Definition3.1). Let N T κ = M T βn and ξ = ξ Π κ ∈ I βn be defined as ξ κ in 3.11(T3), or iflh( T ) = β + 1 and κ = OR( M T β ), then ξ = ξ Π κ = δ β . Also let U Π κ = U βnξ and π Π κ : N T κ → U Π κ the corresponding ultrapower map. ⊣ Let Π : ( T , θ ) ֒ → min X . Let β ∈ θ ∩ lh( T ) − and ξ ∈ I Π β .Then E Π ξ denotes F Q βξ (the lift of E T β in E + ( M X ξ )). ⊣ Given T , X two putative m -maximal trees, X a true tree, the trivial tree embedding Π : ( T , ֒ → min X is that with I Π0 = [0 , T = X , the identity embedding Π : ( T , lh( T )) ֒ → ( T , lh( T )) is that with I Π β = [ β, β ] for all β < lh( T ). ⊣ Let
Π : ( T , α + 1) ֒ → min X where α + 1 < lh( T ) . Then E Π δ Π α is X ↾ ( δ Π α + 1) -normal. The lemma follows easily from the fact that ~F δ Π α is (ex T α , F ξ does not measure more subsets of itscritical point than those in Ult (ex T α , ~F ξ ).)We can propagate minimal tree embeddings T ֒ → min X via ultrapowersanalogously to in [4, 4.23, 4.24], so there are two possibilities: an extender iseither T -copying or T -inflationary . We first consider the T -copying case. Let Π : ( T , α +1) ֒ → min X be of degree m , with α +1 < lh( T ).Let γ α = γ Π α etc. Let X ′ = the m -maximal tree X ↾ ( δ α + 1) b (cid:10) E Π δ α (cid:11) (by 3.18, E Π δ α is X ↾ ( δ α + 1)-normal). Suppose that M X ′ δ α +1 is wellfounded. LetΠ ′ : ( T , α + 2) ֒ → pre X be such that Π ′ ↾ ( α + 1) = Π and I Π ′ α +1 = [ δ α + 1 , δ α + 1] X ′ . We say ( X ′ , Π ′ ) (orjust Π ′ for short) is the one-step copy extension of ( X , Π) (or of Π). ⊣ Adopt the hypotheses of 3.19. Suppose T , X are on an m -standard M . Then Π ′ is standard, so T ↾ ( α + 2) has wellfounded models. Proof. Π ′ is puta-minimal as γ α +1 = δ α +1 , so is minimal and standard byLemma 3.12. (cid:3) We next consider the T -inflationary case. Let Π : ( T , θ ) ֒ → min X , of degree m , with lh( X ) = η + 1.Let γ α = γ Π α , etc. Let E ∈ E + ( M X η ) be X -normal and X ′ be the putative m -maximal tree X b h E i . Let ξ = pred X ′ ( η + 1). Suppose that:23 M X ′ ∞ is wellfounded,– ξ ∈ I β for some β < θ ,– if β + 1 < lh( T ) then E is a Q Π βξ -extender and cr( E ) < e ν ( Q Π βξ ), and– if β + 1 = lh( T ) then η + 1 / ∈ D X ′ deg .The minimal E -inflation of ( X , Π) is ( X ′ , Π ′ ), where Π ′ : ( T , β + 1) ֒ → pre X ′ is such that I Π ′ β = ( I β ∩ ( ξ + 1)) ∪ { η + 1 } and I Π ′ α = I α for every α < β . ⊣ Adopt the hypotheses of 3.21. Suppose T , X are on an m -standard M . Then Π ′ is minimal and standard. The lemma is a direct consequence of the definitions and Lemma 3.12.
We now proceed to the definition of a minimal inflation of a normal iterationtree T . This is almost the exact definition of inflation from [4]; the only differ-ences are that here we are not considering wcpms (coarse structures), and weuse the minimal one-step copy extension and minimal- E -inflation at successorsteps, instead of the non-minimal versions. But we will write out the definitionexplicitly, for convenience. An intuitive introduction can be seen in [4, § (Minimal inflation) . Let M be m -standard and T , X be puta-tive m -maximal trees on M , X a true tree. We say that X is a minimal inflation of T , written T min X , iff there is (cid:0) t, C, C − , f, h Π α i α ∈ C (cid:1) with the followingproperties (which unique the tuple); we will also define further notation:1. We have t : lh( X ) − → { , } . The value of t ( α ) indicates the type of E X α ,either T -copying (if t ( α ) = 0) or T -inflationary (if t ( α ) = 1).2. C ⊆ lh( X ) and C ∩ [0 , α ] X is a closed initial segment of [0 , α ] X .3. We have f : C → lh( T ) and C − = { α ∈ C (cid:12)(cid:12) f ( α ) + 1 < lh( T ) } .4. For α ∈ C we have Π α : ( T , f ( α ) + 1) ֒ → min X ↾ ( α + 1), with δ α ; f ( α ) = α ,where we write δ α ; β = δ Π α β , etc.5. 0 ∈ C and f (0) = 0 and Π : ( T , ֒ → min X ↾ α + 1 < lh( X ). Then: – If α ∈ C − then lh( E X α ) ≤ lh( E Π α α ).– t ( α ) = 0 iff [ α ∈ C − and E X α = E Π α α ]. One could consider dropping the closure requirement here; cf. [4, Footnote q/17]. One might also consider weakening these conditions; cf. [4, Footnotes r/18, s/19].
24. Let α + 1 < lh( X ) be such that t ( α ) = 0. Then we interpret E X α = E Π α α as a copy from T , as follows:– α + 1 ∈ C and f ( α + 1) = f ( α ) + 1.– ( X ↾ α + 2 , Π α +1 ) is the minimal one-step copy extension of ( X ↾ α + 1 , Π α ).8. Let α + 1 < lh( X ) be such that t ( α ) = 1. Then we interpret E X α as T -inflationary, as follows. Let ξ = pred X ( α + 1). Then:– α + 1 ∈ C iff:– ξ ∈ C − and Q ξ ; f ( ξ ) E M ∗X α +1 , or– ξ ∈ C \ C − and α + 1 / ∈ D X deg .– If α + 1 ∈ C then:– f ( α + 1) = f ( ξ ).– ( X ↾ α + 2 , Π α +1 ) is the minimal E X α -inflation of ( X ↾ α + 1 , Π ξ ).9. Let α ∈ C and β ∈ I α ; γ for some γ ≤ f ( α ). Then:– β ∈ C and f ( β ) = γ .