Approachable Free Subsets and Fine Structure Derived Scales
aa r X i v : . [ m a t h . L O ] J a n Approachable Free Subsets and Fine StructureDerived Scales ∗ Dominik Adolf and Omer Ben-NeriaFebruary 1, 2021
Abstract
Shelah showed that the existence of free subsets over internally ap-proachable subalgebras follows from the failure of the PCF conjecture onintervals of regular cardinals. We show that a stronger property called theApproachable Bounded Subset Property can be forced from the assump-tion of a cardinal λ for which the set of Mitchell orders { o ( µ ) | µ < λ } isunbounded in λ . Furthermore, we study the related notion of continuoustree-like scales, and show that such scales must exist on all products incanonical inner models. We use this result, together with a covering-typeargument, to show that the large cardinal hypothesis from the forcing partis optimal. The study of set theoretic algebras has been central in many areas, with manyapplications to compactness principles, cardinal arithmetic, and combinatorialset theory.An algebra on a set X is a tuple A = h X, f n i n<ω where f n : X k n → X is afunction. A sub-algebra is a subset M ⊆ X such that f n ( x , . . . , x k n − ) ∈ M for all ( x , . . . , x k n − ) ∈ M k n and n < ω . The set of sub-algebras of A is knownas a club (in P ( X )). The characteristic function χ M of M is defined on theordinals of M by χ M ( τ ) = sup( M ∩ τ ).Shelah’s celebrated bound in cardinal arithmetic ([26]) states that if ℵ ω is astrong limit cardinal then 2 ℵ ω < min {ℵ ω , ℵ (2 ℵ ) + } . Starting from a supercompact cardinal, Shelah proved that for every α < ω ,there exists a generic extension in which 2 ℵ ω = ℵ α +1 (see [15]). It is a centralopen problem in cardinal arithmetic if 2 ℵ ω ≥ ℵ ω is consistent. A major break-through towards a possible solution is the work of Gitik ([14],[10]) on the failure ∗ Mathematics Subject Classification.
Primary 03E04, 03E45, 03E55
1f the PCF-conjecture. Shelah’s PCF conjecture states that | pcf( A ) | ≤ | A | for every progressive set A of regular cardinals. In [27], Shelah has extractedremarkable freeness properties of sets over subalgrbras, from the assumptionof 2 ℵ ω ≥ ℵ ω , or more generally, from the assumption | pcf( A ) | > | A | for aprogressive interval of regular cardinals | A | . Definition 1.
Let A = h X, f n i n be an algebra and x ⊂ X . We say that x is free with respect to A if for every δ ∈ x and n < ω , δ f n “( x \ { δ } ) <ω .More generally, x is free over a subalgebra N ⊆ A if for every δ ∈ x and n < ω , δ f n “( N ∪ ( x \ { δ } )) <ω .A cardinal λ has the Free Subset Property if every algebra A on λ or a bigger H θ ,has a free subset x ⊆ λ which is cofinal in λ . A regular cardinal λ with the FreeSubset Property is Jonsson. Koepke [19] has shown that the free subset propertyat ℵ ω is equiconsistent with the existence of a measurable cardinal. For asingular limit λ of a progressive interval | A | , it is shown in [27] that if | pcf( A ) | > | A | then λ satisfies the Free Subset Property. In his PhD thesis ([23]), Pereira hasisolated the notion of the Approachable Free Subset Property (AFSP) to playa critical role in the result from [27]. The Approachable Free Subset Propertyfor a singular cardinal λ asserts that there exists some sufficiently large H θ , θ > λ and an algebra A on H θ such that for every internally approachablesubstructure N ≺ A with | N | < λ , there exists an infinite sequence of regularcardinal h τ i | i < cof( λ ) i ∈ N such that the set x = { χ N ( τ i ) | i < cof( λ ) } is freeover N .Pereira showed that Shelah’s proof yields that if λ is a limit of a progressiveinterval A or regular cardinals and | pcf( A ) | > | A | then the Approachable FreeSubset Property holds at λ .Working with fixed sequences h τ n | n < ω i of regular cardinal, we considerhere the following version of this property. Definition 2.
The
Approachable Free Subset Property (AFSP) with re-spect to h τ n i n asserts that for every sufficiently large regular θ > λ = ( ∪ n τ n ) andfor every internally approachable subalgebra N ≺ A , of an algebra A extending( H θ , ∈ , h τ n i n ), satisfying | N | < λ there exists a cofinite set x ⊆ { χ N ( τ n ) | n <ω } which is free over N .By moving from one cardinal θ to θ ′ > θ if needed, it is routine to verifythe definition of AFSP with respect to a sequence h τ n i n can be replaced with asimilar assertion in which the requirement of “every internally approachable N ”is replaced with “ for every internally approachable in some closed unboundedsubset of P λ ( A )”. Clearly, if AFSP holds with respect to a sequence h τ n i n thenAFSP holds with respect to the singular limit λ = ∪ n τ n , as in the original def-inition of [23]. I.e., min( A ) > | A | . See Definition 10 ℵ ω < ℵ ω . I.e., proving (in ZFC) that AFSPmust fail at ℵ ω (or AFSP fails w.r.t every subsequence h τ n i n of {ℵ k | k < ω } )would imply that 2 ℵ ω < ℵ ω . To this end, Pereira ([23]) has isolated the notionof tree-like scales , as a potential tool of proving AFSP must fail. Definition 3.
