aa r X i v : . [ m a t h . L O ] J a n Background construction for λ -indexed mice Farmer Schlutzenberg ∗† WWU M¨unsterJanuary 12, 2021
Abstract
Let M be a λ -indexed (that is, Jensen indexed) premouse. We provethat M is iterable with respect to standard λ -iteration rules iff M is iter-able with respect to a natural version of Mitchell-Steel iteration rules. Us-ing this equivalence, we describe a background construction for λ -indexedmice, analogous to traditional background constructions for Mitchell-Steelindexed mice, and which absorbs Woodin cardinals from the backgrounduniverse.We also prove some facts regarding the correspondence between stan-dard iteration trees and u-iteration trees on premice with Mitchell-Steelindexing; these facts were stated and used without proof in [6]. There are two standard forms of fine structural premice commonly used in the in-ner model theoretic literature: those with
Mitchell-Steel indexing (MS-indexing,MS-premice) , and those with λ -indexing or Jensen indexing (J-indexing, J-premice) . Let M be an active premouse of either kind, F = F M the activeextender of M , κ = cr( F ) its critical point, ν = ν ( F ) the strict sup of genera-tors, and λ = λ ( F ) = i MF ( κ ) where i MF : M → U = Ult( M, F ) is the ultrapowermap. We have ν ≤ λ . If M is an MS-premouse then OR M = ( ν + ) U , whereas if M is a J-premouse then OR M = ( λ + ) U . Also, iteration trees on these premiceare usually formed according to different rules: let T be a normal tree on M . If M is MS-indexed, this conventionally mean that pred T ( α +1) is the least β suchthat cr( E T α ) < ν ( E T β ), whereas if M is J-indexed, it conventionally means thatpred T ( α + 1) is the least β such that cr( E T α ) < λ ( E T β ). Let us call these tworules MS-rules and
J-rules respectively. If M is J-indexed, it also makes senseto form trees with MS-rules, although this is not normally done. (Actually, we ∗ Partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun-dation) under Germany’s Excellence Strategy EXC 2044-390685587, Mathematics M¨unster:Dynamics–Geometry–Structure. † [email protected], https://sites.google.com/site/schlutzenberg L [ E ]-constructions for MS-premice absorb all Woodincardinals from the background universe R (assuming R is sufficient iterabilityin some larger universe), whereas constructions for J-premice require strongerlarge cardinals in R in order to reach Woodin cardinals. This is because in orderto get iterability with respect to J-rules, the traditional methods have requiredthat when an active J-premouse M has F M induced by background extender F ∗ , F ∗ should be at least λ ( F M )-strong; this ensures in particular that theusual procedure for lifting trees on M to trees on R makes sense.Also of relevance is Fuchs’ translation between MS-mice and J-mice in [2],and respective iteration strategies for them in [1]. His translation of iterationstrategies works, however, only with strategies for MS-rules, and thus does notgive the translation between the standard forms of iterability of the respectiveforms of mice.In this note we show that for k -sound J-premice, for example, ( k, ω + 1)-iterability with respect to J-rules is equivalent to that with respect to a naturalform of MS-rules ( lifting-MS-rules ), and in fact, there is a one-to-one correspon-dence between witnessing strategies. In fact, even without iterability, there isin fact a direct translation between iteration trees of the two kinds, given thatthey satisfy some slight extra properties (which follow immediately if they areaccording to ( k, ω + 1)-strategies). Moreover, the correspondence extends eas-ily to k -maximal stacks. A straightforward variant also establishes equivalencewith strategies for MS-rules as described earlier.Using this, we then define a form of backgrounded L [ E ]-construction forJ-premice, and show that it absorbs Woodin cardinals from the backgrounduniverse, like that for MS-premice. The method for the background constructionis similar to that in [7, § See [6] for a guide to notation, and in particular [6, §
2] for background on u-finestructure (which is only relevant in § MS-premouse or is
MS-indexed iff it is a premouse withMitchell-Steel indexing, but allowing superstrong extenders as discussed in [6]. The basic idea for the tree conversion process for J-trees was observed by the author in2014 or slightly earlier. After mentioning the basic method to Steel and Schindler during alunch at the
A2 am See in mid 2015, it seemed apparent that it had escaped notice prior tothat point, and in particular was not covered in Fuchs [1]. So it seemed it should be writtenup. The variant for MS-trees was observed in 2015, and the background construction forJ-mice during 2018.
J-premouse or is
J-indexed iff it has Jensen indexing (alsoknown as λ -indexing). A seg-pm M is internally MS-indexed iff M pv is MS-indexed, and is internally J-indexed iff M pv is J-indexed. We write ι ( M ) =max( ν ( F M ) , lgcd( M )) for an active seg-pm M .For normal iteration trees T (where normality depends on context), we write e ν T β for the exchange ordinal associated to M T β ; that is, pred T ( α + 1) is the least β such that cr( E T α ) < e ν T β .We will often deal with padded fine structural trees T . When E T α = ∅ , wealways set pred T ( α + 1) = α , M T α +1 = M T α , deg T α +1 = deg T α , and i T α,α +1 = id. Let M be an m -sound J-premouse. An J- m -maximal tree T on M is one formed with the usual exchange ordinals for iterating such mice( e ν T α = λ ( E T α )), and otherwise, the usual conditions for m -maximality. A J- ( m, Ω + 1) -iteration strategy for M is an ( m, Ω + 1)-iteration strategy for M with respect to such trees. A weakly-MS- m -maximal tree T on M is likewise,but with exchange ordinals e ν T α = ν ( E T α ). An weakly-MS- ( m, Ω + 1) -iterationstrategy for M is one for such trees. ⊣ Let us start with describing the main point of the tree conversion processto be presented. Let U be a weakly-MS- m -maximal tree on an m -sound J-premouse M . Let ν = ν ( E U ) and λ = λ ( E U ) and κ = cr( E U ), and supposethat ν ≤ κ < λ and 2 / ∈ D U . Then pred U (2) = 1, and note that in fact 2 / ∈ D U deg ,since letting k = deg U (1), we have λ < ρ k ( M U ). Now suppose we form a J- m -maximal tree T with E T = E U and E T = E U . Then we would instead setpred T (2) = 0, and noting that ρ (ex T ) ≤ ν (where ex T α = M T α | lh( E T α )), we get M ∗T = ex T and deg T = 0, so M T = Ult (ex T , E T ). But (ex T ) pv is a cardinalsegment of M U and lh( E T ) ≤ ρ k ( M U ), and it follows that Ult (ex T , E T ) is acardinal segment of M U , and moreover, i ∗T ⊆ i ∗U : M U → M U . Also, a standard computation shows that F ( M T ) is equivalent to the two-step iteration given by ( E T , E T ) = ( E U , E U ). (That is, forming a degree k ultrapower U with F ( M T ) gives the same result as first forming an intermediatedegree k ultrapower U ′ with E T , then forming U as the degree k ultrapowerof U ′ with E T , including agreement of the overall ultrapower maps.) So let T ′ = T b F ( M T ) (formed as a J- m -maximal tree). Then note that M ∗T ′ = M ∗U (since F ( M T ) and E T = E U have the same critical point and measure space)and deg T ′ = deg U = deg U = k , and so M T ′ = M U and i ∗T ′ = i U ◦ i ∗U .The correspondence described above is the key component of the conversionprocess in general. In more generality, the nesting of extenders (like how E T sitsinside F ( M T ) above) can be arbitrarily finitely deep, and we need to keep trackof transfinite concatenations of such things, and the resulting modifications in3ree order between an (appropriate) J- m -maximal tree T and the corresponding U . The entire conversion process is explicit and locally computed (in bothdirections, T to U and U to T ).We will actually not discuss the process explicitly for weakly-MS- m -maximaltrees, but instead for another form, lifting-MS- m -maximal trees, which arisenaturally in the background construction. But a very similar process can alsobe used for weakly-MS- m -maximal trees. For the conversion between J-trees T and lifting-MS-trees U , the disagreements in tree order and models etc also arisein a slightly different fashion (because U uses extra extenders, which appear inthe dropdown sequence of extenders used in T ). But the basic point of thecorrespondence is still that described above. The rest of the work is mostly amatter of bookkeeping, but somehow when one tries to write things down indetail, that bookkeeping seems to run to some length. Let M be a J-pm and N E M with N active.The mod-dropdown of ( M, N ) is the sequence h N i i i ≤ k , with k < ω as largeas possible, such that N = N and N i +1 is the least N ′ such that N i ⊳ N ′ E M and either (i) N ′ ⊳ M and ρ N ′ ω < ρ N i ω , or (ii) N ′ = M .Let h N i i i ≤ k be the mod-dropdown of ( M, N ). The reverse-e-dropdown ( rev-e-dropdown ) of ( M, F N ) is the sequence h E i i i ≤ n enumerating all extenders ofthe form F N i where i ≤ k and N i is active and if i > ν ( F N i ) < ρ N i − ω ,in order of decreasing index OR N i . ⊣ Let h E i i i ≤ n be the rev-e-dropdown of ( M, E ) (so E = E n ). Then:1. Let i < n . Let N ′ be the largest element of the mod-dropdown of ( M, E n ) such that N ′ ⊳ M | lh( E i ) (so M | lh( E i +1 ) E N ′ ). Then ν ( E i ) < λ ( E i ) ≤ ρ N ′ ω ≤ ν ( E i +1 ) . In fact, writing ν k = ν ( E k ) and λ k = λ ( E k ) and lh k = lh( E k ) , either:– ν i < λ i = ρ N ′ ω ≤ ρ M | lh i +1 ω ≤ ν i +1 ≤ λ i +1 < lh i +1 ≤ OR N ′ < lh i , or– ν i < ρ N ′ ω ≤ ρ M | lh i +1 ω ≤ ν i +1 ≤ λ i +1 < lh i +1 ≤ OR N ′ < λ i < lh i .2. For each i ≤ n , the reverse-e-dropdown of ( M, E i ) is h E k i k ≤ i .3. Let i < n and U = Ult k ( N, E i ) , where N is a k -sound J-premouse and E i an N -extender with cr( E i ) < ρ Nk , and if N is active then cr( E i ) < λ ( F N ) .Suppose U is wellfounded. Then the reverse-e-dropdown of ( U, E n ) is:(a) (cid:10) F U (cid:11) b h E k i i +1 ≤ k ≤ n , if N is active and ν ( F N ) ≤ cr( E i ) ,(b) h E k i i +1 ≤ k ≤ n , otherwise. Proof.