– I α ; ε = I β ; ε for all ε < f ( β ) = γ ,– I β ; f ( β ) = I α ; f ( β ) ∩ ( β + 1).10. If α ∈ C is a limit then f ( α ) = sup β< X α f ( β ). ⊣ Note that in the definition of minimal inflation, we assume that M T is m -standard, where T , X are m -maximal.Adopt the hypotheses and notation of condition 9 above. Note that U α ; f ( β )0 = M X γ α ; f ( β ) = M X γ β ; f ( β ) = U β ; f ( β )0 and π α ; f ( β )0 = π β ; f ( β )0 . By 3.4(v), if e β ≤ X α then e β ∈ I α ; e γ for some e γ ≤ T f ( α ),so condition 9 applies to e β, e γ , and therefore f ( e β ) ≤ T f ( α ).Let α ∈ C be a limit. As in [4], Π α is determined by h Π β i β<α and T and X ↾ ( α + 1). For suppose f ( α ) > f ( β ) for all β < X α . From condition 9, for ξ < f ( α ), it follows that I α ; ξ = ( lim β< X α I β ; ξ ) = unique value of I β ; ξ for sufficiently large β < X α. So α = (cid:0) lim ξ Let T min X , of degree m ( so M T is m -standard ) , with X ofsuccessor length β + 1 . Let C − = ( C − ) T min X . Then:1. If β ∈ C − then E T min X β is X -normal.2. Let E ∈ E + ( M X β ) be X -normal, with lh( E ) ≤ lh( E T min X β ) if β ∈ C − .Let X ′ be the putative m -maximal tree X b h E i , and suppose M X ′ ∞ iswellfounded. Then X ′ is a minimal inflation of T . Proof. Part 1 follows from 3.18, and part 2 from 3.20 and 3.22. (cid:3) However, just as in [4], at limit stages we need to assume some condensationholds of Σ, in order to extend. See [4, § Let Ω > ω be regular. Let Σ be an ( m, Ω + 1)-strategy foran m -standard pm M . Then Σ has minimal inflation condensation or is minimal-inflationary iff for all trees T , X , if– T , X are via Σ,– X is a minimal inflation of T , as witnessed by ( f, C, . . . ),– X has limit length and lh( X ) ≤ Ω,– b = def Σ( X ) ⊆ C and f “ b has limit ordertype,then letting η = sup f “ b , we have f “ b = Σ( T ↾ η ). ⊣ Like in [4], the definition immediately gives that minimal inflations viaminimal-inflationary Σ can be continued at limit stages: Let Ω > ω be regular. Let Σ be a minimal-inflationary ( m, Ω+1) -strategy for an m -standard M . Let T , X be such that X is via Σ , X is a minimalinflation of T , as witnessed by ( f, C, . . . ) , and lh( T ) = sup α ∈ C ( f ( α ) + 1) . Then T is via Σ .Suppose also that X has limit length λ and let X ′ = ( X , Σ( X )) . Then thereis T ′ via Σ such that T E T ′ and X ′ is an inflation of T ′ , as witnessed by ( C ′ , f ′ , . . . ) . Moreover, we may take T ′ such that either: (***Could add the strategy condensation for partial strategies here) T ′ = T and if λ ∈ C ′ then f ′ ( λ ) < lh( T ) , or– T has limit length ¯ λ , T ′ = ( T , Σ( T )) , λ ∈ C ′ , f ′ ( λ ) = ¯ λ and γ ′ λ ;¯ λ = λ .Further, the choice of T ′ is uniqued by adding these requirements. Also as in [4], we have a simple characterization of when T min X , giventhat T , X are via a common minimal-inflationary strategy: Let Ω > ω be regular. Let Σ be a minimal-inflationary ( m, Ω+1) -strategy for an m -standard M and T , X be via Σ .Then ( i ) T min X iff:– X satisfies the bounding requirements on extender indices imposed by T ;that is, for each α + 1 < lh( X ) , if T min X ↾ ( α + 1) and α ∈ ( C − ) T min X ↾ ( α +1) then lh( E X α ) ≤ lh( E T min ( X ↾ α +1) α ) , and– if T has limit length then X does not determine a T -cofinal branch; thatis, for each limit η < lh( X ) , if T min X ↾ η and ( f, C ) = ( f, C ) T min X ↾ η and [0 , η ) X ⊆ C then lh( T ) > sup α< X η f ( α ) .Moreover, ( ii ) suppose T min X and lh( X ) is a limit. Let X ′ = ( X , Σ( X )) .Then either T min X ′ or [ T has limit length and ( T , Σ( T )) min X ′ ] . We can also define the minimal analogue of strong hull condensation. It eas-ily implies minimal inflation condensation; we do not know whether the converseholds. Let Ω > ω be regular. Let Σ be an ( m, Ω + 1)-strategy foran m -standard M . We say that Σ has strong minimal hull condensation iff whenever X is via Σ and Π : T ֒ → min X is a minimal tree embedding, then T is also via Σ. ⊣ One can also define the minimal analogue of extra inflationary from [4], butwe don’t need it. We now give some important examples of strategies withminimal strong hull condensation. The proofs are just direct translations of [4,Lemma 4.45, Theorem 4.47]. Let Σ be an ( m, Ω + 1) -strategy for an m -standard M . Supposethat Σ is the unique ( m, Ω + 1) -strategy for M . Then Σ has strong minimal hullcondensation. The second result deals with strategies with the (weak) Dodd-Jensen property. We abbreviate Dodd-Jensen with DJ . Note that only the firstpart of the proof of [4, Theorem 4.47], which regards λ -indexing, is relevanthere; in our setting it adapts immediately to give the proof for both indexings.For this result, we assume that M is a pure extender mouse, thus, not astrategy mouse. This is important because the proof involves a comparison.27 .34 Theorem. Let Ω > ω be regular. Let M be an m -standard pure L [ E ] -premouse with card( M ) < Ω . Let Σ be an ( m, Ω + 1) -strategy for M such thateither Σ has the DJ property, or M is countable and Σ has weak DJ. Then Σ has strong minimal hull condensation. Proof. A routine adaptation of the first part of the proof of [4, Theorem 4.47]. (cid:3) We adapt some further terminology from [4]: Let T min X . Let( t, C, C − , f, ~ Π) = ( t, C, C − , f, ~ Π) T min X and let γ α ; β , etc, be as in 3.23. Suppose that X has successor length α + 1.We say that X is:– ( T ) -pending iff α ∈ C − .– non- ( T ) -pending iff α / ∈ C − .