Let h τ n i n<ω be an increasing sequence of regular cardinals. Ascale ~f = h f α | α < η i is a tree-like scale on Q n τ n if for every α = β < η and n < ω , f α ( n + 1) = f β ( n + 1) implies f α ( n ) = f β ( n ).Pereira shows in [24] that the existence of a continuous tree-like scale on aproduct Q n<ω τ n guarantees the failure of AFSP with respect to h τ n i n (see alsoLemma 15), and further proves that continuous tree-like scales, unlike otherwell-known types of scales, such as good scales, can exist in models with someof the strongest large cardinal notions, e.g. I -cardinals. Moreover, Cummings[5] proved that tree-like scales can exist above supercompact cardinals. Theseresults show that as opposed to other well-known properties of scales such asgood and very-good scales, which exhibit desirable ”local” behaviour but cannotexist in the presence of certain large cardinals ([6]), the notion of continuous tree-like scales may coexist with the some of the strongest large cardinals hypothesis.The consistency of the inexistence of a continuous tree-like scale on a product Q n τ n of regular cardinal has been established by Gitik in [12], from the consis-tency assumption of a cardinal κ satisfying o ( κ ) = κ ++ +1.The argument makesa sophisticated use of the key features of Gitik’s extender based Prikry forcingby a ( κ, κ ++ )-extender. Concerning the possible consistency of the Approach-able Free Subset Property, Welch ([31]) has shown that AFSP with respect to asequence h τ n i n implies that the large cardinal assumption of Theorem 4 holdsin an inner model.It remained open whether AFSP with respect to some sequence h τ n | n < ω i isconsistent at all, and if so, whether its consistency strength is strictly strongerthan the (seemingly) weaker property, of no continuous tree-like scale on Q n τ n .The current work answers both questions: Theorem 4.
It is consistent relative to the existence of a cardinal λ such thatthe set of Mitchell orders { o ( µ ) | µ < λ } is unbounded in λ , that the Approach-able Free Subset Property holds with respect to some sequence of regular cardinals ~τ = h τ n i n .Moreover,the sequence τ n can be made to be a subsequence of the first uncount-able cardinals, in a model where λ = ℵ ω . Theorem 5.
Let λ be a singular cardinal of countable cofinality such that thereis no inner model M with λ = sup { o M ( µ ) | µ < λ } . Let h τ n | n < ω i be a see Definition 9 for the definition of a continuous scale E.g., on the fact that there are unboundedly many pairs ( α, α ∗ ) ∈ [ κ ] , sharing the sameRudin-Keisler projection map π α ∗ ,α . equence of regular cardinals cofinal in λ . Then Q n<ω τ n carries a continuoustree-like scale. To achieve the proof of Theorem 5, we establish a result of an independentinterest, that the continuous tree-like scales naturally appear in fine-structuralcanonical inner models. Thus obtaining complementary result to aforemen-tioned theorems by Pereira and Cummings, i.e. we know that no large cardinalproperty that can consistently appear in canonical inner models disproves theexistence of products with continuous tree-like scales (e.g., Woodin cardinals).
Theorem 6.
Let M be a premouse such that each countable hull has an ω -maximal ( ω + 1) -iteration strategy. Let λ ∈ M be a singular cardinal of count-able cofinality. Let h κ i : i < ω i be a sequence of regular cardinals cofinal in λ .Then Q i<ω κ i /J bd carries a continuous tree-like scale. Continuous tree-like scales on products of successor cardinals in L whereimplicitly constructed by Donder, Jensen, and Stanly in [7]. In the course ofproving Theorem 4, we establish the consistency of a principle stronger thanAFSP, which we call the Approchable Bounded Subset Property.Let N be a subalgebra of A = h H θ , f n i n and ~τ = h τ n i n be an increasing sequeneof cardinals. Given a set x ⊆ H θ , we define N [ x ] to be the A -closure of theset ( x ∪ N ). We say that N satisfies the Bounded Appending Property with respect to ~τ if for every n < ω , setting x = { χ N ( τ n ) | n = n } thenthe addition of x to N does not increase the supremum below τ n , namely χ N [ x ] ( τ n ) = χ N ( τ n ). Definition 7.
The
Approachable Bounded Subset Property holds withrespect to h τ n i n if for every sufficiently large regular θ > λ = ( ∪ n τ n ) andinternally approachable subalgebra N ≺ A , of an algebra A extending ( H θ , ∈ , h τ n i n ), that satisfies | N | < λ , then N satisfies the bounded appending propertywith respect to a tail of h τ n i n .We show in Lemma 15 ABSP with respect to a sequence h τ n i n implies AFSPwith respect to the same sequence, as well as the inexistence of a continuous es-sentially tree-like scale; a weakening of tree-like scale introduced by Pereira(see Definition 9). The proof of the forcing Theorem 4, stated above, goesthrough proving that ABSP is consistent with respect to a sequence of regulars h τ n i n .The following summarizes the main results of this paper: Corollary 8.
The following principles are equiconsistent:1. There exists a sequence of regular cardinals h τ n | n < ω i for which theApproachable Bounded Subset Property holds.2. There exists a sequence of regular cardinals h τ n | n < ω i for which theApproachable Free Subset Property holds. . There exists a sequence of regular cardinals h τ n | n < ω i for which theproduct Q n τ n does not carry a continuous Tree-Like scale.4. There exists a cardinal λ such that the set of Mitchell orders { o ( µ ) | µ < λ } is unbounded in λ . The paper is organized as follows:
The remainder of this section will bededicated to discussing preliminary material in PCF theory and the theory ofinner models.
Section 2 will dedicated to the forcing argument establishingthe proof Theorem 4. In
Section 3 we discuss how to construct tree-like scalesfrom the fine structure of canonical inner models. In
Section 4 we will usethese fine structural scales to derive scales on products in V using a covering-like argument. Finally, in Section 5 we finish with a list of open problems.