This is basically a straightforward consequence of the definitions, and leftto the reader. In part 3, if N is active with ν ( F N ) ≤ cr( E i ), use coherence andthat lh( E i ) is a U -cardinal to get appropriate agreement of the mod-dropdowns4f M and U , and use that ν ( F U ) = ν ( E i ), which is by a standard calculation (seefor example [4, Lemma 2.11***]). If instead N is active with cr( E i ) < ν ( F N ),and γ is a generator of F N , then j ( γ ) is a generator of F U , where j : N → U isthe ultrapower map, and hence ν ( F U ) ≥ j “ ν ( F N ) > λ ( E i ) ≥ ρ N ′ ω , where N ′ isas in part 1. Note N ′ is in the mod-dropdown of ( U, E n ), so F U is not in therev-e-dropdown in this case. (cid:3) Let M be an m -sound J-pm and T a putative iteration treeon M . We say that T is lifting-MS- m -maximal iff there is an ordinal λ > h η α i α<λ , h n α i α +1 <λ such that:1. η = 0 and h η α i α<λ is continuous.2. n α < ω and η α +1 = η α + n α + 1.3. If λ = α + 1 then n α = 0.4. T has length η = sup α<λ ( η α + 1).5. If γ + 1 < lh( T ) then E T γ ∈ E + ( M T γ ).6. If α + 1 < λ then (cid:10) E T η α , E T η α +1 , . . . , E T α α + n α (cid:11) is the rev-e-dropdown of( M T η α , E T η α + n α ).7. If α + 1 < β + 1 < λ then lh( E T η α + n α ) < lh( E T η β + n β ).8. If γ + 1 < lh( T ) then pred T ( γ + 1) is the least δ such that cr( E T γ ) <ν ( E T δ ), and M ∗T γ +1 E M T δ and deg T ( γ + 1) are determined as usual for m -maximality. ⊣ Note that given T ↾ ( η α + 1), we can choose any E ∈ E + ( M T η α )such that lh( E T η β + n β ) < lh( E ) for all β < α , and then extend T by satisfyingcondition 6 with E T η α + n α = E , given that the resulting ultrapowers are well-founded. In particular, E T η α + i ∈ E + ( M T η α + i ) for each i ≤ n α , by coherence.Also note that by 2.3 and coherence, for all j, k with j ≤ k ≤ n α , (cid:10) E T η α + j , E T η α + j +1 , . . . , E T η α + k (cid:11) is a tail segment of the rev-e-dropdown of ( M T η α + j , E T η α + k ). It follows fromLemma 2.6 below that it is actually the full rev-e-dropdown. Let M be an m -sound J-premouse and T be lifting-MS- m -maximalon M . Then:1. ν ( E T γ ) < ν ( E T δ ) for all δ + 1 < γ + 1 < lh( T ) .2. Let γ < lh( T ) be such that F = F M T γ = ∅ . Then λ ( E T η α + n α ) < ν ( F ) forall α such that η α + n α < γ , and if γ = δ + 1 then λ ( E T δ ) < ν ( F ) .3. For all γ + 1 < lh( T ) , if M ∗T γ +1 is active then cr( E T γ ) < ν ( F ( M ∗T γ +1 )) . roof. We prove that the lemma holds for T ↾ ξ by induction on ξ . Supposethat it holds for T ↾ ξ ; we prove it for T ↾ ξ + 1.If ξ is a limit then we just have to verify part 2 holds when γ = ξ , but thisis easy.So suppose ξ = γ + 1.Part 1: Let δ < γ . If there is α such that δ, γ ∈ [ η α , η α + n α ], then ν ( E T δ ) <ν ( E T γ ) by definition of the reverse-e-dropdown. So we may assume that δ = η α + n α for some α . Then lh( E T δ ) is a cardinal of M T γ and lh( E T δ ) < lh( E T γ ).So if E T γ = F ( M T γ ) then easily ν ( E T δ ) < lh( E T δ ) ≤ ν ( E T γ ), so we are done. Andif E T γ = F ( M T γ ) then by induction part 2, we have ν ( E T δ ) ≤ λ ( E T δ ) < ν ( E T γ ),as required.Part 3: Suppose not. Let δ = pred T ( γ + 1) and κ = cr( E T γ ). So ν ( F ( M ∗T γ +1 )) ≤ κ < ν ( E T δ ) , so ex T δ ⊳ M ∗T γ +1 . There is no N such that ex T δ E N ⊳ M ∗T γ +1 and ρ Nω ≤ κ . Butthen it follows that F ( M ∗T γ +1 ) is in the reverse-e-dropdown of ( M T δ , ex T δ ).Now fix α such that δ ∈ [ η α , η α + n α ]. If δ = η α , then by 2.3, we shouldhave used F ( M ∗T α +1 ) in T before E T δ , contradiction. So δ = η α + i + 1 for some i + 1 ≤ n α . Let F = E T η α + i . Then λ ( F ) and lh( F ) are cardinals of M T δ , and wecan’t have lh( F ) < OR( M ∗T γ +1 ) by coherence. But since ρ ( M ∗T γ +1 ) ≤ ν ( M ∗T γ +1 ) < lh( F ) , it follows that M ∗T γ +1 = M T δ . By induction, λ ( F ) < ν ( F ( M T δ )). So λ ( F ) < ν ( F ( M ∗T γ +1 )) ≤ κ < lh( E T δ ) < lh( F ) . But lh( F ) = ( λ ( F ) + ) M T δ , so there is some N such thatex T δ E N ⊳ M T δ | lh( F )and ρ Nω ≤ κ . So OR( M ∗T γ +1 ) < lh( F ), contradiction.Part 2 follows from part 3 as usual. (cid:3) Let M be an m -sound J-pm. A lifting-MS- ( m, ω + 1)-strategyfor M is a winning strategy for player II in the iteration game producing (pu-tative) lifting-MS- m -maximal trees, in which, given T ↾ ( η α + 1) (notationas above), player I chooses E ∈ E + ( M T η α ) with lh( E T η β + n β ) < lh( E ) for all β < α , then T is extended with the reverse-e-dropdown of ( M T α , E ), producing T ↾ ( η α +1 + 1); and player II chooses cofinal branches at limit stages. Player IImust ensure every model produced is wellfounded, in order to win.We say that M is lifting-MS- ( m, ω +1) -iterable iff there is a such an iterationstrategy for M . ⊣ Let Ω > ω be regular. Let M be an m -sound J-pm. Then thefollowing are equivalent: M is J- ( m, Ω + 1) -iterable,– M is weakly-MS- ( m, Ω + 1) -iterable,– M is lifting-MS- ( m, Ω + 1) -iterable.Moreover, there is a one-to-one correspondence between strategies of one kindand those of another, and given such a pair (Λ , Σ) of strategies, there is a direct(and local) translation between trees T via Λ and corresponding trees U via Σ . The precise meaning of statements at the end is clarified in what follows.
Let T be a J- m -maximal tree on an m -sound J-premouse. Let α < lh( T ). We say α is T -special iff M T α is active and there is β + 1 ≤ T α suchthat E T β = ∅ and ( β + 1 , α ] T ∩ D T = ∅ and ν ( F ( M ∗T β +1 )) ≤ cr( E T β ). We say α is T -very-special ( T -vs ) iff T -special and E T α = F ( M T α ). If α is T -special, and β is the least witness, let C ( F ( M T α )) denote F ( M ∗T β +1 ).Let E ∈ E + ( M T α ). Let h E i i i ≤ n be the rev-e-dropdown of ( M T α , E ). The T -despecialized-reverse-e-dropdown ( T -despec-rev-e-dropdown ) of ( M T α , E ) is h E i i ≤ i ≤ n ,if α is T -special, and is E i = F ( M T α ), and is h E i i i ≤ n otherwise. ⊣ Note α < ext , T β = ⇒ α < β . So < ext , T is wellfounded. Let T be a J- m -maximal tree on an m -sound J-premouse. Let α be T -special, β the least witness and γ = pred T ( β + 1) . Let F = C ( F ( M T α )) = F ( M ∗T β +1 ) . Then:1. F is in the reverse-e-dropdown of ( M T γ , E T γ ) , which is identical with the T -despecialized-reverse-e-dropdown of ( M T γ , E T γ ) , and γ is the unique suchordinal.2. ν ( F ) ≤ cr( E T β ) < min( λ ( E T γ ) , λ ( F )) ,3. deg T β +1 = deg T α = 0 ,4. F ( M T α ) is equivalent to the composition of the extenders h F i b (cid:10) E T δ (cid:11) δ +1 ∈ [ β +1 ,α ] T and for δ , δ ∈ [ β + 1 , α ] T with δ < δ , we have λ ( E T δ ) ≤ cr( E T δ ) . Proof sketch.
Part 1: It is a standard fact (and easy to see) that M ∗T β +1 is in themod-dropdown of ( M T γ , E T γ ), so this part follows easily from the definitions.Part 2: The fact that ν ( F ) ≤ cr( E T β ) < λ ( E T γ ) is by definition, and the factthat cr( E T β ) < λ ( F ) is standard.Part 4 is as in [4, Lemma 2.27***] (extended to transfinite iterations in aroutine manner). (cid:3) So when T -despecializing, we just remove E from the rev-e-dropdown if E = F ( M T α )and α is T -special. .11 Definition. Let T be a padded J- m -maximal tree on an m -sound J-premouse M . We say that T is nicely padded iff there is an ordinal λ > h η α i α<λ , h n α i α +1 <λ such that:1. η = 0 and h η α i α<λ is continuous and T has length η = sup α<λ ( η α + 1).2. If α + 1 < λ then n α < ω and η α +1 = η α + n α + 1.3. If η α is T -vs or α + 1 = λ then n α = 0.4. Let α +1 < λ with η α non- T -vs. Then n α >
0. Moreover, let E = E T η α + n α .Then:– E = ∅ and e ν T η α + n α = λ ( E ).– E T η α + i = ∅ and pred T ( η α + i + 1) = η α + i for each i < n α (so M T η α = M T η α + n α and deg T η α = deg T η α + n α ).– The T -despec-rev-e-dropdown of ( M T η α , E ) has length n α ; let it be (cid:10) G T η α , G T η α +1 , . . . , G T η α + n α − (cid:11) , so G T η α + n α − = E .– e ν T η α + i = ν ( G T η α + i ) for each i < n α . Write η T α = η α , n T α = n α , n ′T α = n ′ α .We say that ( M, m, T ) is suitable iff M is an m -sound J-premouse and T isa nicely padded J- m -maximal tree on M . ⊣ Let (
M, m, T ) be suitable. Let α + 1 < lh( T ) with E T α = ∅ ,so α = η ξ + n ξ for some ξ (with notation as above). If α is T -vs and β is leastwitnessing that α is T -special, then let e α denote pred T ( β + 1) − T ( β + 1) = η χ + i + 1 for some χ, i ), and let C ( E T α ) denote F ( M ∗T β +1 ) (also by that lemma, F ( M ∗T β +1 ) = G T η χ + i ).If α is non- T -vs (so n ξ >
0) then let e α = η ξ + n ξ − C ( E T α ) = E T α = G T e α ). ⊣ We make some basic observations on the interaction between T -special or-dinals and nice pudding: Let ( M, m, T ) be suitable and adopt notation as above. Let η + 1 < lh( T ) with E T η = ∅ . Then:1. Suppose pred T ( η + 1) = η α + i + 1 . Then:(a) M ∗T η +1 E M T η α | lh( G T η α + i ) .(b) η + 1 is T -special iff M ∗T η +1 = M T η α | lh( G T η α + i ) .(c) If η + 1 is T -special then: So if ν ( E ) = λ ( E ) then e ν T η α + n α − = ν ( E ) = λ ( E ) = e ν T η α + n α . In this situation it wouldhave been in some ways more natural to make E T η α + n α − = E instead of E T η α + n α − = ∅ and E T η α + n α = E , but for uniformity of notation it turns out simpler to keep the separation. C ( F ( M T η +1 )) = G T η α + i .– If η + 1 / ∈ D T then i = 0 and G T η α = F ( M T η α ) and η α is non- T -special.2. Suppose pred T ( η + 1) = η α . Then:(a) The following are equivalent:(i) η + 1 is T -special,(ii) η + 1 / ∈ D T and M T η α is active and ν ( F ( M T η α )) ≤ cr( E T η ) ,(iii) η α is T -special and η + 1 / ∈ D T .(b) If η + 1 is T -special then C ( F ( M T η +1 )) = C ( F ( M T η α )) . Proof sketch.
Suppose pred T ( η + 1) = η α . A short unravelling of definitionsshows (i) ⇔ (ii). Suppose η + 1 is T -special; let us observe that η α is T -special.Let F = F ( M T η α ). By (ii), η + 1 / ∈ D T and F = ∅ and ν ( F ) ≤ cr( E T η ). Itfollows easily that F is in the rev-e-dropdown of ( M T η α , E T η α ). But then if η α is non- T -special, the T -despec-rev-e-dropdown is just the rev-e-dropdown, so G T η α = F , so e ν T η α = ν ( F ) ≤ cr( E T η ), so pred T ( η + 1) > η α , contradiction. (cid:3) Let (
M, m, T ) be suitable. For γ + 1 , α + 1 < lh( T ) suchthat E T γ = ∅ 6 = E T α , write γ ≤ ext , T direct α iff either γ = α or α is T -vs and γ + 1 ∈ ( e α + 1 , α ] T . Let ≤ ext , T be the transitive closure of ≤ ext , T . Let < ext , T direct and < ext , T be the strict parts.The standard decomposition of E T α is the enumeration of { C ( E T β ) (cid:12)(cid:12) β ≤ ext , T α } , in order of increasing critical point. ⊣ Note here that if β < ext , T β then β < β but cr( C ( E T β )) < cr( C ( E T β )). Let ( M, m, T ) be suitable and α + 1 < lh( T ) .The standard decomposition of E T α is well-defined. That is, if β , β ≤ ext , T α and β = β then κ β = κ β where κ β i = cr( C ( E T β i )) = cr( E T β i ) ; moreover, if β = α then κ β < κ β , and if κ β < κ β then ν ( C ( E T β )) ≤ κ β .Further, E T α is equivalent to composition of the extenders in the standarddecomposition of E T α (in order of increasing critical point). Proof.