– ( T ) -terminal iff T has successor length and X is non- T -pending.We say that X is: – T -terminally-non-dropping iff T -terminal and α ∈ C ; hence, f ( α ) + 1 =lh( T ) and ( γ α ; f ( α ) , δ α ; f ( α ) ] X ∩ D X deg = ∅ , – T -terminally-dropping iff T -terminal and α / ∈ C .Suppose X is T -terminally-non-dropping and let α +1 = lh( X ) and β = f ( α ).Then we define π T min X∞ : M T β → M X α by π T min X∞ = π α ; β α . ⊣ Suppose X is T -terminally-non-dropping and T , X are m -maximal. Note that π ∞ = π T min X∞ is an n -embedding, where n = deg X ( ∞ ).If X is T is also terminally non-dropping, then note that X is terminally non-dropping, n = m and π ∞ ◦ i T = i X . X / T We now discuss the minimal analogue of the factor tree of [4, § The terminology here is slightly different to that in [4], because we only deal with non-dropping , as opposed to both non-dropping and non-model-dropping , and here, α ∈ C \ C − requires no drop in model or degree, whereas only no drop in model in [4]. .1 The factor tree order < X / T Define T -unravelling , minimal- T -good (or just good ), as in [4, Definition 8.1](with minimal inflations replacing inflations throughout). For a good minimalinflation X of T , define the associated objects λ α , ζ α , L α η δ , X α , ( t α , C α , . . . ), θ α , ( λ α , ζ α , L α , X α , t α , . . . ) T min X , ( I αξ ) T min X , ( π αξi ) T min X , etc, as in [4, Def-inition 8.2]. Define < X / T as in [4, 8.3], and V ≥ α , < ( α )0 as in [4, 8.5]. Then [4,Lemmas 8.4, 8.6] hold, after replacing inflations with minimal inflations. Werestate [4, 8.7], because we use ξ ακ here, as opposed to the γ αθκ of [4] (recall ξ ακ = ξ Π α κ and π ακ = π Π α κ were specified in Definition 3.15): Let T min X be good. Adopt notation as above. Let α ≤ X / T β < lh( X / T ) with λ β ∈ C β ( so λ α ∈ C α by [4, 8.6] ) . Then:1. < X / T is an iteration tree order on lh( X / T ) .2. For all µ < X λ < lh( X ) , we have η µ ≤ X / T η λ .3. For all ( θ, κ ) with θ < lh( T ) and κ ≤ OR( M T θ ) and θ = α T κ , either ξ ακ ∈ [ λ α , lh( X α )) or ξ ακ ∈ L δ for some δ < X / T α .4. Suppose α < β . Let ξ + 1 = succ X / T ( α, β ) and χ = pred X ( λ ξ +1 ) . Then: ( a ) χ ∈ L α and θ α ≤ θ = def f α ( χ ) ≤ θ β . ( b ) For θ ′ < θ , we have I αθ ′ = I βθ ′ ⊆ χ and for κ ≤ OR( M T θ ′ ) with θ ′ = α T κ ,we have ξ ακ = ξ βκ < χ , ( c ) γ αθ = γ βθ but δ αθ = χ < X δ βθ , ( d ) If θ + 1 < lh( T ) then for κ < e ν (ex T θ ) with θ = α T κ , if π ακ ( κ ) < cr( E X ζ ξ ) then ξ ακ = ξ βκ , and if π ακ ( κ ) ≥ cr( E X ζ ξ ) then ξ ακ = χ < X ξ βκ . Proof. See the proof of [4, Lemma 8.7]. (cid:3) Let T min X be good. Adopt notation as above. Let α < lh( X / T ) with λ α ∈ C T min X . Then ( i ) for λ ∈ [ λ α , lh( X α )) , ~F α ∞ = def ~F αλ = (cid:10) E X ζ γ (cid:11) γ +1 ≤ X / T α . Therefore ( ii ) M X α ∞ = Ult n ( M T∞ , ~F α ∞ ) where n = deg T ( ∞ ) . Proof. Recall here that ~F αλ = ~F Π α λ was defined in 3.6. Part (i) is verified by aneasy induction on lh( X / T ). Part (ii) follows via Lemma 3.12. (cid:3) Let T min X be good, C = C T min X and n = deg T ( ∞ ).Adopt notation as above. Given α ≤ X / T β with λ α , λ β ∈ C , then π αβ : M X α ∞ → M X β ∞ denotes the natural factor map given by 4.2; that is, π αβ = i M X α ,n~F β ∞ \ ~F α ∞ .29uppose instead α ≤ X / T β < lh( X / T ) with λ α / ∈ C . So by [4, 8.6], lh( X α ) = λ α + 1 and lh( X β ) = λ β + 1 and λ α ≤ X λ β . If ( λ α , λ β ] X ∩ D X deg = ∅ , let π αβ = i X λ α λ β : M X α ∞ → M X β ∞ . If α is also a successor ordinal, let π ∗ αβ = i ∗X λ α λ β : M ∗X α λ α → M X β ∞ ;note that M ∗X α λ α E M X γ λ for some λ ∈ L γ where γ = pred X / T ( α ), and if λ γ ∈ ( C − ) γ then possibly λ γ < ζ γ . ⊣ We can now easily describe the full factor tree X / T : Let T min X be good and adopt notation as before. Let( N, n ) = ( M T∞ , deg T∞ ). Then the factor tree X / T (or the flattening of ( T , X ))is the n -maximal tree U on N such that lh( U ) = lh( X / T ), < U = < X / T , and E U α = E X ζ α for each α + 1 < lh( U ) (see Lemma 4.5). ⊣ Let T min X be good. Then:1. The factor tree U = X / T exists ( in particular, U has wellfounded models ) ,and is unique.2. [0 , α ] U drops in model or degree iff λ α / ∈ C α , for all α < lh( U ) .3. ( M U α , deg U α ) = ( M X α ∞ , deg X α ∞ ) for all α < lh( U ) .4. For α ≤ U β , we have ( α, β ] U ∩ D deg = ∅ iff π αβ is defined, and in this case, i U αβ = π αβ .5. If α + 1 ≤ U β and ( α + 1 , β ] U ∩ D deg = ∅ then i ∗U α +1 ,β = π ∗ α +1 ,β .6. Suppose T , U have successor length and X is non- T -pending, so X also hassuccessor length. Let ~ T = ( T , U ) . Then b ~ T ∩ D ~ T deg = ∅ iff b X ∩ D X deg = ∅ .If b ~ T ∩ D ~ T deg = ∅ then i ~ T ∞ = i X ∞ .where in parts 4 and 5, “ τ = σ ” means “ τ is defined iff σ is, and when defined,they are equal”. Proof. The uniqueness in part 1 is clear. Parts 2–5 are by induction on segments( X / T ) ↾ η and U ↾ η of X / T and U (see below), and then part 6 follows easilyfrom those things and the commutativity properties of minimal tree embeddings(which we leave to the reader).If η = 1, everything is trivial, as we have X = T .Now suppose we have the inductive hypotheses at η = α + 1 (so M U α = M X α ∞ etc); we want to extend to α + 2. Let E = E X ζ α . We have E ∈ E + ( M X α ζ α ),but lh( E ) < lh( E X α ζ α ) if ζ α + 1 < lh( X α ), as E is T -inflationary. So E ∈ + ( M X α ∞ ) = E + ( M U α ). And E is U ↾ ( α + 1)-normal, since ζ β < ζ α for β < α ,so lh( E U β ) = lh( E X ζ β ) ≤ lh( E ).Let β = pred U ( α + 1), so β is least such that cr( E ) < e ν (ex U β ). So by [4, 8.4] β = pred X / T ( α + 1), also as desired. Let λ = pred X ( λ α +1 ). So λ ∈ L β . If λ + 1 = lh( X β ) then by induction,( M U β , deg U β ) = ( M X β ∞ , deg X β ∞ ) = ( M X λ , deg X λ ) , and since E U α = E = E X α ζ α , the inductive hypotheses are immediately maintained(note there can be a drop in model or degree in this case). So suppose λ +1 < lh( X β ), so λ β , λ ∈ ( C − ) β and [0 , β ] U ∩ D U deg = ∅ . But either (ex X β λ ) pv isa cardinal proper segment of M X β ∞ = M U β , or we have MS-indexing, E X β λ issuperstrong, lh( X β ) = λ + 2 and OR( M X β λ +1 ) = lh( E X β λ )). So either:1. E is total over ex X β λ , λ α +1 ∈ ( C − ) α +1 , and U does not drop in model ordegree at α + 1, or2. λ α +1 ∈ D X and α + 1 ∈ D U and M ∗X λ α +1 = M ∗U α +1 ⊳ ex