Acknowledgments:
The work on this project was initiated following a suggestion by Assaf Rinot tostudy the consistency of the Approachable Free Subset Property. The authorsare grateful for this suggestion and for supporting the first author during theacademic year of 2018-2019 at Bar-Ilan University under a grant from the Eu-ropean Research Council (grant agreement ERC-2018-StG 802756). The initialidea for the inner model construction of continuous tree-like scale was conceivedduring the Berkeley conference on Inner Model theory in July 2019. The authorswould like to thank Ralf Schindler and John Steel for organizing the meetingand creating the opportunity for this collaboration. The first author would liketo thank Grigor Sargsyan for his generous support and warm hospitality duringthe Spring of 2020 (NSF career award DMS-1352034). During that time thefirst author had the opportunity to travel to Pittsburgh. It was there that somesignificant improvements were made to the lower bound argument, and wouldlike to thank James Cummings for the opportunity to present this research andinsightful conversations. The second author was partially supported by the Is-rael Science Foundation (Grant 1832/19). He would like thank Luis Pereirafor insightful discussions on the subject and many valuable remarks on this pa-per, and to Spencer Unger and Philip Welch for many valuable comments andsuggestions.
For a set X and a cardinals λ , P λ ( X ) denotes the collections of all subsets a ⊆ X of size | a | < λ . J bd denotes the ideal of bounded subsets of ω . Let I be an ideal on ω and f, g two functions from ω to ordinals. We write f < I g if { n < ω | f ( n ) ≥ g ( n ) } ∈ I . We write f < ∗ g for f < J bd g . Let h τ n | n < ω i be a sequence of ordinals of strictly increasing cofinalities. Asequence of functions ~f = h f α | α < η i ⊆ Q n τ n of a regular length η , is a pre-scale in ( Q n τ n , < I ) if ~f is strictly increasing in the ordering < I . A prescale5s a scale if it is cofinal in Q n τ n . As we focus on J bd from this point forward,we will frequently say that ~f is a (pre-)scale in Q n τ n , without mentioning theideal J bd . Definition 9.
Suppose that ~f = h f α | α < η i is a (pre-)scale in Q n τ n .1. ~f is continuous if for every limit ordinal δ < η of uncountable cofinality,the sequence ~f ↾ δ is < ∗ -cofinal in Q n f δ ( n ).2. ~f is Tree-like if for every α = β < η and n < ω , if f α ( n + 1) = f β ( n + 1)then f α ( n ) = f β ( n ).3. ~f is Essentially Tree-like if for every n < ω and µ ∈ [ τ n , τ n +1 ) the set { µ ′ < τ n | ∃ β < η, f β ( n + 1) = µ and f β ( n ) = µ ′ } is nonstationary in τ n .If a product Q n τ n carries a scale, it is not difficult to find another scale onit with the tree-like property (see Pereira [23]), but such a scale need not becontinuous. Considering notions such as the Approachable Free Subset Property or the Ap-proachable Bounded Subset Property with respect to subalgebras of Algebras A = ( θ, f n ) n , there is no harm in replacing the domain θ with another set of thesame size, such as H θ in cases relevant to us, and adding more structure to thealgebra. Therefore, from this point on, we will only restrict ourselves to set the-oretic algebras A of the form A = ( H θ , ∈ , f n ) n , which extend the model ( H θ , ∈ )in the language of set theory, and include Skolem functions. In particular, asubalgebra N ≺ A will always be an elementary substructure.This allows us to reformulate our notion of freeness. Assuming the algebra A is rich enough to satisfy a fraction of ZFC , and N ⊆ A is sufficiently closedso it is an elementary substructure N ≺ A , then the fact that a set x is freeover N is equivalent to having that for every δ ∈ x and a function f ∈ N , δ f “( x \ { δ } ) <ω .The notion of internally approachable structures was formally introducedin [9]. We refer the reader to [8] for further exposition. The definition belowis similar to the standard ones, with the addition that here, we will focus oninternally approachable unions of uncountable cofinality. Definition 10.
An elementary subalgebra (substructure) N ≺ A of an algebra A = ( H θ ; ∈ , f n ) n is said to be internally approachable of length ρ if N = S i<ρ N i is a union of a sequence ~N = h N i | i < ρ i of elementary subalgebras N i ≺ N , we will always be abe to assume so specifically, the Replacement property j < ρ , ~N ↾ j = h N i | i < j i belongs to N .We say that N ≺ A is internally approachable if it is internally approach-able of length ρ for some ρ of uncountable cofinality cof( ρ ) > ℵ . Notation 11.
Let N ≺ ( H θ ; ∈ ) for a regular cardinal θ . • For every regular cardinal τ ∈ N , define χ N ( τ ) = sup( N ∩ τ ). • Given a sequence ~τ = h τ n | n < ω i ⊆ N define the function χ ~τN ∈ Q n τ n by χ ~τN ( n ) = χ N ( τ n ) if the last ordinal is strictly smaller than τ n , and 0otherwise.The following folklore result connects continuous scales with characteristicfunctions of internally approachable structures. We include a proof for com-pleteness. Lemma 12.