By induction using Lemma 2.10, again as in [4, Lemma 2.27***]. (cid:3)
Let (
M, m, T ) be suitable. Let α < lh( T ). Given γ ≤ T α , ~E T γα denotes the sequence D E T β E γ< T β +1 ≤ T α (so ~E T γα corresponds to i T γα whenthe latter exists), and ~F T γα denotes the sequence D E T β E β +1 ∈ [ ξ +1 ,α ] T , where ξ isleast such that γ < T ξ + 1 ≤ T α and ( ξ + 1 , α ] T does not drop in model. Write ~E T <α = ~E T α and ~F T <α = ~F T α . 9iven a sequence ~E = h E α i α<λ of short extenders, we define h U α , k α i α ≤ λ ,if possible, by induction on λ , as follows. Set U = M and k = k . Given U η and k η ≤ ω are well-defined and U η is a k η -sound seg-pm and η < λ ,then: if cr( E η ) < OR( U η ) and there is N E U η such that E η measures exactly P ([ κ ] <ω ) ∩ N , then letting N E U η be the largest such, and letting n ≤ ω belargest such that ( N, n ) E ( U η , k η ) and cr( E η ) < ρ Nn , if such n exists, then U η +1 = Ult n ( N, E η ) and k η +1 = n . We say there is a drop in model at η + 1 iff N ⊳ U η . Given a limit η such that U α is well-defined for each α < η , then U η is well-defined iff there are only finitely many drops in model < η , and then U η is the resulting direct limit and k η = lim α<η k α . We define Ult k ( M, ~E ) = U λ ,if this is well-defined, and if so, and there is no drop in model, we define the iteration map i M,k~E : M → U λ resulting naturally from the ultrapower maps.Also, if there is no drop in model, or the only drop in model occurs at 1, thendefine ¯ i M,k~E : N → U λ where N E M is as above. ⊣ Let (
M, m, T ) be suitable and β ≤ T α < lh( T ). Then clearly M T α = Ult k ( M T β , ~E T βα ) and i T βα = i M T β ,k~E T βα where k = deg T β (with one map definediff the other is), and likewise i ∗T γ +1 ,α = ¯ i M T β ,k~E T βα when pred T ( γ + 1) = β . Let (
M, m, T ) be suitable.We say that T is unravelled iff, if lh( T ) = α + 1 then α is non- T -special.The unravelling unrvl( T ) of T is the unique unravelled J- m -maximal tree S on M , if it exists, such that (i) T E S , (ii) if lh( T ) is a limit then T = S , and(iii) if lh( T ) = α + 1 then β is S -vs for each β + 1 < lh( S ) with α ≤ β . (So E S β = F ( M S β ) for each β ≥ α , and cr( E S α + i +1 ) < cr( E S α + i ), so lh( S ) < lh( T )+ ω ;the existence of S just depends on the wellfoundedness of the resulting models.)We say that T is everywhere unravelable iff (i) unrvl( T ↾ ( η T α + 1)) exists forevery α < λ T , and (ii) for each α + 1 < λ T and i < n T α , letting W = ( T ↾ ( η α + i + 2)) b G T η α + i (a nicely padded putative J- m -maximal tree) , W has wellfounded models (so isactually a nicely padded J- m -maximal tree) and unrvl( W ) exists (another suchtree). ⊣ It might be that N = U η is a type 3 MS-premouse with cr( E η ) = lgcd( U η ), in which case n does not exist. That is, it satisfies all the requirements of a nicely padded J- m -maximal tree excludingthe wellfoundedness of the last model. Note that we use T ↾ ( η α + i + 2) (which has lastmodel M T η α + i +1 ) as opposed to T ↾ ( η α + i + 1) or even T ↾ ( η α + 1), because although G = G T η α + i ∈ E + ( M T η α ) = E + ( M T η α + i ), using exit extender G at stage η α + i in T ′ would notgive a nicely padded tree T ′ . By Lemma 2.3, we do get a nicely padded tree by using it atstage η α + i + 1. When α ′ = α + 1, clause (i) for α ′ replacing α is just the same as the instance of clause(ii) with α and i = n T α −
1, but clause (i) for limit α is not covered by clause (ii). .19 Definition. Let (
M, m, T ) be suitable with T unravelled and every-where unravelable. The conversion conv( T ) is the unique padded lifting-MS- m -maximal tree U on M , with exchange ordinals e ν U α , satisfying (we verify laterthat this works, writing λ = λ T , η α = η T α , etc):1. lh( U ) = lh( T ) and e ν U γ = e ν T γ for all γ + 1 < lh( T ).2. If α + 1 < λ and η α is T -vs (so n α = 0) then E U η α = ∅ , pred U ( η α +1 ) = η α .3. If α + 1 < λ and η α is non- T -vs then E U η α + i = G T η α + i for all i < n α , and E U η α + n α = ∅ , with pred U ( η α +1 ) = η α + n α .
4. Let η < lh( T ) be a limit. Fix γ < T η such that ( γ, η ) T ∩ D T = ∅ and ifany ξ ∈ ( γ, η ) T is T -special then γ is T -special. Let X be the set of all β < η such that β ≤ ext , T α for some α + 1 ∈ ( γ, η ) T with E T α = ∅ . Then:(a) { e β + 1 (cid:12)(cid:12) β ∈ X } is cofinal in η ;(b) E U e β = C ( E T β ) for each β ∈ X ;(c) for all β , β ∈ X , either f β + 1 ≤ U f β + 1 or vice versa;(d) [0 , η ) U is the ≤ U -downward closure of { e β + 1 (cid:12)(cid:12) β ∈ X } . ⊣ Let ( M, m, T ) be suitable with T unravelled and everywhereunravelable, and T non-trivial. Write λ = λ T etc. Then:1. U = conv( T ) is a well-defined padded MS-lifting- m -maximal tree on M .2. Suppose lh( T ) = η + 1 (so η is non- T -special). Let ε T + 1 ≤ T η be leastsuch that ( ε T + 1 , η ] T ∩ D T = ∅ , and ε U likewise for U . Then (and let δ, N ∗ , k be defined by):(a) M U η = M T η ,(b) [0 , η ] T ∩ D T = ∅ ⇐⇒ [0 , η ] U ∩ D U = ∅ .(c) δ = pred T ( ε T + 1) = pred U ( ε U + 1) ,(d) N ∗ = M ∗U ε U +1 = M ∗T ε T +1 ,(e) ~F U <η is equivalent to ~F T <η , and in fact, ~F U <η is the enumeration of { C ( E T β ) (cid:12)(cid:12) ∃ γ (cid:2) ε T + 1 ≤ T γ + 1 ≤ T η and β ≤ ex , T γ (cid:3) } in order of increasing critical point,(f) k = deg T ( η ) = deg U ( η ) ,(g) M T η = Ult k ( N ∗ , ~F T <η ) = Ult k ( N ∗ , ~F U <η ) = M T η ,(h) i ∗U ε U +1 ,η = i ∗T ε T +1 ,α . So note E U η α + n α − = E T η α + n α . . We have:(a) For α < λ , conv(unrvl( T ↾ η α + 1)) = ( U ↾ η α + 1) b ( ∅ , . . . , ∅ ) (where ( U ↾ η α + 1) b ( ∅ , . . . , ∅ ) is an extension of U ↾ η α + 1 by justpadding; the extension is finitely long), and(b) For each α + 1 < λ and i < n α , conv (cid:16) unrvl (cid:16) ( T ↾ η α + i + 2) b G T η α + i ) (cid:17)(cid:17) = ( U ↾ η α + i + 2) b ( ∅ , . . . , ∅ ) .
4. Let η + 1 < lh( T ) with E T η = ∅ and X = { β (cid:12)(cid:12) β ≤ ext , T η } . Then for each β ∈ X , we have E U e β = C ( E T β ) , and for all β , β ∈ X , if cr( E T β ) ≤ cr( E T β ) then f β + 1 ≤ U f β + 1 ≤ U η + 1 and ( f β + 1 , η + 1] U ∩ D U deg = ∅ .5. Let η + 1 < T η ′ + 1 < lh( T ) be such that E T η = ∅ 6 = E T η ′ and η ′ + 1 isnon- T -special and ( η + 1 , η ′ + 1] T does not drop in model. Let β ≤ ext , T η and β ′ ≤ ext , T η ′ . Then:– e η + 1 ≤ U e β + 1 ≤ U η + 1 ≤ U e η ′ + 1 ≤ U e β ′ + 1 ≤ U η ′ + 1 ,– ( e η + 1 , η ′ + 1] U ∩ D U = ∅ , and– if ( η + 1 , η ′ + 1] T ∩ D T deg = ∅ then ( e η + 1 , η ′ + 1] U ∩ D U deg = ∅ . Proof.
The proof is by induction on the unravelled trees which appear in part3. For lh( T ) = 1 it is trivial and for lh( T ) a limit, it follows immediately byinduction. So suppose lh( T ) = ε + 1 for some ε > Case . lh( T ) = η ξ + n ξ +2+ n where n < ω and η ξ is non- T -vs but η ξ + n ξ +1+ i is T -vs for all i < n .So E = E U η ξ + n ξ − = E T η ξ + n ξ = ∅ , this is the last non-empty extender used in U , η ξ +1 = η ξ + n ξ + 1, and T = unrvl( T ↾ ( η ξ +1 + 1)). Let µ ≤ ξ be least suchthat E T η is T -vs for each η ∈ [ η µ , η ξ ). Let¯ T = unrvl( T ↾ ( η µ + 1)) , and say lh( ¯ T ) = η µ + ℓ + 1 (so ℓ < ω ). Let ¯ U = conv( ¯ T ). So η ξ ∈ [ η µ , η µ + ℓ ]and by induction, we have M ¯ T η µ + ℓ = M ¯ U η µ + ℓ = M ¯ U η µ = M U η µ = M U η ξ . It follows that lh( E U η α + n α ) ≤ lh( G T η ξ + n ξ ) for each α < ξ and (by induction on i ) G T η ξ + i ∈ E + ( M U η ξ + i ) for each i ≤ n ξ , and we can set E U η ξ + i = G T η ξ + i for each i ≤ n ξ . By induction, the lemma also holds forunrvl (cid:16) T ↾ ( η ξ + i + 2) b G T η ξ + i (cid:17) i + 1 < n ξ . Let κ = cr( E ) and η χ + j = pred T ( η ξ +1 ) = pred T ( η ξ + n ξ + 1) = pred U ( η ξ + n ξ ) , (recall e ν T β = e ν U β for all β + 1 < lh( T ), by 2.19). Recall E U η ξ + n ξ = ∅ andpred U ( η ξ +1 ) = η ξ + n ξ . Subcase . η ξ +1 is non- T -special.Let N ∗ = M ∗T η ξ +1 and d = deg T η ξ +1 . We claim that M ∗U η ξ + n ξ +1 = N ∗ anddeg U η ξ + n ξ +1 = d . For if η χ is T -special and j = 0 then η ξ +1 ∈ D T (as η ξ +1 isnon- T -special), but then since ( M T η χ ) pv E M U η χ and OR( M T η χ ) is a cardinal of M U η χ , this gives the claim in this case. Suppose η χ is non- T -special or j >
0. If j = 0 then M T η χ = M U η χ and deg T η χ = deg U η χ , which suffices. Suppose j > G = G T η χ + j − , so ρ M T ηχ | lh( G )1 ≤ ν ( G ) ≤ κ. Now ex T η χ + n χ E M T η χ | lh( G ), and as η ξ +1 is non- T -special, therefore N ∗ ⊳M T η χ | lh( G )(and in particular η ξ +1 ∈ D T ), and by coherence, we have N ∗ ⊳ M U η χ + j , so N ∗ = M ∗U η ξ + n ξ +1 , which suffices.It follows that M T η ξ +1 = M U η ξ +1 , and (combined with induction if η ξ +1 / ∈ D T ,noting that j = 0 in this case) there is appropriate agreement of iteration maps.(As η ξ +1 is non- T -special, E is the last extender of T ′′ = unrvl( T ↾ ( η ξ +1 + 1)),and conv( T ′′ ) = U ↾ ( η ξ +1 + 1).)The remaining properties in this subsubcase are now straightforward to ver-ify by induction. Subcase . η ξ +1 is T -special and j > G = G T η χ + j − , we have N ∗ = M ∗T η ξ +1 = M T η χ | lh( G ),and e ν T η χ + j − = ν ( G ) ≤ κ < λ ( G ). Also deg T η ξ +1 = 0. So M T η ξ +1 = Ult ( N ∗ , E ),and note then that F ( M T η ξ +1 ) is equivalent to the two-step iteration ( G, E ) . (1)Let T ′ = unrvl( T ↾ ( η χ + j + 1) b G ) (nicely padded J- m -maximal), lh( T ′ ) = η χ + j + 2 + ℓ (so ℓ < ω ) and U ′ = conv( T ′ ) = U ↾ ( η χ + j + 1) b ( ∅ ) b ( ∅ , . . . , ∅ ) , noting the lemma applies to T ′ by induction (note E U ′ η χ + j − = G and E U ′ η χ + j = ∅ and η T ′ χ +1 = η χ + j + 1). Letting k = deg T ′ ∞ = deg U ′ ∞ = deg U ( η χ + j ), then M T ′ ∞ = M U ′ ∞ = M U η χ + j = Ult k ( M ∗U η χ + j , G ) . Note κ = cr( E ) < λ ( G ) < ρ k ( M U η χ + j ) and since E is total over M T η χ | lh( G ) andby coherence etc, E is total over M U η χ + j . So deg U η ξ +1 = deg U η ξ + n ξ = k and M U η ξ +1 = M U η ξ + n ξ = Ult k ( M U η χ + j , E )= Ult k (Ult k ( M ∗U η χ + j , G ) , E )= Ult k ( M ∗U η χ + j , F ( M T η ξ +1 )) , T ′′ = unrvl( T ↾ ( η ξ +1 + 1)) and U ′′ = conv( T ′′ ) = U ↾ ( η ξ +1 + 1) b ( ∅ , . . . , ∅ ) . Then η ξ +1 is T -special, hence T ′′ -special, M T ′′ η ξ +1 = Ult ( M T η χ | lh( G ) , E )and F ( M T ′′ η ξ +1 ) is equivalent to the two-step iteration ( G, E ), and then an easyinduction gives that for each i ∈ (0 , ℓ ], η ξ +1 + i is T ′′ -special and M T ′′ η ξ +1 + i = Ult ( M T ′ η χ + j + i , E )and F ( M T ′′ η ξ +1 + i ) is equivalent to the two-step iteration ( F ( M T ′ η χ + j + i ) , E ). Fur-ther, lh( T ′′ ) = η ξ +1 + 2 + ℓ , and recalling k = deg T ′ ∞ = deg T ′ η χ + j +1+ ℓ = deg U ′ ∞ ,note k = deg T ′′ ∞ = deg T ′′ η ξ +1 +1+ ℓ , and (letting) P ∗ = M ∗T ′ ∞ = M ∗T ′′ ∞ , we have M T ′′ ∞ = M T ′′ η ξ +1 +1+ ℓ = Ult k ( P ∗ , F ( M T ′′ η ξ +1 + ℓ ))= Ult k (Ult k ( P ∗ , F ( M T ′ η χ + j + ℓ )) , E )= Ult k ( M T ′ η χ + j +1+ ℓ , E )= Ult k ( M U ′ η χ + j , E ) = M U ′′ ∞ , with corresponding ultrapower maps, and since by induction the iteration mapsof T ′ , U ′ match appropriately, so do those of T ′′ , U ′′ .Regarding part 4 for T ′′ and X = { β (cid:12)(cid:12) β ≤ ext , T η ξ +1 } , note X = { η ξ + n ξ , η ξ +1 } and ^ η ξ + n ξ = η ξ + n ξ − g η ξ +1 = η χ + j −
1, and E U ′′ e γ = C ( E T γ )for γ ∈ X (these two extenders are G and E ), and η χ + j ≤ U ′′ η ξ + n ξ < U ′′ η ξ +1 < U ′′ η ξ +1 + 1 + i for i ≤ ℓ , and ( η χ + j, η ξ +1 + 1 + ℓ ] U ′′ ∩ D U ′′ deg = ∅ ( U ′′ only pads after η ξ + n ξ ).Part 5 follows from the above considerations and by induction applied to T ′ . Subcase . η ξ +1 is T -special and j = 0.By Lemma 2.13, η χ is T -special, η ξ +1 / ∈ D T , and ν ( F ) ≤ κ < λ ( F ) where F = F ( M T η χ ). Note then that F ( M T η ξ +1 ) is equivalent to the two-step iteration( F, E ). So things are almost the same as in Subcase 1.2, with F replacing G ,and we leave the details to the reader. Case . lh( T ) = η ζ + ℓ + 1 where ζ is a limit and E T η ζ + i is T -vs for all i < ℓ .Let b = [0 , η ζ ) T . Note that η ζ is T -special iff η α is T -special for all suffi-ciently large η α ∈ b . By induction with parts 4 and 5, b induces a U ↾ λ -cofinalbranch, which has the properties required by Definition 2.19(4). (If λ is T -special then apply part 4 to unrvl( T ↾ ( η α +1)) for sufficiently large η α +1 < T η ζ .)So if η ζ is non- T -special, then induction easily shows that M T η ζ = M U η ζ and iteration maps agree appropriately etc. If η ζ is T -special, then proceedessentially as in Subcase 1.3, but using ~F T η α η ζ and the equivalent ~F U η α η ζ , where η α ∈ b is sufficiently large, in place of single extenders of T , U .14his completes the proof of the lemma. (cid:3) Let M be an m -sound J-premouse and U ′ be a lifting-MS- m -maximal tree on M . Then there is a unique pair ( T , V ) such that ( M, m, T ) is suitable, T is unravelled everywhere unravelable, V = conv( T ) and U ′ is thetree given by removing all padding from V . Proof.