X β λ (so note then that ζ β = λ , as E is total over ex U β ) and deg X ( λ α +1 ) = deg U ( α + 1).In case 2, it is again routine to see that the hypotheses are maintained. Incase 1 (recalling [0 , β ] U ∩ D U deg = ∅ ), M U α +1 = Ult n ( M U β , E ) and i U β,α +1 is theultrapower map. But also, using facts about minimal tree embeddings, M X β ∞ = Ult n ( N, ~F β ∞ ) ,M X α +1 ∞ = Ult n ( N, ~F α +1 ∞ ) , and ~F α +1 ∞ = ~F β ∞ b h E i , so M X α +1 ∞ = Ult n ( M X β ∞ , E ) and π β,α +1 is the ultrapowermap, as desired.Now suppose that we have the inductive hypotheses below a limit η , and weconsider η + 1. Since < X / T is an iteration tree order and < X / T ↾ η = < U ↾ η ,[0 , η ) U = [0 , η ) X / T does indeed give a U ↾ η -cofinal branch. If for some α < U η , we have λ α / ∈ ( C − ) α , then everything is easy, so suppose otherwise. Inparticular, U does not drop in model or degree in [0 , η ) U . Now M U η is just thedirect limit as usual, and i U αη the associated direct limit map. By inductionthen, M U η = dirlim α ≤ U β< U η (cid:16) M X α ∞ , M X β ∞ ; π αβ (cid:17) , and i U αη is the associated direct limit map. But we have λ η ∈ C η , ~F η ∞ =lim α< U η ~F α ∞ and M X η ∞ = Ult n ( N, ~F η ∞ ) and π αη is an associated factor map, andlikewise for the π αβ for β < U η . But these match the direct limit and associatedmaps just described, as desired. (cid:3) In [4, 8.4] it says cr( E ) < ι (ex U β ), but this is equivalent saying cr( E ) < e ν (ex U β ) here, as ι (ex U β ) = e ν (ex U β ) unless ex U β is MS-indexed type 2, in which case ι (ex U β ) = OR(ex U β ). Minimal comparison In this section we quickly adapt the techniques of comparative and genericityinflation from [4, § 5] to minimal inflations. Let Ω > ω be regular. Let M be m -standard. Let T be aset of m -maximal trees on M , with each T ∈ T of length ≤ Ω + 1. Let X be m -maximal on M . We say that X is a minimal comparison of T (withrespect to Ω) iff:– X is a minimal inflation of each T ∈ T ; let t T = t T min X , etc, for T ∈ T ,– for each α + 1 < lh( X ) there is T ∈ T such that t T ( α ) = 0,– X has successor length β + 1 ≤ Ω + 1,– if β + 1 < Ω then there is no T ∈ T with β ∈ ( C − ) T . ⊣ The proof of [4, Lemma 5.2] gives: (Minimal comparison) . Let Ω > ω be regular. Let M be m -standard and Σ be an ( m, Ω + 1) -strategy for M with minimal inflation conden-sation. Let T be as in 5.1, and suppose card( T ) < Ω and each T ∈ T is via Σ with lh( T ) ≤ Ω + 1 . Then there is a unique minimal comparison X of T via Σ .Moreover, there is T ∈ T such that, letting T ′ = T if T has successor length,and T ′ = T b Σ( T ) otherwise, we have– X is T ′ -terminally-non-dropping, and– if lh( X ) = Ω + 1 then lh( T ′ ) = Ω + 1 . We need the adaptation of commutativity of inflation [4, Lemma 6.2]. PropertiesC1–C4 are just as in [4]. But there are some differences in property C5: drops inmodel in [4] correspond more to drops in model or degree here, the conclusionsof C5(e) are crucially different, because of minimality, and C5(f) is new. (Commutativity of minimal inflation) . Let M be m -standard and X , X , X be such that:– each X i is m -maximal on M ,– X , X have successor length, X α M X α M X γ M X α M X γ M X γπ α α π α α ττ π α γ π α α τ Figure 3: Commutativity of minimal inflation. We have α ∈ C , α = f ( α ), α = f ( α ) = f ( α ), γ kℓ = γ kℓα ℓ ; α k , γ = γ α ; γ , τ kℓ = π kℓα ℓ ; α k i kℓ α ℓ where i kℓ = i kℓα ℓ ; α k α ℓ , and τ = π α ; α i α . (So possibly dom( τ ) = dom( τ ), andpossibly τ τ .) Note α = δ α ; α = δ α ; α and α = δ α ; α and γ ≤ X α and γ ≤ X γ ≤ X α . Solid arrows indicate total embeddings, and dottedarrows indicate partial embeddings (with domain and codomain initial segmentsof the models in the figure). The vertical arrows depict ultrapowers by branchextenders; for example, the left-most depicts the ultrapower map correspondingto Ult n ( U α ; α i , ~E X γ α ) where n = m X α i , and we refer to i here (not i )as we are considering factoring τ . The diagram commutes, after restricting tothe relevant domains. 33 each X i +1 is a minimal inflation of X i ,– X is non- X -pending ( but X could have limit length or be X -pending ) . Then X is a minimalinflation of X , and things commute in a reasonable fashion. That is, let ( t ij , C ij , ( C − ) ij , f ij , (cid:10) Π ijα (cid:11) α ∈ C ij ) = ( t, C, . . . ) X i min X j for i < j ; we also use analogous notation for other associated objects. Let α < lh( X ) . If k < and α ∈ C k let α k = f k ( α ) . Then ( cf. Figure 3,which depicts a key case of the lemma ) :C1. If α ∈ C then α ∈ C , α ∈ C and α = f ( α ) = f ( f ( α )) = f ( α ) .C2. α ∈ ( C − ) and t ( α ) = 0 iff α ∈ ( C − ) and t ( α ) = 0 and α ∈ ( C − ) and t ( α ) = 0 .C3. Suppose α ∈ C and α ∈ C . Then: ( a ) If α + 1 = lh( X ) then α ∈ C . ( b ) If β ≤ f ( α ) and ξ ∈ I α ; β then γ α ; ξ ∈ C . ( c ) If β < f ( α ) and ξ = δ α ; β then δ α ; ξ ∈ C .C4. Suppose α ∈ C . Then: ( a ) If β ≤ α and γ = γ α ; β then γ α ; β = γ α ; γ and π α ; β = π α ; γ ◦ π α ; β . ( b ) S β ≤ α I α ; β ⊆ S β ≤ α I α ; β ⊆ C . ( c ) If β ≤ α and γ ∈ I α ; β then f ( γ ) ∈ I α ; β .C5. Suppose α ∈ C . For k < ℓ ≤ let:– γ kℓ = γ kℓα ℓ ; α k α ℓ – i kℓ = i kℓα ℓ ; α k α ℓ – τ kℓ = π kℓα ℓ ; α k i kℓ α ℓ : M X k α k i kℓ → M X ℓ α ℓ ( maybe γ = γ α ; γ ) . Then: ( a ) i ≤ i ( so ( M X α i , m α i ) E ( M X α i , m α i ) , with equality iff i = i ) . ( b ) i + i = i . ( c ) i = i γ ; α ; that is, i is the least i ′ such that γ ∈ I α ; α i ′ . ( d ) If i = i ( which holds iff i = 0 iff ( γ , α ] X ∩ D X deg = ∅ ) then τ = τ ◦ τ . e ) Suppose i < i ( which holds iff i > iff ( γ , α ] X ∩ D X deg = ∅ iff ( M X α i , m X α i ) ⊳ ( M X α i , m X α i )) . Then M X α i = U α ; α i α = Ult m X α i ( M X α i , ~F α ; α ) , and τ = τ ◦ π α ; α i α . ( f ) ~F α ; α = ~F α ; α ⊛ ~F α ; α (see Definitions 3.6 and 2.11). Proof of Lemma 6.1. By induction on lh( X ). Fix α + 1 < lh( X ), and supposethe lemma holds for X ↾ ( α + 1). Note first that the lemma for X ↾ ( α + 2)says the same things with respect to α ≤ α as does the lemma for X ↾ ( α + 1),except for C1 when α = α , as α ∈ dom( t X i min X ↾ ( α +2) ) for i = 0 , 1, but α / ∈ dom( t X i min X ′ ). So letting α = α , we just need to verify to verify C1 for α , and then verify the other parts for α + 1. We consider three cases. Case . α is X -copying and α is X -copying; that is, α ∈ ( C − ) and t ( α ) = 0 and α ∈ ( C − ) and t ( α ) = 0.We first establish part C2 for α . (Cf. Figure 3, which is related.) Note that γ ∈ C , by induction with part C3(b), applied with β = α ′ and ξ = δ α ; β = α . So by induction with X ↾ ( γ + 1), we have f ( γ ) = α ′ and (by partC5(f)) ~F γ ; γ = ~F α ; α ⊛ ~F γ ; γ . (3)Since t ( α ) = 0, we have ex X α = Q α ; α ′ = Ult (ex X α ′ , ~F α ; α ), so by line (3), Q = def Q γ ; γ = Ult (ex X α , ~F γ ; γ ) = Ult (ex X α ′ , ~F γ ; γ ) = Q γ ; γ . Moreover, since α = δ α ; α , ~E = def ~E X γ α is ( Q, X α = Ult ( Q, ~E ).But since t ( β ) = 1 for every β + 1 ∈ ( γ , α ] X , by induction t ( β ) = 1also, and it follows that α ∈ C and α ′ = f ( α ) and δ α ; α ′ = α and Q α ; α ′ = ex X α , establishing part C2.So let α = α ′ . It follows that α + 1 ∈ C ∩ C and α + 1 ∈ C and I ijα i +1; α j +1 = { α j + 1 } and f ij ( α j + 1) = α i + 1 for 0 ≤ i < j ≤ 2. This givespart C1 (for for X ↾ ( α + 2), with α + 1 replacing α ). Part C2 for α + 1 istrivial (for the reasons discussed at the start of the proof). Part C3 is clear byinduction. For part C4, use induction and the considerations just mentioned,and to see that π α +1; α +1 = π α +1; α +1 ◦ π α +1; α +1 , (4)just use that π ijα j +1; α i +1 is the ultrapower map i M X α , deg X α ~F where ~F = ~F ijα j +1; α j +1 = ~F ijα j +1; α j = ~F ijα j ; α j , which by part C5(f) gives line (4). These considerations also give part C5 for α + 1 (in this case, with notation as there, we have i = i = i = 0).35 ase . α is X -inflationary (that is, t ( α ) = 1).Part C2 for α : We claim t ( α ) = 1. For if α ∈ ( C − ) then by induction, α ∈ C and α ∈ C and f ( α ) = α , hence α ∈ ( C − ) , but then since X is non- X -pending, α + 1 < lh( X ) and lh( E X α ) ≤ lh( E α ), so α ∈ ( C − ) and (as t ( α ) = 1 and considering part C5(f), much as in the previous case)lh( E X α ) < lh( E α ) ≤ lh( E α ), so t ( α ) = 1.Let ξ = pred X ( α + 1).Part C1: Suppose α + 1 ∈ C . Then ξ ∈ C ; let ξ = f ( ξ ) and ξ = f ( ξ ), so also ξ ∈ C and ξ = f ( ξ ).Suppose ξ + 1 < lh( X ). Then ξ + 1 < lh( X ), since ξ ∈ C and X isnon- X -pending. So ex X ξ E Q ξ ; ξ , which implies Q ξ ; ξ E Q ξ ; ξ (again usingpart C5(f)), and since α + 1 ∈ C , E X α is ( Q ξ ; ξ , Q ξ ; ξ , α + 1 ∈ C , and f ( α + 1) = f ( ξ ) = ξ , and since ξ ∈ C and f ( ξ ) = ξ , this gives part C1.If instead ξ + 1 = lh( X ) then α + 1 / ∈ D X deg , and it is straightforward (andsimilar to before).Parts C3 and C4 for α + 1 are easy by induction.Part C5: Suppose α + 1 ∈ C . Then Π i α +1 is the minimal E X α -inflationof Π i ξ for i = 0 , 1, and part C5 at α + 1 follows that part at ξ . In particular,Figure 3 at stage α + 1 is derived easily from the corresponding figure at ξ , bysimply adding one further step of iteration above M ∗X α +1 E M X ξ , and regardingpart C5(f), we have ~F i α +1; α +1 = ~F i ξ ; ξ b h E i where E = E X α for i = 0 , 1, andby induction, ~F ξ ; ξ ⊛ ~F ξ ; ξ = ~F ξ ; ξ , and note that by normality of X (and that all inflated ~F ξ ; ξ -images of theextenders in ~F ξ ; ξ are either used along (0 , ξ ] X or are nested into some otherextender used along that branch), E is non-nested in the ⊛ -product ~F ξ ; ξ ⊛ ( ~F ξ ; ξ b h E i ) , so this ⊛ -product is just ( ~F ξ ; ξ ⊛ ~F ξ ; ξ ) b h E i = ~F α +1; α +1 . Case . α is X -copying but α is X -inflationary ( α ∈ ( C − ) and t ( α ) =0 and t ( α ) = 1).So α + 1 ∈ C and f ( α + 1) = α + 1 and γ α +1; α +1 = α + 1. Note t ( α ) = 1, so C2 holds for α . Let ξ i = pred X i ( α i + 1) for i = 1 , 2. Then ξ ∈ C and f ( ξ ) = ξ . Applying induction to stage ξ , and with calculationssimilar to before, we get that α + 1 ∈ C iff α + 1 ∈ C ; and if α + 1 ∈ C then, letting ξ = f ( ξ ) = f ( ξ ), we have f ( α + 1) = ξ = f ( α + 1).So part C1 holds.Parts C3 and C4 for α + 1 follow easily from the preceding remarks andinduction. Part C5: Suppose α + 1 ∈ C , so as discussed above, α + 1 ∈ C ,and ξ i ∈ C i for i = 1 , 2. Note i = 0 (notation as in C5). The diagram at α + 1 is given by adding a commuting square to the top of the diagram at ξ ,using E X α and E X α . Consider part C5(f). Since t i ( α i ) = 1 for i = 1 , 2, we have36 F iα i +1; α i +1 = ~F iξ i ; ξ i b (cid:10) E X i α i (cid:11) for i = 1 , 2. By induction, ~F ξ ; ξ = ~F ξ ; ξ ⊛ ~F ξ ; ξ ,so ~F α +1; α +1 = ( ~F ξ ; ξ ⊛ ~F ξ ; ξ ) b (cid:10) E X α (cid:11) , and letting ~F α ; α = ~F ξ ; ξ b ~F (note ~F ξ ; ξ E ~F α ; α ), note that cr( E ) < cr( E X α ) < cr( F ) for all E ∈ ~F ξ ; ξ and F ∈ ~F ,ex X α = Ult (ex X α , ~F ξ ; ξ b ~F ) , and all F ∈ ~F are nested in this product, so ~F α +1; α +1 ⊛ ~F α +1; α +1 = ( ~F ξ ; ξ ⊛ ~F ξ ; ξ ) b (cid:10) E X α (cid:11) = ~F α +1; α +1 , as