Suppose that ~τ = h τ n | n < ω i ∈ N is a strictly increasing sequenceof regular cardinals for which Q n τ n carries a continuous scale ~f = h f α | α < η i .For every N ≺ H θ which is internally approachable of size | N | < S n τ n , with ~f ∈ N , if δ = χ N ( η ) then χ ~τN ( n ) = f δ ( n ) for all but finitely many n < ω .Proof. Let ~N = h N i | i < ρ i be a sequence witnessing N = ∪ i N i is internallyapproachable of length ρ which has uncountable cofinality. Since ~f is continuous,it suffices to show that ~f ↾ δ is < ∗ -cofinally interleaved with the functions in Q n χ ~τN ( n ) to prove that χ ~τN ( n ) = f δ ( n ) for almost all n < ω . First,for every f α ∈ ~f ↾ δ there exists some β ∈ N ∩ δ so that α < β , and thus f α < ∗ f β .But f β ∈ N since β ∈ N , which means that f β ∈ Q n χ ~τN ( n ). Next, fix g ∈ Q n χ N ( τ n ). We show that g < ∗ f α for some α < δ . To this end, N = S i<ρ N i guarantees that for each n < ω there is i < ρ such that g ( n ) < χ N i ( τ n ). Sincecof( ρ ) > ℵ there is i < ρ such that g ( n ) < χ N i ( τ n ) for all n , and in particular, g < ∗ χ ~τN i . Since ~f ∈ N is < ∗ -cofinal in Q n τ n and χ ~τN i ∈ N , there is some α ∈ N ∩ η ⊆ δ so that χ ~τN i < ∗ f α , and thus g < ∗ f α . Lemma 13.
Let λ < θ be cardinals with θ regular, and ⊳ be a well-orderingof H θ . Suppose that S ⊆ P λ ( H θ ) is a stationary set of internally approachablestructures N ≺ ( H θ ; ∈ , ⊳ ) , and X ∈ H θ is a set which belongs to all N ∈ S ,and satisfies that | X ω | ≤ η is a regular cardinal and ρ ℵ < η for every cardinal ρ < λ . Then, for every assignment which maps each N ∈ S to a countablesequence h x Nn | n < ω i ∈ X ω which is contained in N , there exists a stationarysubset S ∗ ⊆ η and a constant sequence h x n | n < ω i such that for every δ ∈ S ∗ there is N ∈ S satisfying χ N ( η ) = δ and h x Nn i n = h x n i n .Proof. Let h ~x α | α < η i be the ⊳ -least enumeration of X ω in H θ , where each ~x α is of the form h x αn | n < ω i . For each N ∈ S let α N < η be such that h x Nn | n <ω i = ~x α N . Note that α N need not be a member of N since h x Nn | n < ω i neednot. Since each N ∈ S is the union of a sequence h N i | i < ρ i with cof( ρ ) > ℵ ,and h x Nn | n < ω i ⊆ N there is some i < ρ so that h x Nn | n < ω i ⊂ N i , and7hus h x Nn | n < ω i ∈ ( X ∩ N i ) ω ∈ N . Moreover, as | X ∩ N i | < λ , we havethat | ( X ∩ N i ) ω | < η , and therefore there exists some β N ∈ η ∩ N so that( X ∩ N i ) ω ⊂ h ~x α | α < β N i . We conclude that, α N < β N < χ N ( η ). Next,define S = { χ N ( η ) | N ∈ S} . S ⊆ η is stationary, and by choosing for each δ ∈ S a specific structure N δ ∈ S with δ = χ N ( η ), we can form a pressing downassignment taking each δ ∈ S to α N δ < δ . Let α ∗ < η and S ∗ ⊆ S be so that α N δ = α ∗ for all δ ∈ S ∗ . The claim follows for S ∗ and h x n | n < ω i = ~x α ∗ .Let ~τ = h τ n | n < ω i be an increasing sequence of regular cardinals, λ = ∪ n τ n , and θ > λ + regular. A set C ⊆ P λ ( H θ ) is a closed unbounded setif it contains all elementary substructures M ≺ A of size | M | < λ of somealgebra A = ( H θ , ∈ , f n ) n on H θ . We reformulate the definitions of ApproachableFree Subset Property and Approachable Bounded Subset Property from theintroduction. Definition 14.
1. Let F : [ λ ] <ω → λ be a function. We say that a subset X ⊆ λ is free with respect to F if for every γ ∈ X , γ F [ X \ { γ } ] <ω .2. The Approachable Free Subset Property (AFSP) with respect to ~τ asserts that there exists a closed unbounded set C ⊆ P λ ( H θ ) of structures N ≺ ( H θ ; ∈ ) so that for every internally approachable structure N ∈ C there exists some m < ω such that the set { χ N ( τ n ) | m ≤ n < ω } is freewith respect to every function F ∈ N
3. The
Approachable Bounded Subset Property (ABSP) with respectto ~τ asserts that there exists a closed unbounded set C ⊆ P λ ( H θ ) ofstructures N ≺ ( H θ ; ∈ ) so that for every internally approachable structure N ∈ C there exists some m < ω such that for every F ∈ N , F : [ λ ] k → λ of finite arity k < ω , and distinct numbers d, d , d , . . . , d k ∈ ω \ m , if F ( χ N ( τ d ) , . . . χ N ( τ d k )) < τ d then F ( χ N ( τ d ) , . . . χ N ( τ c d )) < χ N ( τ d ) . To see that the formulations in Definition 14 are equivalent to the onesgiven in the introduction, note that if θ > λ = ∪ n τ n is the first for which thatthere exists a club C ⊆ P λ ( H θ ) which is definable in ~τ consisting of subalgebra M ⊆ A = ( H θ , ∈ , f n ) n , then for every θ ′ > θ and M ′ ≺ H θ ′ , if ~τ ∈ M ′ then θ, C ∈ M ′ and M ′ ∩ C ∈ C . Lemma 15.