We proceed by induction on lh( U ′ ). The induction is an easy consequenceof Lemma 2.20 except for the case that lh( U ′ ) = ζ + 1 for some limit ζ . Soconsider this case, assuming that the lemma holds for trees U ′′ of length ≤ ζ .So we have trees T ↾ ζ and U ↾ ζ corresponding to U ′ ↾ ζ , where writing λ = λ T ↾ ζ , η α = η T ↾ ζα etc, (note) ζ = λ = sup α<ζ η α . Write η ζ = ζ . Set U = ( U ↾ η ζ ) b b ,where b is the branch determined by [0 , ζ ) U ′ . (So the desired V will be of theform U b ( ∅ , . . . , ∅ ).) Claim . There is α < U η ζ such that for all ξ < ζ and i < n ξ with α < U η ξ + i + 1 < U η ζ , letting δ = λ ( G T η ξ + i ) = λ ( E U η ξ + i ), there is χ ∈ [ η ξ + i + 1 , η ζ ) U with i U η ξ + i +1 ,χ ( δ ) ≤ cr( i U χη ζ ) . Proof.
If not, then select a sequence h ( ξ n , i n , δ n ) i n<ω of witnessing triples ( ξ, i, δ )such that writing γ n = η ξ n + i n , we have γ n + 1 < U γ n +1 + 1. Then since δ n = i ∗U γ n +1 (cr( E U γ n )) , we get i U γ n +1 ,η ζ ( δ n ) > i U γ n +1 +1 ,η ζ ( δ n +1 ) for each n < ω , so M U η ζ is illfounded, acontradiction. (cid:3) We now break into cases, mostly in order that we can discuss a simpler casefirst as a warm-up:
Case . For all sufficiently large η + 1 < U η ζ with E U η = ∅ , we have i U η +1 ,η ζ exists and λ ( E U η ) ≤ cr( i U η +1 ,η ζ ).Note then that for all sufficiently large η + 1 < U η ζ with E U η = ∅ , there is α such that pred U ( η + 1) = η α (as when pred U ( η + 1) = η α + j with j ∈ (0 , n α ],and i U η α + j,η ζ exists, then cr( i U η α + j,η ζ ) < λ ( E U η α + j − )).Fix ξ such that:– η ξ < U η ζ and [ η ξ , η ζ ) U ∩ D U deg = ∅ – pred U ( η + 1) = η α for some α whenever η + 1 ∈ [ η ξ , η ζ ) U with E U η = ∅ (but don’t demand that ξ is least such). Let b = [ η ξ , η ζ ) U .Let η + 1 ∈ b with E U η = ∅ , and η χ = pred U ( η + 1). Then η = η ξ + n ξ − ξ , so E T η = ∅ , E T η +1 = E U η , E U η +1 = ∅ and pred U ( η +1) = η = pred T ( η +2),and note that by choice of ξ ,succ U ( η + 1 , η ζ ) = η ξ +1 = η + 2 . η + 2 / ∈ D T deg . For because U ↾ η ζ = conv( T ↾ η ζ ), if η χ is non- T -special then M T η ζ = M U η ζ and deg T η ζ = deg U η ζ , and since η + 1 / ∈ D U deg , therefore η + 2 / ∈ D T deg . And if η χ is T -special then ( M T η χ ) pv is a cardinal segment of M U η χ and deg T η χ = 0, and so η + 2 / ∈ D T deg . Note that also λ ( E U η ) ≤ cr( i U η +1 ,η ζ ) in thissituation.Let χ be least such that either χ = η ζ or η ξ is T -special and η ξ ≤ U χ < U η ζ and letting δ = lgcd( M T η ξ ), we have i U η ξ χ ( δ ) ≤ cr( i U χη ζ ) and E U η = ∅ where η + 1 = succ U ( χ, η ζ ). Let b ′ be the < η ζ -closure of[0 , η ξ ] T ∪ n η + 2 ∈ b ∩ ( χ + 1) (cid:12)(cid:12)(cid:12) E U η = ∅ o . Claim . b ′ is a branch of T ↾ η ζ (cofinal iff χ = η ζ ), and ( b ′ \ ( η ξ +1)) ∩ D T deg = ∅ . Proof.
We observe that this holds for the initial segments of b ′ , by induction.For b ′ ∩ ( η ξ + 1) it is trivial. Suppose it holds for b ′ ∩ ( β + 1) where β ∈ b ′ .Note that β = η γ for some γ , and η γ is T -special iff η ξ is T -special. Suppose η γ < χ and let η + 1 = succ U ( η γ , η ζ ).Suppose η is T -vs. Then η γ = pred U ( η + 1) = η , so η γ , η ξ are T -special,and e ν U η = λ ( E T η ) = i T η ξ η ( δ ) = i U η ξ η ( δ ) ≤ cr( i U η +1 ,η ζ ) = cr( i U ηη ζ ) , so χ ≤ η , contradiction. So η is non- T -vs. But then E U η = ∅ and by the remarksabove, pred T ( η + 2) = η etc, and the claim holds for b ′ ∩ (( η + 2) + 1).Now let γ > ξ be a limit with η γ ∈ b and suppose the claim holds below η γ .Then [0 , η γ ) U is determined from [0 , η γ ) T by Definition 2.19, and by induction,it easily follows that [0 , η γ ] T = b ′ ∩ ( η γ + 1). (cid:3) Now if χ = η ζ , so b ′ is T ↾ η ζ -cofinal, then set T ′ = T ↾ λ b b ′ , a well-definedputative tree.Suppose now that η γ = χ < η ζ and let η + 1 = succ U ( η γ , η ζ ). Arguingas above, it follows that η γ is T -vs, E U η γ = ∅ pred U ( η γ + 1) = η γ and η γ +1 = η γ + 1 < U η ζ . Set ξ = γ + 1, and define b ′ from ξ like b ′ was defined from ξ .Proceed in this way defining ξ n , b ′ n as far as possible.For α < lh( U ), let ℓ α be the ℓ < ω such that unrvl( T ↾ ( η α + 1)) has length η α + 1 + ℓ .Note that ℓ α is constant over all η α ∈ b ′ n , and if η α ∈ b ′ n and η β ∈ b ′ n +1 then ℓ β = ℓ α −
1. (This uses the properties of b ′ n , including that it is a branchof T .) So we reach n < ω such that b ′ n is indeed T ↾ η ζ -cofinal, and define T ′ = T ↾ η ζ b b ′ n . Set b ′ = b ′ n and ℓ = ℓ β for η β ∈ b ′ . Claim . T ′ has wellfounded last model, and moreover, unrvl( T ′ ) is well-definedwith wellfounded models, andconv(unrvl( T ′ )) = U ↾ ( η ζ + 1) b ( ∅ , . . . , ∅ ) (2)and ℓ T ′ ζ = ℓ , i.e. the tuple ( ∅ , . . . , ∅ ) in (2) has length ℓ .16 roof. Suppose ℓ = 0. Then η β is non- T -special for η β ∈ b ′ , so note that T ′ is already unravelled, and M T ′ η ζ = M U η ζ . So the fact that (2) holds in this casefollows easily by induction and because our definition of b ′ ensures that [0 , η ζ ) U is recovered from b ′ via Definition 2.19.If ℓ > η ζ is T ′ -special, but it is similar, with computation much asin the proof of Lemma 2.20. (We have that ( M T η ξn ) pv is a cardinal segment of M U η ξn and OR( M T η ξ + n ) ≤ ρ k ( M U η ξn ) where k = deg U η ξn , so note that ( M T ′ b ′ ) pv is a cardinal segment of M U η ζ , and in particular is wellfounded. Now letting T ′′ = unrvl( T ′ ), proceed by induction on i < ℓ , rearranging extenders as in theproof of Lemma 2.20, to show that ( M T ′′ η ζ + i ) pv is a cardinal segment of M U η ζ , andfinally that M T ′′ ∞ = M U η ζ and deg T ′′ η ζ + ℓ = deg U η ζ .) (cid:3) Case . Otherwise: There are cofinally many η + 1 < U η ζ such that E U η = ∅ and i U η +1 ,η ζ exists and cr( i U η +1 ,η ζ ) < λ ( E U η ).This case is an embellishment of the previous one. Note that if η + 1 < U η ζ and E U = ∅ and and i U η +1 ,η ζ exists and γ + 1 = succ U ( η + 1 , η ζ ), thencr( i U η +1 ,η ζ ) < λ ( E U η ) iff η = η α + j − α and j ∈ (0 , n α ] and it is notthe case that γ = η + 1 = η α + n α .Appealing to Claim 1 for existence, fix any ξ such that:(a) η ξ < U η ζ and ( η ξ , η ζ ) U ∩ D U deg = ∅ ,(b) E U η = ∅ where η + 1 = succ U ( η ξ , η ζ ),(c) for all η + 1 ∈ ( η ξ , η ζ ) U with E U η = ∅ , there is χ ∈ ( η + 1 , η ζ ) U such that i U η +1 ,χ ( λ ( E U η )) ≤ cr( i U χη ζ ).For η + 1 ∈ ( η ξ , η ζ ) with E U η = ∅ , let χ η be the least witness χ as above andsuch that E U γ = ∅ where γ + 1 = succ U ( χ, η ζ ). Note that if η < η ′ are such and η ′ + 1 ≤ χ η then χ η ′ ≤ χ η .Now let χ ∈ [ η ξ , η ζ ] U be least such that either χ = η ζ or– η ξ is T -special and i U η ξ χ ( δ ) ≤ cr( i U χη ζ ) where δ = lgcd( M T η ξ ) (so actually χ > η ξ , by (b)), and– E U η = ∅ where η + 1 = succ U ( χ, η ζ ).Note that if χ < η ζ then χ η ≤ χ for all η + 1 ∈ ( η ξ , χ ] U with E U η = ∅ .Let b ′ = [0 , η ξ ] T ∪ b ′ where b ′ is the < χ -closure of χ ∩ n χ η (cid:12)(cid:12)(cid:12) η + 1 ∈ ( η ξ , η ζ ) U and E U η = ∅ o . Note b ′ \ η ξ ⊆ [ η ξ , η ζ ) U . Claim . b ′ is a branch of T ↾ λ and b ′ \ ( η ξ + 1) ∩ D T deg = ∅ .17 roof. We establish the claim regarding b ′ ∩ ( β + 1) by induction on β ∈ b ′ with β ≥ η ξ . If β = η ξ it is trivial. Suppose the claim holds for b ′ ∩ ( β + 1)where β ∈ b ′ and β ≥ η ξ and let η + 1 = succ U ( β, η ζ ). We must see that χ η is a successor ordinal, pred T ( χ η ) = β and χ η / ∈ D T deg . If cr( i U η +1 ,η ζ ) ≥ λ ( E U η )then we just get χ η = η + 2 (note E U η +1 = ∅ ) and things are easy, so supposecr( i U η +1 ,η ζ ) < λ ( E U η ). Let χ ′ ≤ χ η be least such that χ ′ = η γ for some γ and i U η +1 ,χ ′ ( λ ( E U η )) ≤ cr( i U χ ′ η ζ ), so in fact M U χ ′ = M U χ η , and α is T -vs for each α ∈ [ χ ′ , χ η ). Let T ′ = unrvl( T ↾ ( χ ′ + 1)), so η χ + 1 ≤ lh( T ′ ) and T ′ ↾ ( η χ + 1) E T .Let U ′ = conv( T ′ ), so M U ′ ∞ = M U χ ′ = M U χ η etc. Now by Lemma 2.20 applied to T ′ , U ′ , ~F T ′ < ∞ is equivalent to ~F U ′ < ∞ , and E U η b (cid:10) E U α (cid:11) α +1 ∈ ( η +1 ,χ ′ ] (3)is a tail segment of the latter. But then the choice of χ η easily gives thatthere is α ≥ χ ′ which is T -vs and E T α is equivalent to the concatenation in3, in particular with η = e α (computed relative to T ′ , U ′ ) and E U η = C ( E T ′ α ).Therefore pred T ′ ( α + 1) = pred U ( η + 1) = β , and it is straightforward to seethat α + 1 / ∈ D T ′ deg . Finally, we claim that α + 1 = χ η and T ′ ↾ ( χ η + 1) E T . Forif χ η < α + 1 then just note that cr( i U χ η ,η ζ ) < i U η +1 ,χ η ( λ ( E U η ), and if α + 1 < χ η then η ξ is T -special and i U η ξ χ η ( δ ) ≤ cr( i U χ η η ζ ), and then it easily follows that χ η = χ , contradicting that χ η < χ (by definition of b ′ ). So χ η = α + 1, and itfollows that T ′ ↾ ( χ η + 1) E T .Now let γ be a limit with η γ ≤ χ and η γ ∈ ( η ξ , η ζ ] U , and suppose we havethe claim below γ . The fact that [0 , η γ ] T determines [0 , η γ ) U as in