desired.This completes the successor case. The limit case is analogous, and like inthe proof of [4, Lemma 6.2], so the reader should refer there. (cid:3) An easy consequence is (for terminology etc see Definition 3.35): Let X , X , X be as in 6.1. Suppose that X is X -terminaland X is X -terminal. Then X is X -terminal. Moreover, X is X -terminally-dropping iff either X is X -terminally-dropping or X is X -terminally-dropping.Moreover, if X is X -terminally-non-dropping then π ∞ = π ∞ ◦ π ∞ . Let M be m -standard and ~ X = hX α i α<λ a sequence of m -maximal trees on M . We say ~ X is a ( degree m , on M ) terminal minimal inflationstack iff X β is an X α -terminal minimal inflation of X α for all α < β < λ , andlh( X α ) a successor for each α < λ . If ~ X is a terminal minimal inflation stack, ~ X is continuous iff for each limit η < λ , X η is a minimal comparison of hX α i α<η . ⊣ Let ~ X be a continuous terminal minimal inflation stack. For ν < η < lh( ~ X ) , write C νη = C X ν min X η , etc. Let η < lh( ~ X ) be a limit. Then:1. For every ξ < lh( X η ) , there is ν < η such that for all α ∈ [ ν, η ) :(a) ξ ∈ C αη and ξ ∈ I αηξ ; f αη ( ξ )0 , and(b) if ξ + 1 < lh( X η ) then t αη ( ξ ) = 0 .2. There is ν < η such that X η is X ν -terminally-non-dropping (and hence, X η is X α -terminally-non-dropping for each α ∈ [ ν, η ) ).3. For all β < γ < lh( ~ X ) , Y γ is Y β -terminally-non-dropping iff Y α +1 is Y α -terminally-non-dropping for all α ∈ [ β, γ ) . For the first two parts we only use the continuity of the stack at η , not otherlimits. 37 roof. Part 1: The proof is by induction on ξ . In general it suffices to find ν witnessing part 1a, because then if ξ + 1 < lh( X η ) and we take ν ′ < η with t ν ′ η ( ξ ) = 0, then max( ν, ν ′ ) works, because by commutativity of inflation 6.1,then t αη ( ξ ) = 0 for all α ∈ [ ν ′ , η ). Part 1a for ξ = 0 is trivial. For ξ = ζ + 1,note that if ν witnesses both parts for ζ , then ν witnesses part 1a for ξ + 1. Forlimit ξ , let ξ ′ < X η ξ be such that ( ξ ′ , ξ ) X η ∩ D X η deg = ∅ and let ν be a witness for ξ ′ ; then ν also works for part 1a for ξ .Part 2: Since X η is to be X ν -terminal for all ν < η , this follows immediatelyfrom part 1a at stage ξ where ξ + 1 = lh( X η ).Part 3: If Y γ is Y β -terminally-non-dropping then it follows that for each α ∈ [ β, γ ), Y α +1 is Y α -terminally-non-dropping, by iterated application of Corollary6.2 to the stack ( Y β , Y α , Y α +1 , Y γ ). Conversely, supposing that Y α +1 is Y α -terminally-non-dropping for each α ∈ [ β, γ ), proceed by induction on η ∈ ( β, γ ]to show that Y η is Y β -terminally-non-dropping, again using the same corollary,together with part 2 to handle limits η . (cid:3) Let ~ X be a continuous terminal minimal inflation stack oflength λ . Write C νη , etc, as above. Let η < λ with η a limit and ξ < lh( X η ).Fix ν < η with ξ ∈ C νη . For α ∈ [ ν, η ] (note ξ ∈ C αη , by commutativity ofminimal inflation 6.1) let ξ α = f αη ( ξ ) (so ξ = ξ η and ξ α ∈ C να and ξ α = δ ναξ α ; ξ ν ).Let ν ≤ α ≤ β ≤ ε ≤ η . Now(i) If (*i) ξ η ∈ ( C − ) νη and t νη ( ξ η ) = 0, then let ω αβ = ω αβξ β ; ξ α ξ β : ex X α ξ α → ex X β ξ β (so ω αε = ω βε ◦ ω αβ ).(ii) If (*ii) ( γ νηξ η ; ξ ν , ξ η ] X η does not drop in model or degree (so neither does( γ αβξ β ; ξ α , ξ β ] X β ), then let π αβ = π αβξ β ; ξ α ξ β : M X α ξ α → M X β ξ β (so π αε = π βε ◦ π αβ ).Given such η, ξ, ν, h ξ α i ν ≤ α ≤ η , we say that ~ X is exit-good at ( η, ξ, ν ) iff, if ( ∗ i)holds then ex X η ξ η = dirlim ν ≤ α ≤ β<η (cid:16) ex X α ξ α , ex X β ξ β ; ω αβ (cid:17) and ω αη is the direct limit map for ν ≤ α < η, and model-good at ( η, ξ, ν ) iff, if ( ∗ ii) holds then M X η ξ η = dirlim ν ≤ α ≤ β<η (cid:16) M X α ξ α , M X β ξ β ; π αβ (cid:17) and π αη is the direct limit map for ν ≤ α < η. We say that ~ X is good iff ~ X is exit- and model-good at all such ( η, ξ, ν ). ⊣ Let ~ X = hX α i α<λ be a continuous terminal minimal inflationstack. Then ~ X is good. roof. We prove exit- and model-goodness at each ( η, ξ, ν ), by induction onlimits η < lh( ~ X ), with a sub-induction ξ < lh( X η ). So fix η and ξ < lh( X η ) and ν < η with ξ η = ξ ∈ C νη and adopt notation as in 6.3. Case . ξ = 0This case is trivial. Case . ξ = υ + 1.Suppose 6.3( ∗ ii) holds; that is, ( γ νηξ η ; ξ ν , ξ η ] X η does not drop in model or de-gree; we must verify model-goodness at ( η, ξ, ν ).Suppose first that in fact, ( ∗ ) υ ∈ ( C − ) νη and t νη ( υ ) = 0. Let υ α = f αη ( υ )for ν ≤ α ≤ η . Then by exit-goodness at ( η, υ, ν ),ex X η υ η = dirlim ν ≤ α ≤ β<η (cid:16) ex X α υ α , ex X β υ β ; ω αβυ β ; υ α υ β (cid:17) and ω αηυ η ; υ α υ η is the direct limit map for ν ≤ α < η. But by properties of minimal inflation, for ν ≤ α ≤ β ≤ η ,ex X β υ β = Ult (ex X α υ α , ~F αβυ β ; υ β ) and ω αβυ β ; υ α υ β is the ultrapower map , and letting k = deg X α ( ξ α ) (note k is independent of α ), M X β ξ β = Ult k ( M X α ξ α , ~F αβξ β ; ξ β ) and ~F αβξ β ; ξ β = ~F αβυ β ; υ β and π αβ is the ultrapower map . So let ¯ M = dirlim ν ≤ α ≤ β<η (cid:16) M X α ξ α , M X β ξ β ; π αβ (cid:17) , ¯ π α : M X α ξ α → ¯ M be the direct limit map and σ : ¯ M → M X η ξ η the map with σ ◦ ¯ π α = π αη for all α. (the latter existing since π βη ◦ π αβ = π αη ). Note that σ ↾ ν ( E X η υ η ) = id, andso by commutativity (and the degree of elementarity of the maps), therefore¯ M = M X η ξ η and σ = id and ¯ π α = π αη , which gives model-goodness at ( η, ξ, ν ).Next suppose instead that ( ∗ ) above fails. Let ν ′ ∈ ( ν, η ) be such that ( ∗ )holds at ν ′ . Still ξ ν ∈ C νν ′ and [ γ νν ′ ξ ν ; ξ ν ′ , ξ ν ′ ) X ν ′ does not drop in model or degree(and ξ ν ′ = δ νν ′ ξ ν ; ξ ν ′ ). By model-goodness at ( η, ξ, ν ′ ), it suffices to verify that π νη is an appropriate direct limit map. But by commutativity, π νη = π ν ′ η ◦ π νν ′ ,and since π ν ′ η is an appropriate direct limit map, so is π νη .Note that by model-goodness, for ν ≤ α < η such that 6.3( ∗ ii) holds for ν ,( † ) ~F αηξ η ; ξ η is derived from π αη . 