Suppose that ~τ = h τ n | n < ω i is an increasing sequence of regularcardinals.1. If there is no continuous essentially tree-like scale on Q n τ n then there isno continuous tree-like scale on Q n τ n .2. AFSP w.r.t ~τ implies that there is no continuous tree-like scale on Q n τ n . . ABSP w.r.t ~τ implies both(i) AFSP w.r.t ~τ , and(ii) there is no continuous essentially tree-like scale on Q n τ n .Proof.
1. This is an immediate consequence of the definitions of an essentiallytree-like scale and a tree-like scale on Q n τ n .2. We prove the contrapositive statement, that if there exists a continuoustree-like scale on Q n τ n then AFSP fails with respect to ~τ . Suppose that ~f is a continuous tree-like scale on Q n τ n . Since ~f is tree-like, we can assignto it a function F : λ → λ , λ = ∪ n τ n , defined as follows: For every n < ω and µ , τ n ≤ µ < τ n +1 , define F ( µ ) = ( f α ( n ) if µ = f α ( n + 1) for some α < η F ( µ ) is well defined, i.e., does not depend on the choice of α such that µ = f α ( n + 1), since ~f is tree-like. It is clear from the definition of F thatfor every δ < η and n < ω , F ( f δ ( n + 1)) = f δ ( n ). Now, if C ⊆ P λ ( H θ ) is aclosed unbounded subset, N ∈ C is an internally approachable structurewith F ∈ N , and δ = χ N ( η ), then χ ~τN ( n ) = f δ ( n ) for all but finitely many n < ω . Hence, for all but finitely many n < ω , F ( χ N ( τ n +1 )) = χ N ( τ n ),which means that { χ N ( τ n +1 ) , χ N ( τ n ) } is not free with respect to F ∈ N .Since C was an arbitrary closed and unbounded subset, AFSP with respectto ~τ fails.3. The fact that ABSP implies AFSP is immediate from the definition ofthe two properties. To show that ABSP w.r.t ~τ implies that there isno continuous scale on Q n τ n which is essentially tree-like, we prove thecontrapositive statement. Suppose that h f α | α < η i is a continuousessentially tree-like scale on a product Q n τ n . Then by Definition 9 forevery n < ω , there is a function C n : τ n +1 → P ( τ n ) so that for every µ <τ n +1 , C n ( µ ) is a closed and unbounded subset of τ n which is disjoint from { µ n < τ n | ∃ β < η, f β ( n + 1) = µ and f β ( n ) = µ n } . Let C be any clubof elementary substructures of ( H θ ; ∈ ). Take an internally approachablesubstructure N ∈ C and of size | N | < λ = ∪ n τ n , so that both h τ n | n < ω i and h C n | n < ω i belong to N . Define δ = χ N ( η ) and let m < ω so that f δ ( n ) = χ N ( τ n ) for all n ≥ m . Fixing n ≥ m and examiningthe elementary extension N ′ = N [ { f δ ( n + 1) } ] = { F ( f δ ( n + 1)) | F ∈ N } ≺ ( H θ ; ∈ ) of N , we have that C n ( f δ ( n + 1)) ∈ N ′ since C n ∈ N .Now, as C n ( f δ ( n + 1)) ⊆ τ n is closed unbounded, we must have that χ N ′ ( τ n ) ∈ C n ( f δ ( n + 1)). However χ N ( τ n ) = f δ ( n ) C n ( f δ ( n + 1))by the definition of C n . This implies that χ N ′ ( τ n ) > χ N ( τ n ) = f δ ( n ),which in turn, implies that F ( f δ ( n + 1)) > f δ ( n ) for some F ∈ N . Since N ∈ C where C is an arbitrary closed unbounded collection, and n is anarbitrarily large finite ordinal, we conclude that ABSP fails with respectto h τ n | n < ω i . 9 .2 Fine structure primer We shall take our fine structure from [30]. Our result almost certainly alsoapplies to different forms of fine structure such as the fine structure theory of[17], in fact, the proof of Theorem 6 in particular would be greatly simplified,but at the cost of significantly complicating the arguments in the core modelpart of this paper. As there is currently no account of the covering lemma for λ -indexing, we think it prudent to choose Mitchell-Steel mice at this time. Wedon’t use [32], as ¬ O ¶ is much too strong a limitation for this section. (Whiletechnically Mitchell and Steel operate under the assumption of ¬ M in [30], itis well understood by now that their fine structure theory functions well pastthis point.)For our purposes an extender F is a directed system of ultrafilters { ( a, X ) | a ∈ [lh( F )] <ω , X ⊂ [crit( F )] | a | } as described in [16, p. 384]. The individual ultra-filters will be denoted as F a := { X ⊂ crit( F ) | a | | ( a, X ) ∈ F } . For a ⊂ b and f a function with domain [crit F ] | a | , we let f a,b be the function with domain[crit F ] | b | determined by f a,b (¯ b ) = f (¯ a ) where ¯ a is the unique subset of ¯ b deter-mined by the type of a and b . This gives rise to an embeddings from Ult( M , F a )into Ult( M , F b ). The direct limit along those embeddings is the extender ul-trapower Ult( M , F ), elements of which we will present as pairs [ f, a ] M F where f ∈ M is a function with domain [crit( F )] | a | and a ∈ [lh( F )] <ω . The directlimit map shall be denoted ι M F : M →