Definition2.19, together with induction, easily implies that [0 , η γ ) T = b ′ ∩ η γ . (cid:3) Now suppose that χ < η ζ . Then η ξ is T -special, and letting δ = lgcd( M T η ξ ),arguing as above (and almost by the claim), we get χ ′ < U χ such that M U χ ′ = M U χ and η ξ ≤ T χ ′ and α is T -vs for each α ∈ [ χ ′ , χ ). Hence χ ′ = η α for some α .Define ℓ α like before (so unrvl( T ↾ ( η α + 1)) has length η α + 1 + ℓ α ). Then ℓ ξ = ℓ χ ′ > ℓ χ . Now let ξ = χ . Given ξ n , define b ′ n from ξ n like b ′ was definedfrom ξ . Then there is n < ω with b ′ n cofinal in η ζ .The rest is just like Claim 3 of Case 1.This completes the proof of the lemma. (cid:3) In this section we describe a background construction for J-mice which is basedon the traditional kind for MS-mice, and absorbs Woodin cardinals from thebackground universe just like traditionally for MS-mice. Other than being basedon those for MS-mice, it however incorporates some features analogous to theconstruction of [7, § .1 Definition. Let M be a J-pm. We say that M satisfies the MS-ISC iffeither M is passive, or letting κ = cr( F M ), then for every ν ∈ [( κ + ) M , ν ( F M ))such that ν is the natural length of ¯ F = F M ↾ ν , either:1. ¯ F is non-type Z and either:(a) ¯ F ∈ E M , or(b) ν is a limit of generators of F M and M | ν is active and¯ F ∈ E (Ult( M | ν, F M | ν )) , or2. ¯ F is type Z, ν = µ + 1, µ is a cardinal of M , and there is a normal measure G ∈ E (Ult( M, F M ↾ µ )) with cr( G ) = µ , with ¯ F = G ◦ ( F M ↾ µ ). ⊣ Let E be some class of extenders. The J- E -good maximal L [ E ] -construction of length λ is the unique sequence h N α i α<λ of J-pms N α suchthat:1. N = V ω ,2. for limit η < λ , N η = lim inf α<η N α ,3. given α + 1 < λ , if N α has largest cardinal θ and there is N such that:(a) N is an active J-pm satisfying the MS-ISC,(b) either N pv = N α , or N α ⊳ N with OR N α = ( θ + ) N ,(c) θ ≤ ν ( F N ) < OR N α ,(d) there is E ∈ E such that F N ↾ ν ( F N ) ⊆ E and strength( E ) = ν ( F N ),then letting ν = ν ( F N ) be least among all such N , there is a uniquesuch N with ν ( F N ) = ν , and this is N α +1 ; otherwise there is no N satisfying conditions 3a–3d, and N α is ω -solid and ω -universal and N α +1 = J ( C ω ( N )).So if we reach α such that N α is non- ω -solid or non- ω -universal, or thereare two distinct J-pms N, N ′ with ν ( F N ) = ν ( F N ′ ) minimal for satisfyingconditions 3a–3d, then λ = α + 1. ⊣ Let h N α i α<λ be a J- E -good maximal L [ E ] -construction. Let α <λ . Then:1. Let M ⊳ N = N α be such that ρ Mω is an N -cardinal. Suppose that if N is active then ρ Mω ≤ ν ( F N ) . Then there is a unique β < λ such that C ω ( N β ) = M , and moreover, β < α .2. If β < α and ρ N β ω ≤ ρ N γ ω for all γ ∈ ( β, α ) then C ω ( N β ) ⊳ N α and ρ N β ω isan N α -cardinal. roof. By induction on α . Part 2 is proved as usual, so we omit it.Suppose α = β + 1.If N α is active, just apply induction to N β ; this works as θ ≤ ν ( F N α ) where θ is the largest cardinal of N β (this is the key difference to the standard versionof this lemma). Otherwise N α = J ( C ω ( N β )). So if M ⊳ N α and ρ = ρ Mω isan N α -cardinal then ρ ≤ ρ N β ω . Use induction and universality of the standardparameter basically as usual, noting that if N β is active then ρ N β ω ≤ ν ( F N β ).The limit case is as usual. (cid:3) A reasonable structure is a transitive structure R = ( S, E )such that S is transitive, R | = ZFC , E ⊆ S is a class of S -extenders, and foreach E ∈ E and R -cardinal λ , we have E ↾ λ ∈ E . ;Definability over reasonable structures R = ( S, E ) is with respect to thepredicate E , including the fact that R | = ZFC . And iterability for such R is foriteration trees which only use extenders from E and its images. ⊣ Let R = ( S, E ) be an ( ω + 1)-iterable reasonable structure.Let h N α i α<λ be a J- E -good maximal L [ E ]-construction of R and α < λ and m < ω be such that N α is m -solid, and suppose that all proper segments of N α satisfy standard condensation facts. Let N = C m ( N α ). Let π : M → N bean m -lifting embedding. Let E ∈ E M + be such that the reverse-e-dropdown of( M, E ) is just h E i . Let h M i i i ≤ k be the reverse-mod-dropdown of ( M, M | lh( E )).We define the ( α, m, π, E ) -resurrection h α j , m j , π j , σ j i i ≤ k of R , with:– α j ≤ α and m j ≤ ω ,– π j : M j → C m j ( N Rα j ) is an m j -lifting embedding,– ( α , m , π ) = ( α, m, π ),– if j > m j < ω then ρ M j m j +1 = ρ M j ω ,– σ j = τ Rα j m j ◦ π j : M j → N Rα j , where τ Rβkℓ : C k ( N Rβ ) → C ℓ ( N Rβ ) is the coreembedding,– if j < k then σ j ↾ ρ M j +1 ω ⊆ π j +1 .Suppose we have defined α j , m j , π j , σ j , and k > j . We have σ j : M j → N Rα j .Let ρ = ρ M j +1 ω . So ρ is an M j -cardinal, and if M j is active then ρ ≤ ν ( F M j ).If M j is passive or σ j ( ρ ) ≤ ν ( F ( N Rα j )) then by 3.3 we can set α j +1 to bethe unique α ′ such that σ j ( M j +1 ) = C ω ( N Rα ′ ), and set π j +1 = σ j ↾ M j +1 and m j +1 = ω . ZFC itself is not particularly important; it will be clear that we could make do with muchless. M j is active and σ j ( ρ ) > ν = ν ( F ( N Rα j )). Let m < ω be least suchthat ρ M j +1 m +1 = ρ . Let M + = σ j ( M j +1 ),¯ M = cHull M + m +1 ( ν ∪ { ~p M + m +1 } )and ¯ π : ¯ M → M + be the uncollapse, noting that ¯ M is ( m + 1)-sound with ν = ρ ¯ Mm +1 , because ¯ M ∈ N Rα j , where ν is a cardinal, and σ j “ M j +1 ⊆ rg(¯ π ).By condensation, ¯ M ⊳ N Rα j , and ρ ¯ Mω = ν . Therefore we can set α j +1 to be theunique α ′ such that ¯ M = C ω ( N Rα ′ ), and set m j +1 = m and π j +1 = ¯ π − ◦ σ j ↾ M j +1 . ⊣ Let R = ( S, E ) be an ( ω + 1) -iterable reasonable structure.Then R has a J- E -good maximal L [ E ] -construction h N α i α<λ of length λ =OR R + 1 , and for every α < λ and m < ω , C m ( N α ) is J- ( m, ω , ω + 1) ∗ -iterable.The final model N λ is a proper class of R which models ZFC .Moreover, if δ ∈ OR R and R | = “ δ is Woodin as witnessed by extenders in E ”, then N λ | = “ δ is Woodin”. Proof sketch.
The overall proof is mostly standard, with just some little differ-ences. We describe enough to mention these and give the main structure; theremaining details will be very routine.We first consider the iterability. By Theorem 2.8 it suffices to see that M = C m ( N α ) is lifting-MS-( m, ω + 1)-iterable. Fix an ( ω + 1)-strategy Γ for R (with respect to the class E ). We will define a lifting-MS-( m, ω + 1) strategyΣ for M , lifting to trees on R via Γ. We mostly keep track of the usual kindof data and maintain the usual kinds of inductive properties. But some details,particularly to do with resurrection, are a little different than usual. Write C = (cid:10) N Rα (cid:11) α<λ , where λ is as large as possible that this is defined. Fix ξ < λ , m ≤ ω such that N Rξ is m -solid, an m -sound premouse M and an m -liftingembedding π : M → C m ( N Rξ ).We will define trees T , U on M, R respectively, with T being lifting-MS- m -maximal and U a coarse tree, via Γ, with models M α = M T α and R α = R U α ,degrees d α = deg T α , and embeddings i αβ = i T αβ , i ∗ α = i ∗T α , j αβ = i U αβ , j ∗ α = i ∗U α ,and for α < lh( T ), the objects C α , ξ α , d α , π α , ξ ∗ α , π ∗ α , σ α , such that:1. < T = < U ,2. C α = i U α ( C ) and ξ α < lh( C α ),3. ( ξ , π ) = ( ξ, π ) (and d = m ),4. π α : M α → C d α ( N R α ξ α ) is a d α -lifting embedding,5. If α is a successor and β = pred T ( α ) and α / ∈ D T then ξ ∗ α = ξ β and π ∗ α = τ R β ξ β d β d α : M ∗T α → C d α ( N R β ξ ∗ α ) .
21. If α is a successor and β = pred T ( α ) and α ∈ D T , then letting( e α i , e m i , e π i , e σ i ) i ≤ k be the ( ξ β , d β , π β , E T β )-resurrection of R β , and D f M i E i ≤ k be the rev-mod-dropdown of ( M T β , E T β ), and i such that M ∗T α = f M i , then ξ ∗ α = e α i and π ∗ α = τ R β ξ ∗ α e m i d α ◦ e π i : M ∗T α → C d α ( N R β ξ ∗ α ) , noting that d α ≤ e m i since ρ e m i +1 ( M ∗T α ) ≤ cr( E T α ) < ρ e m i ( M ∗T α ).7. If α + 1 < lh( T ) then letting ( e α i , e m i , e π i , e σ i ) i ≤ k be the ( ξ α , d α , π α , E T α )-resurrection of R α , then σ α = e σ k : ex T α → N R α e α k , a 0-lifting embedding, and E U α ∈ E R α is a background for F N where N = N R α e α k , and R α | =“ E U α has strength exactly ν ( F N )”.8. If α + 1 < lh( T ) and β = pred T ( α + 1) then ξ α +1 = i U β,α +1 ( ξ ∗ α +1 ) and π α +1 (cid:16)h a, f M ∗ τq i M ∗ E T α (cid:17) = h σ α ( a ) , f M ′ τq ′ i R β E U α , where M ′ = C d α +1 ( N R β ξ ∗ α +1 ) and q ′ = π ∗ α +1 ( q ), τ is an rΣ d α +1 term, and for N a d α +1 -sound premouse and q ∈ N , f Nτq is the function x τ N ( q, x ).Here letting κ = cr( E T α ), we have σ α ↾ P ( κ ) ∩ ex T α = π ∗ α ↾ P ( κ ) ∩ M ∗T α .9. If α < β < lh( T ) and ν = ν ( E T α ) then σ α ↾ ( M T α | ν ) ⊆ π β and σ α ( ν ) ≤ π β ( ν ) , σ β ( ν )and if ν is not a cardinal of ex T α and ι = ( ν + ) ex T α then σ α ↾ ( M α | ι ) ⊆ π β and σ α ( ι ) ≤ π β ( ι ) , σ β ( ι ) .