39ow suppose that 6.3( ∗ i) holds, that is, ξ η ∈ ( C − ) νη and t νη ( ξ η ) = 0; wemust verify exit-goodness at ( η, ξ, ν ). If 6.3( ∗ ii) also holds, then exit-goodnessfollows from model-goodness, becauseex X η ξ η = Ult (ex X α ξ α , ~F αηξ η ; ξ η )and by ( † ), and because of the agreement between the direct limit maps relevantto exit-goodness with those relevant to model-goodness. Now suppose that6.3( ∗ ii) fails. Let ν ′ ∈ ( ν, η ) be such that 6.3( ∗ ii) holds at ( η, ξ, ν ′ ). Then bycommutativity, much as in the previous paragraph, model-goodness at ( η, ξ, ν ′ )implies exit-goodness at ( η, ξ, ν ). Case . ξ is a limit.Suppose that 6.3( ∗ ii) holds, that is, ( γ νηξ η ; ξ ν , ξ η ] X η does not drop in model ordegree; we must verify model-goodness at ( η, ξ, ν ).Suppose first that γ νηξ η ; ξ ν = ξ η . Then model-goodness at ( η, ξ, ν ) followseasily by induction, using the commutativity of the various maps and the factthat M X η ξ η is the direct limit under iteration maps of X η . Here is some moredetail. For ν ≤ α ≤ β ≤ ε ≤ η and π αβ as before, we have π αε = π βε ◦ π αβ , (5)and for υ ν < X ν ξ ν such that ( υ ν , ξ ν ] X ν does not drop in model or degree, and υ α = γ ναυ α ; υ ν , so υ β = γ αβυ β ; υ α and¯ π αβ = def π αβυ β ; υ α υ β : M X α υ α → M X β υ β , we have υ α < X α ξ α and ( υ α , ξ α ] X α does not drop in model or degree and π αβ ◦ i X α υ α ξ α = i X β υ β ξ β ◦ ¯ π αβ and ¯ π αε = ¯ π βε ◦ ¯ π αβ . By induction (using model-goodness) we also have M X η υ η = [ α<η rg(¯ π αη ) . But then since M X η ξ η = S ρ< X η ξ η rg( i X η ρξ η ), and Γ νηξ η “[0 , ξ ν ) X ν is cofinal in ξ η ,therefore M X η ξ η = [ α<η rg( π αη )and so also by line (5), π αη is the direct limit map, giving model-goodness at( η, ξ, ν ), as desired.Now suppose instead that γ νηξ η ; ξ ν < ξ η . If there is ν ′ ∈ ( ν, η ) such that γ ν ′ ηξ η ; ξ ν ′ = ξ η then we can deduce model-goodness at ( η, ξ, ν ) from model-goodnessat ( η, ξ, ν ′ ) as in the successor case. So suppose there is no such ν ′ . The40rgument here is fairly similar to the previous subcase. We have π αβ for ν ≤ α ≤ β ≤ η , commuting like before, and it suffices to see M X η ξ η = [ α<η rg( π αη ) . So let x ∈ M X η ξ η . Let υ η < X η ξ η and ¯ x be such that x = i X η υ η ξ η (¯ x ) and υ η = γ αηξ η ; ξ α for some α < η . Write υ β = γ αβξ β ; ξ α for α ≤ β ≤ η , so υ β ≤ X β ξ β and ( υ β , ξ β ] X β does not drop in model or degree, and υ ε = γ βεξ ε ; υ β for β ≤ ε ≤ η . Write¯ π βε = π βεξ ε ; υ β υ ε : M X β υ β → M X η υ ε . By induction (with model-goodness) we can fix β ∈ [ α, η ) and ¯¯ x such that¯ x = ¯ π βη (¯¯ x ). Then by commutativity, π βη ( i X β υ β ξ β (¯¯ x )) = i X η υ η ξ η (¯ π βη (¯¯ x )) = x, so x ∈ rg( π βη ), which suffices.The rest of the limit case is dealt with like in the successor case.This completes the proof. (cid:3) In this section we put things together to prove Theorem 1.1, and also Theorem7.2 below. The proof will in fact give an explicit construction of a specificsuch strategy Σ ∗ from Σ, and we denote this Σ ∗ by Σ stmin (or just Σ st for short,though it seems this may be in conflict with the notation in [4]). Given ~ T , X asabove, we denote X by X Σ ( ~ T ) (note that X is uniquely determined by ~ T , Σ). Let Ω > ω be regular. Let m ≤ ω , M be m -standard and Σ be an ( m, Ω) -strategy for M with minimal inflation condensation. Then thereis an optimal- ( m, < ω, Ω) -strategy Σ ∗ for M with Σ ⊆ Σ ∗ , such that for everystack ~ T = hT i i i Proof of Theorems 1.1, 7.2. We will construct an appropriate stacks strategyΣ ∗ for M , extending Σ. Modulo what we have already established regardingminimal inflation, the construction of the strategy is a simplification of theanalogous construction in [4]. We start with the successor case: converting a stack of two normal trees intoa single normal tree. Let T be an m -maximal tree on M of successor length < Ω,via Σ. Let N = M T∞ and n = deg T∞ . Note that we get a unique ( n, Ω)-strategy(( n, Ω + 1)-strategy respectively) Ψ = Ψ Σ T for N by demanding that whenever U is via Ψ with lh( U ) < Ω, there is a tree Y on M via Σ such that T min Y and U is the factor tree Y / T (note also that Y is determined uniquely by thisrequirement); and if Σ is an ( m, Ω + 1)-strategy and U has length Ω + 1, thenthere is likewise such a Y , except that now we can only demand that Y ↾ Ω + 1 isvia Σ (and Y is the T -unravelling of Y ↾ (Ω + 1), which has wellfounded modelsas cof(Ω) > ω ).We can repeat this process finitely often, and using Lemma 4.5 part 6 forthe commutativity etc, this yields Theorem 7.2.We now complete the proof of Theorem 1.1. Assume Σ is an ( m, Ω + 1)-strategy. We define an optimal-( m, Ω , Ω + 1) ∗ -strategy Σ ∗ for M . Given α < Ω,at the start of round α , neither player having yet lost, we will have sequences ~ T = hT β i β<α , ~ Y = hY β i β ≤ α such that:S1. ~ T is an optimal m -maximal stack on M ,S2. ~ Y is a continuous terminal minimal inflation stack of degree m on M , witheach Y β via Σ,S3. for each β < α , lh( T β ) and lh( Y β ) are