Ult( M , F ). We will generally omit thesuperscript in this notation. This should not lead to confusion. Note that wewill later form ultrapowers where some functions involved in the constructionare not elements of the structure but merely definable over it. β < lh( F ) is a generator of F if it cannot be represented as [ f, a ] F for any f ∈ crit( F )crit( F ) ∩ M and a ∈ [ β ] <ω , i.e. { b ∪ { ξ }| f ( b ) = ξ } / ∈ F a ∪{ β } . Letgen( F ) denote the strict supremum of the generators of F . Also let ν ( F ) =max { gen( F ) , (crit( F ) + ) M } .For a subset A of α we will write F ↾ A := { ( a, X ) ∈ F | a ⊂ A } . We willconsider this an extender, forming ultrapowers etc, even if A is not an ordinal.Let η < α be such that η = gen( F ↾ η ), then the trivial completion is the(crit( F ) , ( η + ) Ult( M ; F ↾ η ) )-extender derived from ι F ↾ η . A potential premouse is a structure of the form M = h J ~Eα ; ∈ , ~E, F i where J ~Eα isa model constructed from a sequence of extenders ~E using the Jensen hierarchy.For β ≤ α we define M| β := ( J ~E ↾ ββ ; ∈ , ~E ↾ β, ~E β ) and M|| β := ( J ~E ↾ ββ ; ∈ , ~E ↾ β ).(The difference between the two notations lies in including a top predicate.) If N is of one of the above forms then we write N E M and say N is an initialsegment of M . ~E must be good, i.e. it has the following properties:10Idx) for all β < α if ~E β = ∅ , then β = ( ν ( ~E β ) + ) Ult( M| β ; ~E β ) ;(Coh) for all β < α if ~E β = ∅ , then M|| β = Ult( M| β ; ~E β ) | β ;(ISC) for all β < α if ~E β = ∅ , then for all η < α such that η = gen( ~E β ↾ η ) thetrivial completion of ~E β ↾ η is on ~E or ~E η = ∅ and it is on ι ~E η ( ~E ).Note that ~E β measures exactly those subsets of its critical point that arein M|| β for any β < α such that ~E β = ∅ . F the top extender must be suchthat ~E a F remains good. F can be empty in which case M is called passive,otherwise M is active.To an active potential premouse we associate three constants: µ M the criticalpoint of the top extender; ν M the strict supremum of the generators of M ’s topextender or (( µ M ) + ) M whichever is larger; γ M the index of the longest initialsegment of M ’s top extender (if it exists).We distinguish three different types of active potential premouse: M is activetype I if ν M = ( µ M , + ) M ; M is active type II if ν M is a successor ordinal; M is a active type III if it is neither type I or type II, i.e. the set of the generatorsof M ’s top extender has limit type. The big disadvantage of Mitchell-Steel indexing is that we cannot deal directlywith definability over M , but instead need to work with an amenable code ofour original structure. The exact nature of this coding is dependant on the typeof M . We will take inspiration from [29] and use a uniform notation C ( M ) forthis code.If M := h|M| ; ∈ , ~E, F i is an active potential premouse of type I or II, wewill define an alternative predicate F c coding the top extender F : F c consistsof tuples ( γ, ξ, a, X ) such that ξ ∈ (cid:0) µ M , ( µ M , + ) M (cid:1) and γ ∈ ( ν ( F ) , On ∩|M| ) issuch that ( F ∩ ([ ν ( F )] <ω × M|| ξ )) ∈ M|| γ , and ( a, X ) ∈ ( F ∩ ([ γ ] <ω × M|| ξ )).The point is that F c is amenable. We let C ( M ) := h|M| ; ∈ , ~E, F c i .If M on the other hand is active type III we have to make bigger changes.In the language of [30] we have to “squash”, that is remove ordinals from thestructure. (This is to ensure that the initial segment condition is preserved byiterations.) We let C ( M ) := h J ~Eν ( F ) ; ∈ , ~E ↾ ν ( F ) , F ↾ ν ( F ) i .We then define r Σ -formulae to be Σ over C ( M ), and r Σ n +1 -formulae tobe Σ in a predicate coding an appropriate segment of the r Σ n -theory of C ( M ).We will let Th M n ( α, q ) := { ( ⌈ φ ⌉ , b ) | φ is r Σ n , b ∈ [ α ] <ω , C ( M ) | = φ ( b, q ) } .Projecta can then be defined relative to these formulas, i.e. ρ n +1 ( N ) is theleast ordinal such that some r Σ n +1 -definable (in parameters) subset of it is notin C ( M ). ρ ( M ) = On ∩C ( M ) (which might be smaller than On ∩M ).As usual we define p n +1 ( M ), the ( n + 1)-th standard parameter, to be thelexicographically least p ∈ [On ∩C ( M ) /ρ n +1 ( M )] <ω that defines a missingsubset of ρ n +1 ( M ). 11e can also define canonical r Σ n +1 -Skolem function allowing us to formHull M n +1 ( A ) given a subset A of C ( M ). Note that while our notation makes itlook like a hull of M it is a substructure of C ( M ) not M .We say M is n -sound above β relative to p iff C ( M ) = Hull M n ( β ∪ { p } ). Wewill not mention the parameter if M is n -sound above β relative to p n ( M ). If M is n -sound above ρ n ( M ), we simply say that M is n -sound.A potential premouse is then a premouse if all its initial segments are n -sound for all n . We can now also define fine structural ultrapowers. Let M be a premouse and let F be an extender that measure all subsets of its criticalpoint in M . Let n be such that crit( F ) < ρ n ( M ) and M is n -sound. ThenUlt n ( M , F ) is the ultrapower formed using all equivalence classes [ f, a ] F where a ∈ [lh( F )] <ω and f is a function with domain [crit( F )] | a | that is r Σ n -definableover M (in parameters). Lemma 16.