10. If α is a successor then π α ◦ i ∗T α = i ∗U α ◦ π ∗ α .11. If α < U β and ( α, β ] T ∩ D T = ∅ then i U βα ( ξ β ) = ξ α and π α ◦ i T βα = i U βα ◦ τ R β ξ β d β d α ◦ π β = τ R α ξ α d β d α ◦ i U βα ◦ π β . This determines the entire process, and the propagation of the inductive hy-potheses is quite routine, so we leave it to the reader. (As usual, one maintainsin fact a little more agreement between maps π α , π β , etc, than that statedabove.) There is a very similar construction done in detail in [7, § m -maximal trees ensure thatthe rev-e-dropdown of ( M T β , E T β ) is just D E T β E . (That is, suppose we haveproduced T , U through length β + 1, and E ∈ E + ( M T β ), and lh( E T α ) < lh( E )for all stable α , that is, all α such that lh( E T α ) ≤ lh( E T γ ) for all γ ≥ α . Let h G i i i ≤ k be the rev-e-dropdown of ( M T β , E ), so G k = E . Then when we set E T β + i = G i for 0 ≤ i ≤ k , note that the rev-e-dropdown of ( M T β + i , G i ) is just h G i i .)(Also, we are assuming that R is ( ω + 1)-iterable, meaning without restric-tion on the form of the trees. But for our purposes here, it actually suffices toassume that R is ( ω + 1)-iterable for stacks of normal trees. For lifting-MS- m -maximal trees on M lift to normal trees on R by the conditions specified above,and then this extends to stacks as usual; this uses Lemma 2.6.)This gives that the construction does not break down due to cores not ex-isting, but we also need to see it does not break down due to non-uniqueness ofnext extenders. For this, use a typical bicephalus comparison. This is basicallylike in [3], but slightly different, with features as in the bicephalus argumentsin [7], which the reader should consult for more details. We just give a sketchhere. Suppose we reach a passive model N α , and there are two plausible ac-tive backgrounded extensions P, Q of N α (in the sense of the construction) with ν = ν ( F P ) = ν ( F Q ). Then let B = ( P, Q, ν ), and compare B with itself, likein a standard bicephalus comparison. (But note we only know that P, Q agree(strictly) below ( ν + ) P = ( ν + ) Q = OR N α , and possibly OR N α < OR P and/orOR N α < OR Q .) We get the lifting-MS-0-maximal iterability of B just like inthe proof above. Here if B ′ = ( P ′ , Q ′ , ν ′ ) is a non-dropping iterate of B , write ν ′ = sup i “ ν = sup j “ ν where i : P → P ′ and j : Q → Q ′ are the iterationmaps, so ν ′ = ν ( F P ′ ) = ν ( F Q ′ ) and P ′ || (( ν ′ ) + ) P ′ = Q ′ || (( ν ′ ) + ) Q ′ . If we wantto use E ∈ E P ′ + with OR N α ≤ lh( E ), then we use the rev-e-dropdown of ( P ′ , E )to determine the next few extenders. Likewise for E ∈ E Q ′ + . We don’t need toconvert this to J- m -maximal iterability, because it is straightforward to see thatlifting-MS- m -maximal iterability is enough for the comparison argument. Notethat if we are at a stage α (such as α = 0) of the comparison, with trees T , U ,and M T α = B ′ = ( P ′ , Q ′ , ν ′ ) = M U α is a common bicephalus, then P ′ = Q ′ ,(( ν ′ ) + ) P ′ = (( ν ′ ) + ) Q ′ and P ′ , Q ′ project to ν ′ , so P ′ Q ′ P ′ . So there is aleast difference E, F with say E ∈ E P ′ + and/or F ∈ E Q ′ + . We then want to use E, F , but this is preceded by the rev-e-dropdown of ( P ′ , E ) and that of ( Q ′ , F ).Note here that F P ′ and F Q ′ are included in these (since if ν ′ < lgcd( P ′ ) then(( ν ′ ) + ) P ′ < lh( E ) etc). Now a little further consideration shows that there areno α, β such that E T α = ∅ 6 = E U β and E T α , E U β are compatible, and this leads tocontradiction as usual.We also need the fact that iterable pseudo-J-premice satisfy the MS-ISC.Here a structure P = ( N, F ) is a pseudo-J-premouse iff N is a passive J-premouse, F an extender over N , and P satisfies the conditions of being a Further related bicephalus arguments can be seen in [7]. ν = ν ( F ) and δ = card N ( ν ), then F ↾ δ ∈ E P + . Suppose P is such and is lifting-MS-(0 , ω , ω + 1) ∗ -iterable, but F is not type Z. Then P satisfies the MS-ISC; the proof of this is almost identical to that in [3, § , ω +1)-iterability is enough, by essentially the argumentin [5, Theorem 9.4].)The fact that Woodins of R are Woodin in N = N R OR R now follows bythe argument in [3, § (cid:3) In this section we detail a conversion procedure for trees on MS-premice, whichis very similar to that for J-premice. It is used as a black box in [6, 2.12,2.14***], so this section fills in the missing details from there. We formallyassume the reader is familiar with §
2, For the most part we actually give acomplete account, independent of §
2; this just excludes the version of Lemma2.21, which we leave to the reader (and given what we do describe, it is an easyexercise to fill that in).We assume the reader is familiar with the definitions and basic facts in [6, §§ We say (
M, k ) is suitable iff either:– M is an MS-premouse of type ≤ M is k -sound, or– M is a type 3 MS-premouse and k ≥ M is ( k − k = 1 and M is a u-1-sound internally MS-indexed active seg-pm (notethen letting ν = ν ( F M ), M pm is the premouse with the trivial completionof F M ↾ ν active and M || ( ν + ) M = ( M pm ) pv ).Let ( M, k ) be suitable. Let T be a padded u- k -maximal tree on M . Given α < lh( T ), say α is:– T -special iff M T α is a non-premouse and u- deg T α = 0 (equivalently, M T α fails the MS-ISC),– T -very-special ( T -vs) iff T -special and E T α = F ( M T α ).– T -pre-special ( T -ps) iff M T α is a non-premouse and u- deg T α = 1,– T -very-pre-special ( T -vps) iff T -pre-special and E T α = F ( M T α ).Say T is nicely padded iff:1. If α is T -pre-special and not of form α = β + 1 with E T β = ∅ , then either E T α = ∅ or lh( E T α ) < OR(( M T α ) pm ).24. If α + 1 < lh( T ) and E T α = ∅ then α is T -pre-special and not of form α = β +1 with E T β = ∅ , and moreover, pred T ( α +1) = α and M T α +1 = M T α and deg T ( α + 1) = 1 and α + 2 < lh( T ) and OR(( M T α ) pm ) < lh( E T α +1 ).3. If E T α = ∅ then define e ν T α = ι (ex T α ), and if E T α = ∅ then define e ν T α = ν ( F ( M T α )). We use e ν T α as the exchange ordinal associated to α in T ; thatis, if E T β = ∅ then pred T ( β + 1) is the least α such that cr( E T β ) < e ν T α .If α + 1 < lh( T ), we say that α is a transition point of T iff E T α = ∅ .We say that ( M, k, T ) is suitable iff ( M, k ) is suitable and T is a nicelypadded u- k -maximal tree on M . ⊣ Let ( M, k, T ) be suitable. Let γ be T -special. Then there is aunique β < γ such that u - deg T ( β ) = 1 (so β is non- T -special) and C ( F ( M T γ )) = F (( M T β ) pm ) and E T β = ∅ . Moreover, letting F = C ( F ( M T γ )) and ν = ν F :1. β = α + 1 where α is a transition point of T e ν T α = ν and e ν T α ′ < ν for all α ′ < α and lh( E T α ′ ) ≤ ν for all α ′ < α suchthat E T α ′ = ∅ ,3. α + 1 < T γ ; let ξ + 1 = succ T ( α + 1 , γ ) ,4. u - deg T ξ +1 = u - deg T γ = 0 and ( α + 1 , γ ] T ∩ D T = ∅ ,5. ν ≤ cr( E T ξ ) , and F ( M T γ ) is equivalent to the composition of the extenders h F i b (cid:10) E T δ (cid:11) δ +1 ∈ [ ξ +1 ,γ ] T (and for δ , δ ∈ [ ξ + 1 , γ ] T with δ < δ , we have ν ( E T δ ) ≤ cr( E T δ ) ). Proof sketch.
The proof is straightforward. Part 5 is as in [4, Lemma 2.27***](extended to transfinite iterations in a routine manner). (cid:3)
Let (
M, k, T ) be suitable and γ + 1 < lh( T ). If γ is T -vs, write e γ for the unique transition point α as in Lemma 4.2. If γ is T -vps, write e γ = α where α + 1 = γ (so again, α is a transition point). Otherwise write e γ = γ . ⊣ Let (
M, k, T ) be suitable. Write β ≤ extdirect α iff β = α or [ α is T -vs and β + 1 ∈ ( e α + 1 , α ] T ]. Let ≤ ext be the transitive closure of ≤ extdirect . Let < extdirect and < ext be the strict parts. Note that α < extdirect β implies α < β , so < ext is wellfounded. Write ≤ ext , T = ≤ ext , etc. ⊣ Note here that if β < ext β then β < β but cr( C ( E T β )) < cr( C ( E T β )). Let (
M, k, T ) be suitable and α +1 < lh( T ). Then the standarddecomposition of E T α is the enumeration of { C ( E T β ) (cid:12)(cid:12) β ≤ ext , T α } , in order ofincreasing critical point. ⊣ .6 Lemma. Let ( M, k, T ) be suitable and α + 1 < lh( T ) .The standard decomposition of E T α is well-defined. That is, if β , β ≤ ext , T α and β = β then κ β = κ β where κ β i = cr( C ( E T β i )) = cr( E T β i ) ; moreover, if β = α then κ β < κ β , and if κ β < κ β then ν ( C ( E T β )) ≤ κ β .Further, E T α is equivalent to iteration via the extenders in the standarddecomposition of E T α (in order of increasing critical point). Proof.
This is by induction using Lemma 4.2, again as in [4, Lemma 2.27***]. (cid:3)
Let (
M, k, T ) be suitable. Let α < lh( T ). Given γ ≤ T α , ~E T γα denotes the sequence D E T β E γ< T β +1 ≤ T α (so ~E T γα corresponds to i T γα whenthe latter exists), and ~F T γα denotes the sequence D E T β E β +1 ∈ [ ξ +1 ,α ] T , where ξ isleast such that γ < T ξ + 1 ≤ T α and ( ξ + 1 , α ] T does not drop in model. Write ~E T <α = ~E T α and ~F T <α = ~F T α .Given a sequence ~E = h E α i α<λ of short extenders, we define h U α , k α i α ≤ λ ,if possible, by induction on λ , as follows. Set U = M and k = k . Given U η and k η ≤ ω are well-defined and U η is a k η -sound seg-pm and η < λ , then: ifcr( E η ) < OR( U η ) and there is N E U η such that E η measures exactly P ([ κ ] <ω ) ∩ N , then letting N E U η be the largest such, and letting n ≤ ω be largest suchthat ( N, n ) E ( U η , k η ) and cr( E η ) < u- ρ Nn , then U η +1 = Ult u- n ( N, E η ) and k η +1 = n . We say there is a drop in model at η + 1 iff N ⊳ U η . Given alimit η such that U α is well-defined for each α < η , then U η is well-defined iffthere are only finitely many drops in model < η , and then U η is the resultingdirect limit and k η = lim α<η k α . We now define Ult u- k ( M, ~E ) = U λ , if this iswell-defined, and if so, and there is no drop in model, we define the iterationmap i M, u- k~E : M → U λ resulting naturally from the ultrapower maps. Also, ifthere is no drop in model, or the only drop in model occurs at 1, then define¯ i M, u- k~E : N → U λ where N E M is as above.We also make analogous definitions for standard fine structural ultrapowers(as opposed to u-ultrapowers), with notation Ult k ( M, ~E ) and i M,k~E and ¯ i M,k~E . ⊣ Let (
M, k, T ) be suitable and β ≤ T α < lh( T ). Then clearly M T α = Ult u- m ( M T β , ~E T βα ) and i T βα = i M T β ,m~E T βα where m = u- deg T β (with one mapdefined iff the other is), and likewise i ∗T γ +1 ,α = ¯ i M T β ,m~E T βα when pred T ( γ + 1) = β . Let (
M, k, T ) be suitable.We say that T is unravelled iff, if lh( T ) = α + 1 then α is non- T -special.The unravelling unrvl( T ) of T is the unique unravelled u- k -maximal tree S on M , if it exists, such that (i) T E S , (ii) if lh( T ) is a limit then T = S , and (iii)if lh( T ) = α + 1 then β is S -very-special for each β + 1 < lh( S ) with α ≤ β . (So E S β = F ( M S β ) for each β ≥ α , and cr( E S α + i +1 ) < cr( E S α + i ), so lh( S ) < lh( T )+ ω ;the existence of S just depends on the wellfoundedness of the resulting models.)26e say that T is everywhere unravelable iff (i) unrvl( T ↾ β ) exists for every β ≤ lh( T ), and (ii) for each transition point α , letting W = ( T ↾ ( α +2)) b F ( M T α )(a putative nicely padded u- k -maximal tree), W has wellfounded models andunrvl( W ) exists. ⊣ Let (
M, k ) be suitable. Let m = k if M is type ≤
2, and m = k − m M ( k ) = m . We say that ( M, k, m ) is suitable , andsay (
M, k, T , m ) is suitable iff ( M, k, T ) and ( M, k, m ) are suitable.Let U be an m -maximal (not u- m -maximal!) tree on M . We say that U is( M, u) -wellfounded iff for every α < lh( U ), U = Ult u- k ( M, ~E <α ) is wellfounded. ⊣ With notation as above, it is straightforward to see that if M T α is type ≤ U = M T α , and if M T α is type 3 then U pv E Ult( M T α | ( κ + ) M T α , F ( M T α ))where κ = cr( F ( M T α )). Therefore if U is via a reasonable m -maximal strategyfor M , then U is ( M, u)-wellfounded. Let (