successors < Ω,S4. for each β < α , T β is the factor tree Y β +1 / Y β ,S5. M ~ T ↾ α ∞ is well-defined and = M Y α ∞ , deg ~ T ↾ α ∞ = deg Y α ∞ , [ b ~ T ↾ α ∩ D ~ T ↾ α deg = ∅ iff b Y α ∩ D Y α deg = ∅ ], and if b ~ T ↾ α ∩ D ~ T ↾ α deg = ∅ then i ~ T ↾ α ∞ = i Y α ∞ .S6. for all β < γ ≤ α , we have b ~ T ↾ [ β,γ ) ∩ D ~ T ↾ [ β,γ )deg = ∅ ⇐⇒ Y γ is Y β -terminally-non-dropping , and if b ~ T ↾ [ β,γ ) ∩ D ~ T ↾ [ β,γ )deg = ∅ then i ~ T ↾ [ β,γ ) = ( π ∞ ) Y β min Y γ . The absorption maps ̺ α and ς α of [4] reduce here to the identity, which renders certainissues in [4] trivial. T α is produced of lengthΩ + 1, in which case the game is over and player II has won), by the discussionabove and again using Lemma 4.5, Lemma 6.4, by which Y γ is Y β -terminally-dropping iff there is some α ∈ [ β, γ ) such that Y α +1 is Y α -terminally-dropping,and Lemma 6.2, by which π β,γ +1 ∞ = π γ,γ +1 ∞ ◦ π βγ ∞ when Y γ +1 is Y β -terminally-non-dropping.So suppose η < Ω is a limit and we have produced hT α i α<η and hY α i α<η .We must produce Y η and verify the conditions. Let Y η be the unique minimalcomparison of hY α i α<η via Σ. This exists and has length λ + 1 < Ω by Lemma5.2. By Lemma 6.4, there is ν < η such that Y γ is Y β -terminally-non-droppingfor all β, γ such that ν ≤ β < γ ≤ η . So by induction, for all β ∈ [ ν, η ), b T β doesnot drop in model or degree. And by Lemma 6.6, hY α i α ≤ η is good, so M Y η ∞ = dirlim ν ≤ β ≤ γ<η (cid:16) M Y β ∞ , M Y γ ∞ ; π βγ ∞ (cid:17) = dirlim ν ≤ β ≤ γ<η (cid:16) M ~ T ↾ β ∞ , M ~ T ↾ γ ∞ ; i ~ T ↾ [ β,γ ) (cid:17) = M ~ T ↾ η ∞ , and π βη ∞ = i ~ T ↾ [ β,η ) is the direct limit map.This completes the construction and analysis of the strategy through Ωrounds. Finally for the limit stage Ω, because Ω is regular (in fact cof(Ω) > ω suffices), note that player II wins (but in this case we do not try to define anytree Y Ω ). (cid:3) Let Ω > ω be regular. Let M be m -standard and Σ be an ( m, Ω + 1)-strategyfor M . Let T , T be trees on M according to Σ, each of successor length < Ω.Let N i = M T i ∞ and n i = deg T i ∞ . Let Σ i be the ( n i , Ω + 1)-strategy for N i whichis just the second round of Σ stmin following T i . We now analyze the comparisonof ( N , N ) via (Σ , Σ ). Let ( U , U ) be this comparison, with padding asusual (such that if α + 1 < β + 1 < lh( U , U ) and E U i α = ∅ 6 = E U − i β then e ν ( E U i α ) < e ν ( E U − i β )).Let X be the minimal comparison of ( T , T ). So T i min X for i = 0 , C i = C T i min X for i = 0 , 1. For each α < lh( X ) we have α ∈ C ∪ C andif α + 1 < lh( X ) then t i ( α ) = 0 for i = 0 or i = 1.Now we claim that U i is the factor tree X / T i for i = 0 , 1. For given X ↾ ( α +1)where α + 1 < lh( X ), suppose E X α = E T min X α . Then α ∈ ( C − ) and if α ∈ ( C − ) then lh( E T min X α ) ≤ lh( E T min X α ). But if λ = def lh( E T min X α ) =lh( E T min X α ) then λ is a cardinal in the corresponding models M U i β (for theappropriate β ) and M U β | λ = M U β | λ = (ex X α ) pv . λ = def lh( E T min X α ) < lh( E T min X α ) then λ is a cardinal M U β and M U β | λ = (ex X α ) pv , but M U β | λ = ex X α , so E U β = ∅ and E U β = E X α .These considerations easily lead to the fact that U i = X / T i .It is easy to see that the same argument works for an arbitrary collection ofiterates (given we have enough iterability for the comparison).We now mention a simple corollary. Suppose M = M ∈ L [ x ], where x ∈ R . Suppose T , T are ω -maximal trees on M , both are maximal, in thesense that they have limit length and δ ( T i ) is Woodin in L [ M ( T i )], and bothare countable in L [ x ]. Woodin has asked whether the pseudo-comparison of( L [ M ( T )] , L [ M ( T )]) terminates in countably many steps in L [ x ]. It seems onemight hope to use the analysis of comparison above to answer this questionaffirmatively. There is a simple case where this does work: Suppose M ∈ L [ x ] where x ∈ R and T , T are as in theprevious paragraph, and lh( T ) = lh( T ) = ω. Then the pseudo-comparison of ( L [ M ( T )] , L [ M ( T )]) lasts exactly ω manysteps. Proof. Let X be the minimal comparison of ( T , T ). It suffices to see X lastsonly ω many steps. Suppose not. Then note that there is i < ω ∈ C i and f i ( ω ) = ω . Say i = 0. But then by the maximality of T , X is maximal, which implies the pseudo-comparison has finished at stage ω , acontradiction. (cid:3) It follows that the collection of such trees (maximal of length ω ) is closedunder comparison of pairs and Neeman genericity iteration. But note that thedirect limit M ∞ of all such iterates of M is not ⊆ HOD L [ x ] , because the leastmeasurable of M ∞ is < ω L [ x ]1 . ***This to be added (very similar to [4]). 10 Properties of Σ stmin***This to be added. 44 eferences [1] Ronald Jensen. Manuscript on fine structure, in-ner model theory, and the core model below oneWoodin cardinal. Forthcoming book, draft available at .[2] Farmer Schlutzenberg. Background construction for λ -indexed mice.arXiv:2101.00889.[3] Farmer Schlutzenberg. Fine structure from normal iterability. arXiv:2011.10037.[4] Farmer Schlutzenberg. Iterability for (transfinite) stacks. To appear inJournal of Mathematical Logic. arXiv:1811.03880.[5] Farmer Schlutzenberg. Measures in mice . PhD thesis, University of Cali-fornia, Berkeley, 2007. arXiv: 1301.4702.[6] Farmer Schlutzenberg. The definability of E in self-iterable mice. arXiv:1412.0085, 2014.[7] Farmer Schlutzenberg. A premouse inheriting strong cardinals from V . 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