Let M be a premouse, and let κ ∈ C ( M ) be a regular cardinalthere. Assume ρ n +1 ( M ) ≤ β < κ ≤ ρ n ( M ) for some n such that M is ( n + 1) -sound above β . Then cof( κ ) = cof( ρ n ( M )) .Proof. For ξ < ρ n ( M ) we let N ξ be the structure M|| ξ with Th M n ( ξ, p n ( M ))as an additional predicate. Let then κ ξ be the supremum of ordinals less than κ which are Σ -definable over N ξ from p n +1 ( M ) and ordinals less than β . As allobjects involved are elements of M , we must have κ ξ < κ . On the other handsup ξ<ρ n ( M ) κ ξ = κ as M was ( n + 1)-sound above β .An additional fact that we will need is that if M is an active (potential)premouse, then cof(On ∩M ) = cof(( µ M , + ) M ). See the last remark of Chapter1 in [30]. A (normal, ω -maximal) iteration tree on a premouse M is a tuple T := hhM T α : α ≤ lh( T ) i , h E T α : α < lh( T ) i , D T , h ι T α,β : α ≤ T β ≤ lh( T ) ii where M T α is apremouse for all α ≤ lh( T ) ( M T = M ); E T α is an extender from the M T α -sequence for all α < lh( T ), α < β implies lh( E T α ) < lh( E T β ); ι T α,β : C ( M T α ) →C ( M T β ) is the (possibly) partial iteration map for all α ≤ T β ≤ lh( T ), it istotal iff D T ∩ ( α, β ] ≤ T = ∅ ; ≤ T is the tree order onlh( T ) with root 0, if γ + 1 ≤ lh( T ), then the T -predecessor is the least β such that crit( E T γ ) < gen( E T β ), in that case ( M T γ +1 ) ∗ is the segment of M T β towhich E T γ is applied, if λ ≤ lh( T ) is a limit, then b T λ := [0 , λ ) ≤ T is a cofinal branchwhose intersection with D T is finite, M T λ must be the direct limit of hM T α , ι T α,β : α ≤ T β ∈ b T λ i ;finally, γ + 1 ∈ D T if and only if ( M T γ +1 ) ∗ = M T β .A γ iteration strategy Σ for a premouse M is a function such that Σ( T ) isa cofinal and wellfounded branch for every iteration tree on T of limit length <γ and with the property that Σ( T ↾ α ) = [0 , α ) ≤ T for all limit α < lh( T ). M γ -iterable if there exists a γ -iteration strategy for M . We will just say M isiterable if it is γ -iterabe for all ordinals γ .Let M be a premouse, n < ω and let p n +1 ( M ) = h ξ , . . . , ξ k − i . The( n + 1)-th solidity witness w n +1 ( M ) is a tuple h t , . . . , t k − i where t i = Th M n +1 ( ξ i , h ξ , . . . , ξ i − i ) . We say M is ( n + 1)-solid if w n +1 ( M ) ∈ C ( M ).A core result of [30] is that any reasonably iterable n -sound premouse is( n + 1)-solid. Mitchell-Steel also showed the following with similar methods, seethe remark after Theorem 8.2. Note that the requirement for unique branchescan be replaced by the weak Dodd-Jensen property from [29]. Lemma 17 (Condensation Lemma) . Let M := ( |M| ; ∈ , ~E, F ) be a ( n +1) -soundpremouse such that every countable hull of M has a ( ω + 1) -iteration strategy.Let N be a premouse such that there exist an r Σ n +1 -elementary embedding π : C ( N ) → C ( M ) with crit( π ) ≥ ρ n +1 ( N ) . Then N is an initial segment of M or of Ult( M , ~E crit( π ) ) . Both these results use the notion of a phalanx (although this notion wasnot yet fully developed by the time of [30]) of which we too will have need. Aphalanx is a tuple hhM i : i ≤ α i , h κ i : i < α ii where M i agrees with M j up to( κ + i ) M j for all i < j ≤ α .Phalanxes are a natural byproduct of iteration trees, i.e. if T is a normaliteration tree on some premouse, then hhM T i : i ≤ lh( T ) i , h ν ( E T i ) : i < lh( T ) ii is a phalanx.We can then also define iterability on phanlanxes as a natural extension ofthe structure of iteration trees. Given a phalanx hhM i : i ≤ α i , h κ i : i < α ii and an extender E we can extend the phalanx by applying E to M i where i is minimal with crit( E ) < κ i . (Note we have to require that the length of E isabove sup i<α κ i to maintain “normality”.)A notion of iteration then follows naturally. The most critical difference hereis that we have to keep track above which element of the phalanx any givenmodel of the iteration tree lies. The art of phalanx iteration lies in arrangingthings such that the last model of a co-iteration lies above the “right” model. Our forcing notations is mostly standard. We use the Jerusalem forcing conven-tion by which “a condition p extends (is more informative than) q ” is denotedby p ≥ q . In general, names for a set x in a generic extension will be denotedby ˙ x . If x is in the ground model then its canonical name is denoted by ˇ x .We denote our initial ground model by V ′ , which we assume to satisfy thefollowing assumptions: there are two increasing sequences h κ n | n < ω i , h λ n | < ω i of regular cardinals, with λ n < κ n +1 < λ n +1 for all n , and that each λ n is measurable of Mitchell order o ( λ n ) = κ n .For each n < ω , let h U λ n ,α | α < κ n i be a ⊳ -increasing of normal measures on λ n . I.e., U λ n ,α belongs to the ultrapower by U λ n ,β , whenever α < β . Denote λ = ∪ n λ n .In order to apply our main extender-based forcing notion, we first forcewith a preparatory forcing P ′ over V ′ to transform the Mitchell-order increasingsequences h U λ n ,α | α < κ n i of normal measures, to Rudin-Keisler increasingsequences. For this, we force with a Gitik-iteration P ′ ([11]) for changing thecofinality of measurable cardinals between the cardinals ( κ n , λ n ) for all n < ω .Let G ′ ⊆ P ′ be a generic filter over V ′ , and set V = V [ G ′ ]. We list a number offacts concerning the extensions in V of the measures h U λ n ,α | α < κ n i from V ′ .The analysis leading to these facts can be found in [11], or [3] for a similar typeof poset. The Mitchell-order increasing sequence h U λ n ,α | α < κ n i extends toa Rudin-Keisler increasing sequence of λ n -complete measures h U ∗ λ n ,α | α < κ n i ,with Rudin-Keisler projections π nβ,α : λ n → λ n for each α < β < κ n . We notethat the least measure U ∗ λ n , remains normal. We denote for each n < ω thelinear directed system of measures { U ∗ λ n ,α , π nβ,α | α ≤ β < κ n } by E n , andfurther denote each U ∗ λ n ,α by E n ( α ). Let j E n : V → M E n = Ult( V, E n ) = dirlim α<κ n Ult(
V, E n ( α ))Each measure E n ( α ) can be derived from j E n using a generator γ E n α < j E n ( λ n ).The following list summarizes the key properties of the extenders E n : Fact 18.