M, k, T , m ) be suitable with T unravelled and every-where unravelable. We define a padded m -maximal tree U = conv( T ) on M ,the m -maximal conversion of U , with exchange ordinals e ν U α for α + 1 < lh( U ),by requiring (we verify later that this works):1. lh( U ) = lh( T ).2. e ν U α = e ν T α for α + 1 < lh( U ).3. if α is a transition point of T then E U α = F ( M U α ) (which is the trivialcompletion of F ( M T α ) ↾ ν ( F ( M T α ))).4. If α is T -vs or T -vps then E U α = ∅ and pred U ( α + 1) = α and M U α +1 = M U α and deg U α +1 = deg U α .5. If α is a non-transition point, non- T -vs and non- T -vps then E U α = E T α .6. Let η < lh( T ) be a limit. Fix γ < T η such that ( γ, η ) T does not drop anddoes not contain the successor of a transition point. Let X be the set ofall β < η such that β ≤ ext , T α for some α + 1 ∈ ( γ, η ) T with E T α = ∅ .Then:(a) { e β + 1 (cid:12)(cid:12) β ∈ X } is cofinal in η ;(b) E U e β = C ( E T β ) for each β ∈ X ;(c) for all β , β ∈ X , either f β + 1 ≤ U f β + 1 or vice versa;(d) [0 , η ) U is the ≤ U -downward closure of { e β + 1 (cid:12)(cid:12) β ∈ X } . ⊣ Let ( M, k, T , m ) be suitable with T unravelled and everywhereunravelable, and T non-trivial. Then:1. U = conv( T ) is a well-defined padded m -maximal tree on M . . Suppose lh( T ) = α + 1 (so either M T α is type ≤ or u - deg T α ≥ ). Let ε T + 1 ≤ T α be least such that ( ε T + 1 , α ] T ∩ D T = ∅ , and ε U likewise for U . Then (and let δ, N ∗ be defined by):(a) M U α = ( M T α ) pm ,(b) [0 , α ] T ∩ D T = ∅ ⇐⇒ [0 , α ] U ∩ D U = ∅ .(c) δ = pred T ( ε T + 1) = pred U ( ε U + 1) ,(d) ( N ∗ ) pm = M ∗U ε U +1 where N ∗ = M ∗T ε T +1 , and note if [0 , α ] T ∩ D T = ∅ then N ∗ = ( N ∗ ) pm ,(e) ~F U <α = { C ( E T β ) (cid:12)(cid:12) ∃ γ (cid:2) ε T + 1 ≤ T γ + 1 ≤ T α and β ≤ ex , T γ (cid:3) } , so ~F U <α is equivalent to ~F T <α ,(f) letting n = u - deg T ( α ) and n ′ = deg U ( α ) , we have– M T α = Ult u - n ( N ∗ , ~F T <α ) ,– M U α = Ult n ′ (( N ∗ ) pm , ~F U <α ) ,– i ∗U ε U +1 ,α = i ∗T ε T +1 ,α ↾ C (( N ∗ ) pm ) .3. U is ( M, u) -wellfounded, and moreover:(a) For β ∈ [1 , lh( T )] not of form β = ξ + 2 for a transition point ξ , conv(unrvl( T ↾ β )) = ( U ↾ β ) b ( ∅ , . . . , ∅ ) (where ( U ↾ β ) b ( ∅ , . . . , ∅ ) is an extension of U ↾ β by just padding;the extension is finitely long), and(b) For each transition point ξ , conv (cid:16) unrvl (cid:16) ( T ↾ ( ξ + 2)) b F ( M T ξ +1 ) (cid:17)(cid:17) = ( U ↾ ( ξ + 2)) b ( ∅ , . . . , ∅ ) . (The models witnessing ( M, u) -wellfoundedness appear as the lastmodels of the unravelled trees mentioned in the two clauses above,so they are wellfounded.)4. Let α + 1 < lh( T ) with E T α = ∅ and X = { β (cid:12)(cid:12) β ≤ ext , T α } . Then for each β ∈ X , we have E U e β = C ( E T β ) , and for all β , β ∈ X , if cr( E T β ) ≤ cr( E T β ) then f β + 1 ≤ U f β + 1 ≤ U α + 1 and ( f β + 1 , α + 1] U ∩ D U deg = ∅ .5. Let α + 1 < T α ′ + 1 < lh( T ) be such that E T α = ∅ 6 = E T α ′ and α ′ + 1 isnon- T -special and ( α + 1 , α ′ + 1] T does not drop in model. Let β ≤ ext , T α and β ′ ≤ ext , T α ′ . Then:– e α + 1 ≤ U e β + 1 ≤ U α + 1 ≤ U e α ′ + 1 ≤ U e β ′ + 1 ≤ U α ′ + 1 , Note that T ↾ ξ + 2 has last model M T ξ +1 , and so if ξ is a transition point then the last“extender” of T ↾ ( ξ + 2) is E T ξ = ∅ . So E T ξ = ∅ and E U ξ = F ( M U ξ ) and F ( M T ξ ) = F ( M T ξ +1 ) is equivalent to F ( M U ξ ). ( e α + 1 , α ′ + 1] U ∩ D U = ∅ , and– if ( α + 1 , α ′ + 1] T ∩ D T deg = ∅ then ( e α + 1 , α ′ + 1] U ∩ D U deg = ∅ .6. If U has successor length, then for every γ + 1 ∈ b U with E U γ = ∅ , there is β + 1 < lh( T ) with E T β = ∅ and γ = e β , so E U γ = C ( E T β ) . Proof.
The proof is by induction on lh( T ). For lh( T ) = 1 it is trivial and forlh( T ) a limit, it follows immediately by induction. So suppose lh( T ) = ε + 1 forsome ε > Case . lh( T ) = ξ + n + 2 where n < ω and ξ is non- T -vs but ξ + 1 + i is T -vsfor all i < n .Note then that E T ξ = ∅ (according to the rules of nicely padded trees). Subcase . It is not the case that ξ = ε + 1 for a transition point ε .Let µ ≤ ξ be least such that α is T -vs for each α ∈ [ µ, ξ ). Let¯ T = unrvl( T ↾ ( µ + 1)) , and say lh( ¯ T ) = µ + ℓ + 1 (so ℓ < ω ). Let ¯ U = conv( ¯ T ). So ξ ∈ [ µ, µ + ℓ ] andby induction, we have( M ¯ T µ + ℓ ) pm = M ¯ U µ + ℓ = M ¯ U µ = M U µ = M U ξ . Since ξ is non- T -vs and by subcase hypothesis, therefore E T ξ ∈ E + ( M U ξ ), andnote that lh( E U α ) ≤ lh( E T ξ ) for each α < ξ with E U α = ∅ . So we can set E U ξ = E T ξ = E . Let κ = cr( E ). Let χ = pred T ( ξ + 1) = pred U ( ξ + 1) (recall e ν ¯ T β = e ν ¯ U β for all β + 1 < lh( ¯ T ), by 4.11). Subsubcase . χ is non- T -special and not the successor of a transitionpoint.So ( M T χ ) pm = M U χ and κ < e ν T χ = e ν U χ = ν ( E U χ ), and either χ is a transitionpoint and E T χ = ∅ and E U χ = F ( M U χ ) and M U χ is active type 3, or χ is a non-transition point and E T χ = E U χ . If M T χ = M U χ then ( M U χ ) pv = M T χ || OR( M U χ )and OR( M U χ ) is a cardinal of M T χ , and therefore ξ + 1 ∈ D T iff ξ + 1 ∈ D U ,and ξ + 1 ∈ D T u- deg iff ξ + 1 ∈ D U deg , and when there is a drop, the drops are tothe same model and corresponding u-degree and degree respectively. Note that ξ + 1 is non- T -special, so E is the last extender used in T , U , and lh( T ) = ξ + 2.We claim that ( M T ξ +1 ) pm = M U ξ +1 , and there is appropriate agreement ofiteration maps. This is immediate when M ∗U ξ +1 is non-type 3, so suppose it istype 3. Suppose first there is no drop in model at ξ + 1. So possibly M T χ = M U χ ,and in any case, letting d = u- deg T ξ +1 and e = deg U ξ +1 (so either d = e +1 < ω or d = e = ω ), then M T ξ +1 = Ult u- d ( M T χ , E ) (formed without squashing), whereas M U ξ +1 = Ult e ( M U χ , E ) (formed with squashing). By [6, Definition 2.5] and asin [3, Lemma 9.1], we get ( M T ξ +1 ) pm = M U ξ +1 and the ultrapower maps agreeover ( M U χ ) sq . When there is a drop in model, it is likewise, but slightly simpler,because then we have M ∗T ξ +1 = M ∗U ξ +1 . 29he remaining properties in this subsubcase are now straightforward to ver-ify by induction. Subsubcase . χ = α + 1 for a transition point α (so χ is non- T -special).By subcase hypothesis, χ < ξ . So M U α is type 3, u- deg T ( α ) ≥ M T α ) pm = M U α and E U α ↾ ν = F ( M T α ) ↾ ν where ν = e ν T , U α = ν ( E U α ) = ν ( F ( M T α )). With θ = cr( F ( M T α )) = cr( E U α ), note( M T α ) pv = Ult( M T α | ( θ + ) M T α , F ( M T α )) | OR M T α = Ult( M ∗U α +1 | ( θ + ) M ∗U α +1 , E U α ) | OR M T α = M U α +1 || ( δ + ) M U α +1 , where δ = lgcd( M T α ). Moreover, δ = lgcd( M U α +1 ), because otherwise, θ =lgcd( M ∗U α +1 ) and M ∗U α +1 is active type 2 and F ( M ∗U α +1 ) = E U β for some β < α , but M U α is active with cr( F ( M U α )) = θ , and it is easy to see this gives a contradiction.So ( M T α +1 ) pv ⊳ M U α +1 . Since α + 1 < ξ and by induction, letting T ′ = unrvl(( T ↾ ( α + 2)) b F ( M T α +1 ))(note M T α +1 = M T α ), then T ′ exists (with wellfounded models) and letting U ′ = conv( T ′ ) = U ↾ ( α + 2) b ( ∅ , . . . , ∅ ) , then ( M T ′ ∞ ) pm = M U ′ ∞ = M U α +1 and letting e = deg U α +1 = deg U ′ ∞ and d =u- deg T ′ ( ∞ ), then e = m M T ′∞ ( d ) (so e = d or e = d − ν ≤ κ < e ν T α +1 = e ν U α +1 . Since OR( M T α +1 ) = OR( M T α ) is a cardinal of M U α +1 , clearly ξ + 1 ∈ D T iff ξ + 1 ∈ D U , and if ξ + 1 ∈ D T then M ∗T ξ +1 = M ∗U ξ +1 .So in the dropping case, it is easy to maintain the hypotheses (and ξ + 1 isnon- T -special).Suppose ξ + 1 / ∈ D T . Then M ∗T ξ +1 = M T α +1 = M T α and since ν ≤ κ , we getu- deg T ( α + 1) = 0 and M T ξ +1 = Ult u-0 ( M T α +1 , E ) = Ult( M T α +1 , E ) , so ξ + 1 is T -special. Noting δ < ρ e ( M U α +1 ), U does not drop in model or degreeat ξ + 1, and M U ξ +1 = Ult e ( M U α +1 , E ).Now F ( M T ξ +1 ) is equivalent to the two-step iteration ( F ( M T α ) , E ). With T ′ from above, let lh( T ′ ) = α + ℓ + 3 (so ℓ < ω ; note lh( T ′ ) ≥ α + 3 as T ′ pads at α and E T ′ α +1 = F ( M T α +1 ) = F ( M T α )).Let T ′′ = unrvl( T ↾ ( ξ + 2)) and U ′′ = conv( T ′′ ) = U ↾ ( ξ + 2) b ( ∅ , . . . , ∅ ) . Then an easy induction gives that for each i ≤ ℓ , M T ′′ ξ +1+ i = Ult u-0 ( M T ′ α +1+ i , E )30nd F ( M T ′′ ξ +1+ i ) is equivalent to the two-step iteration ( F ( M T ′ α +1+ i ) , E ), andlh( T ′′ ) = ξ + ℓ + 3, and recalling d = u- deg T ′ ( α + 2 + ℓ ), note d = u- deg T ′′ ( ξ +2 + ℓ ), and (letting) N ∗ = M ∗T ′ α +2+ ℓ = M ∗T ′′ ξ +2+ ℓ , we have M T ′′ ξ +2+ ℓ = Ult u- d ( N ∗ , F ( M T ′′ ξ +1+ ℓ ))= Ult u- d (Ult u- d ( N ∗ , F ( M T ′ α +1+ ℓ )) , E )= Ult u- d ( M T ′ α + ℓ +2 , E ) , and since ( M T ′ α + ℓ +2 ) pm = M U α +1 and d, e = m M T ′ α + ℓ +2 ( d ) correspond appro-priately and the ultrapower maps of T ′ , U ′ agree appropriately, and M U ξ +1 =Ult e ( M U α +1 , E ), we get( M T ′′ ξ +2+ ℓ ) pm = (Ult u- d ( M T ′ α + ℓ +2 , E )) pm = M U ξ +1 = M U ′′ ξ +1 = M U ′′ ξ +2+ ℓ ,e = m M T ′′ ξ +2+ ℓ ( d ) and the ultrapower maps of T ′′ , U ′′ agree appropriately also.Regarding part 4 for T ′′ and for X = { β (cid:12)(cid:12) β ≤ ext , T ξ + 1 } , we have X = { ξ, ξ + 1 } , and e ξ = ξ and ] ξ + 1 = α , and α + 1 ≤ U ′′ ξ + 1 ≤ U ′′ ξ + 2, and U does not drop in model or degree at ξ + 1 or ξ + 2 (note that U ′′ pads at ξ + 1, so pred U ′′ ( ξ + 2) = ξ + 1 etc). Parts 5 and 6 now follow from the aboveconsiderations and by induction applied to T ′ . Subsubcase . χ is T -special.So χ is not the successor of a transition point. It is straightforward to see ν ( F ( M T χ )) = sup α<χ ν ( E T α ) ≤ sup α<χ e ν T α ≤ κ. (Note that we can have, for example, χ = α + 1 and ν ( E T α ) < ι ( E T α ) = e ν T α , andin that case, ν ( F ( M T χ )) = ν ( E T α ).) Note that F (Ult u-0 ( M T χ , E )) is equivalentto the two-step iteration ( F ( M T χ ) , E ). So things are almost the same as inSubsubcase 1.1.2, so we leave the details to the reader. Subcase . ξ = ε + 1 for a transition point ε .So with ¯ T as before, and µ + ℓ + 1 = lh( ¯ T ) = lh( ¯ U ), we have ¯ T E T and¯ U E U and µ + ℓ = ε and E T ε = ∅ and E U ε = F ( M U ε ) and OR( M U ε ) < lh( E T ε +1 ).By observations in Subsubcase 1.1.2, either(i) E T ε +1 ∈ E ( M T ε +1 ) ∩ E ( M U ε +1 ), and we set E U ε +1 = E T ε +1 , or(ii) E T ε +1 = F ( M T ε +1 ), and we set E U ε +1 = ∅ .In case (i) we now proceed as before with ξ = ε +1. (Letting χ = pred T ( ε +2) =pred U ( ε + 2), if χ = ε + 1, it is like Subsubcase 1.1.2; and if χ < ε + 1 and wedefine T ′′ , U ′′ much as before, then ε is not of form e β (computed with respectto T ′′ ) for any β + 1 < lh( T ′′ ) with E T ′′ β = ∅ , but note that ε + 1 / ∈ b U ′′ in thiscase.) In case (ii) it is similar, but the role of the pair ( E T ξ , E U ξ ) = ( E, E ) inthe previous cases is replaced by the pair ( E T ε +1 , E U ε ), which works since thesetwo extenders are equivalent to one another, and here we have ] ε + 1 = ε .31 ase . lh( T ) = λ + n + 1 where λ is a limit and E T λ + i is T -vs for all i < n .Let b = [0 , λ ) T . Note that λ is T -special iff α is T -special for all sufficientlylarge α ∈ b . By parts 4 and 5 for trees of length < λ , b induces a U ↾ λ -cofinal branch, which has the properties required by Definition 4.11(6). (If λ is T -special then apply part 4 to unrvl( T ↾ ( α ′ + 1)) for sufficiently large α ′ + 1 < T λ .)So if λ is non- T -special, then induction easily shows that ( M T λ ) pm = M U λ and iteration maps agree appropriately etc. If λ is T -special, then proceedessentially as in Subsubcase 1.1.3, but using ~F T αλ and the equivalent ~F U αλ , where α ∈ b is sufficiently large, in place of single extenders of T , U .This completes the proof of the lemma. (cid:3) Let ( M, k, m ) be suitable and U ′ be an ( M, u) -wellfounded m -maximal tree on M . Then there is a unique pair ( T , U ) such that T is an unrav-elled everywhere unravelable tree T with ( M, k, T , m ) suitable, U = conv( T ) ,and U ′ is given by removing all padding from U . Proof sketch.
The proof is very much like that of Lemma 2.21, and anyway isstraightforward. So we just give a sketch, and the reader should refer to 2.21for more detail.We ignore U ′ itself and just directly discuss U . We proceed by induction onlh( U ). The induction is an easy consequence of Lemma 4.12 except for the casethat lh( U ) = λ + 1 with limit λ , so consider this assuming that the lemma holdsfor trees of length ≤ λ . In particular, we have corresponding trees T ↾ λ and U ↾ λ . Claim.
There is α < U λ such that for all transition points ξ of T ↾ λ with ξ + 1 ∈ ( α, λ ) U , letting δ = lgcd( M T ξ ), there is χ ∈ [ ξ + 1 , λ ) U such that i U ξ +1 ,χ ( δ ) ≤ cr( i U χλ ). Proof.
If not, then select a sequence h ( ξ n , δ n ) i n<ω of witnessing pairs ( ξ, δ ) with ξ n < ξ n +1 . Then just note that since δ n ≤ i ∗U ξ n +1 (cr( E U ξ n )), we get i U ξ n +1 ,λ ( δ n ) > i U ξ n +1 +1 ,λ ( δ n +1 )for each n < ω , so M U λ is illfounded, a contradiction. (cid:3) Now the more complex case is when there are cofinally many η < U λ whichare transition points, so consider this case. Fix ξ < U λ with ( ξ , λ ) U ∩ D U deg = ∅ .For transition points η with η + 1 ∈ ( ξ , λ ) U , let χ η be the least χ such that i U η +1 ,χ ( δ ) ≤ cr( i U χλ ) and E U γ = ∅ , where γ + 1 = succ U ( χ, λ ). For non-transitionpoints η such that η + 1 ∈ ( ξ , λ ) U and E U η = ∅ , let χ η = η + 1.Let χ be least such that either χ = λ or ξ is T -special and letting δ =lgcd( M T ξ ), we have i U ξ χ ( δ ) ≤ cr( i U χλ ) and E U γ = ∅ where γ + 1 = succ U ( χ, λ ).Let b ′ = [0 , ξ ] T ∪ b ′ where b ′ is the < χ -closure of χ ∩ { χ η (cid:12)(cid:12) η + 1 ∈ ( ξ , λ ) U and E U η = ∅} . b ′ is a branch of T ↾ λ with b ′ \ ( ξ + 1) ∩ D T deg = ∅ .Suppose χ < λ , so ξ is T -special. Then much as before, there is χ ′ < U χ such that α is T -vs for each α ∈ [ χ ′ , χ ), and b ′ ∪ { χ ′ } is a branch of T ↾ λ , and( ξ , χ ′ ] T ∩ D T = ∅ . So letting ℓ α be the ℓ < ω such that unrvl( T ↾ ( α + 1)) haslength α + 1 + ℓ , note that ℓ ξ = ℓ χ ′ > ℓ χ . Now set ξ = χ . Given ξ n , define b ′ n from ξ n like b ′ was defined from ξ . Then we reach some n < ω with b ′ n cofinalin λ . Define T ′ = T ↾ λ b b ′ n .One can now show that T ′ has wellfounded models, unrvl( T ′ ) exists andconv(unrvl( T ′ )) = U b ( ∅ , . . . , ∅ ) , which gives what we need.This completes the sketch of the proof of the lemma. (cid:3) Let (
M, k, m ) be suitable. Let T be a u- k -maximal tree on M . Note there is a unique nicely padded tree T ′ on M which is equivalent to T ; write pad( T ) = T ′ . Say that T is unravelled or everywhere unravelable iff T ′ is. For a padded m -maximal tree U on M pm , let unpad( U ) be the tree givenby removing all padding from U . We extend the conv function as follows: For T unravelled everywhere unravelable u- k -maximal (without padding in T ), defineconv( T ) = unpad(conv(pad( T ))) . (There is no ambiguity here, because if T = pad( T ), i.e. pad( T ) contains nopadding, then U = conv( T ) as defined earlier, then U contains no padding,so unpad( U ) = U .) Note here that the padding, depadding, and conversionprocesses preserve tree length modulo + ω ; in fact,– lh( T ) ≤ lh(pad( T )) < lh( T ) + ω ,– lh(conv(pad( T ))) = lh(pad( T )), and– lh(conv( T )) ≤ lh(conv(pad( T ))) < lh(conv( T )) + ω = lh( T ) + ω . ⊣ Let (
M, k, m ) be suitable and η ∈ OR. Let Σ be a (u- k, η + ω )-iteration strategy for M and let T be via Σ, of successor length. Then note that T is arbitrarily finitely extendible (that is, every putative u- k -maximal tree S such that T E S has wellfounded models). It follows that T is everywhereunravelable. Also, if Γ is an ( m, η + ω )-iteration strategy and U is via Γ, then U is M -u-wellfounded. (If U has length α + 1 and M U α is active type 3, then( M + U α ) pv E M U ′ α +1 , where U ′ = U b F ( M U α ).)Now suppose that cof( η ) > ω and let Σ be an ( m, η +1)-iteration strategy for M . Then Σ extends trivially to an ( m, η + ω )-iteration strategy for M . Likewisefor (u- m, η + 1)- and (u- m, η + ω )-strategies. (Any illfoundedness would reflectdown into some tree via Σ of length < η .) Let ( M, k, m ) be suitable and η ∈ OR . Then M is (u - k, η + ω ) -iterable iff M pm is ( m, η + ω ) -iterable. Moreover, there is a bijection Σ conv(Σ)33 rom the (u - k, η + ω ) -iteration strategies Σ for M to the ( m, η + ω ) -iterationstrategies conv(Σ) for M pm such that for each unravelled (and everywhere un-ravelable; see Remark 4.15) u - k -maximal tree T on M , we have T is via Σ ⇐⇒ conv( T ) is via conv(Σ) , and therefore the properties described in Lemma 4.12 holds. So for example, if T has successor length then so does U = conv( T ), and M T∞ = M U∞ , and if [0 , ∞ ] T does not drop, then neither does [0 , ∞ ] U , and i T ∞ ↾ ( M pm ) sq = i U ∞ . Since the proof above functions at a tree-by-tree level, it caneasily be adapted to natural kinds of partial strategies; for example, strategieswhich act on trees based on M | δ for some M -cardinal δ < ρ ( M pm ), or treeswhich use only extenders E ∈ E N + such that ν E is an N -cardinal, etc, as theseproperties are preserved appropriately by the conversion processes. Beyond suchpreservation, we just need to know that all the u- k -maximal trees are everywhereunravelable, and all the m -maximal trees are M -u-wellfounded. Let (
M, k, m ) be suitable. The iteration game G opt ( M pm , m, λ, η ) ∗ is defined in [6] ( opt stands for optimal ). The game G opt ( M, u- k, λ, η ) ∗ iscompletely analogous, but with u-fine structure. The game G unrvlopt ( M, u- k, λ, η ) ∗ makes the restriction that player I may only end rounds with unravelled trees.We define via these games the corresponding iteration strategies and iterabilitynotions ( optimally- ( m, λ, η ) ∗ -iterable , unravelled-optimally- (u - k, λ, η ) ∗ -iterable ,etc). ⊣ Let ( M, k, m ) be suitable and η, λ ∈ OR with η a limit ordinal.Then:1. M is unravelled-optimally- (u - k, λ, η ) ∗ -iterable iff M pm is optimally- ( m, λ, η ) ∗ -iterable, and
2. if cof( η ) > ω then M is unravelled-optimally- (u - k, λ, η + 1) ∗ -iterable iff M pm is optimally- ( m, λ, η + 1) ∗ -iterable.Moreover, there are bijections Σ conv(Σ) between the sets classes of iterationstrategies, whose action in each round (that is, for each normal tree in a stack)by the conversion of Theorem 4.16. Proof.
The iteration games produce putative stacks hT α i α<γ and hU α i α<γ ′ re-spectively. For round 0, producing T , U , we just use Theorem 4.16 (and theconversion process used in its proof). Given ~ T = hT α i α<ξ and ~ U = hU α i α<ξ , The game builds a putative m -maximal stack. The subscript opt means that player I maynot make artificial drops, and the asterisk (introduced in [8]) means that if in some round γ < λ , a tree T γ is produced of length η (with wellfounded well-defined models) then theentire game stops and player II has won. Actually in this case, the asterisk makes no difference, as η is a limit ordinal. M ~ T∞ and M ~ U∞ well-defined and wellfounded, we will have that ( M ~ T∞ , k ′ , m ′ )is suitable, where k ′ = u- deg ~ T∞ and m ′ = deg ~ U∞ , and that ( M ~ T∞ ) pm = M ~ U∞ .Thus, for round ξ , we can again use Theorem 4.16. The agreement betweenmodels, degrees and iteration maps given by Theorem 4.16 and Lemma 4.12ensures that the inductive hypotheses carry through limit stages. (cid:3) Like for normal trees, this also adapts easily to partial strategiesfor stacks.
References [1] Gunter Fuchs. λ -structures and s -structures: Translating the iterationstrategies. Annals of Pure and Applied Logic , 162(9):710–751, 2011.[2] Gunter Fuchs. λ -structures and s -structures: Translating the models. An-nals of Pure and Applied Logic , 162(4):257–317, 2011.[3] William J. Mitchell and John R. Steel.
Fine structure and iteration trees ,volume 3 of
Lecture Notes in Logic . Springer-Verlag, Berlin, 1994.[4] Farmer Schlutzenberg. The definability of E in self-iterable mice. arXiv:1412.0085.[5] Farmer Schlutzenberg. Fine structure from normal iterability.arXiv:2011.10037.[6] Farmer Schlutzenberg. Iterability for (transifnite) stacks. To appear inJournal of Mathematical Logic, arXiv:1811.03880.[7] Farmer Schlutzenberg. A premouse inheriting strong cardinals from V . An-nals of Pure and Applied Logic , 171(9), 2020.[8] John R. Steel. Core models with more Woodin cardinals.