1. cp( j E n ) = λ n and M <κ n E n ⊆ M E n γ E n = λ n and h γ α | α < κ n i is a strictly increasing and continuoussequence3. γ E n = sup α<κ n γ E n α is strongly inaccessible in M E n , and we may assumethat there exists a function g n : λ n → λ n such that γ E n = j E n ( g n )( λ n )4. for each α < β < κ n , E n ( α ) is strictly weaker than E n ( β ) in the Rudin-Keisler order. I.e., for every A ∈ E n ( α ) there is ν ∈ A such that π − β,α ( { ν } )is unbounded in λ n .5. for every α < κ n and h : λ n → λ n such that j E n ( h )( γ E n α ) < γ E n . j E n ( h )( γ E n α ) < γ E n β for all β > α .Next, we force over V with a short extender-based-type forcing P , associatedwith the extenders E n , n < ω . P is a variant of the forcing in [2] . Extendingthe arguments of [2], we focus here on the generic scale associated with theextender-based-forcing, and use it to analyze the possible internally approach-able structures in the generic extensions. This approach follows the one takenin [1], where an extender-based forcing has been used to obtain results concern-ing internally-approachable structures witnessing that ground model sequences h S n | n < ω i being tightly-stationary. 14 efinition 19. Conditions p ∈ P are sequences p = h p n | n < ω i such thatthere is some ℓ < ω for which the following requirements hold:1. for n < ℓ , p n = h f n i , where f n : λ + → λ n is a partial function of size | f n | ≤ λ , with 0 ∈ dom( f n ) and both f n (0) , g n ( f n (0)) < λ n are stronglyinaccessible cardinals2. For n ≥ ℓ , p n = h f n , a n , A n i , where f n is as above, a n : λ + → κ n is apartial continuous and order-preserving function, whose domain is a closedand bounded set of λ + of has size | a n | < κ n .We define mc( a n ) to be a n (max( d n )) = max(rng( a n )), and require thatthe set A n to be contained in λ n \ λ n − and belong to E n (mc( a n )).3. dom( a n ) ∩ dom( f n ) = ∅ and dom( a n ) ⊆ dom( a n +1 ) for every n ≥ ℓ , a n (0) = 0, and for every δ ∈ ∪ n dom( f n ) there exists some m < ω suchthat δ ∈ dom( a m ).For a condition p ∈ P as above, we denote ℓ, f n , a n , A n by ℓ p , f pn , a pn , A pn respectively. Direct extensions and end-extensions of conditions are defined asfollows. A condition p ∗ is a direct extension of p , if ℓ p ∗ = ℓ p , f pn ⊆ f p ∗ n for all n < ω , and a pn ⊆ a p ∗ n , A p ∗ n ⊆ ( π n mc( a p ∗ n ) , mc( a pn ) ) − A pn for all n ≥ ℓ p .For every ν ∈ A pn , define p n⌢ h ν i = h f ′ n i , where f ′ n = f p n ∪ {h α, π n mc( a pn ) ,a n ( α ) ( ν ) i | α ∈ dom( a pn ) } . If ~ν = h ν ℓ p , . . . , ν n − i belong to Q n − i = ℓ p A pi , we define the end extension of p by ~ν , denoted p ⌢ ~ν , to be the condition p ′ = h p ′ n | n < ω i , defined by p ′ k = p k for every k
6∈ { ℓ p , . . . , n − } , and p ′ k = p k⌢ h ν k i otherwise. A condition q ∈ P extends p if q is obtained from p by a finite sequence of end-extensions anddirect extensions. Equivalently, q is a direct extension of an end-extension p ⌢ ~ν of p . Following the Jerusalem forcing convention, we write p ≥ q if p extends q ,and p ≥ ∗ q if p is a direct extension of q . Notation 20.
We introduce the following notational convention for the Rudin-Keisler projections π nα,β to be applied in the context of the forcing P . Let p be acondition and ν ∈ A pn for some n ≥ ℓ p and α ∈ dom( a pn ). We write π p mc( p ) ,α ( ν )for π n mc( a pn ) ,a n ( α ) ( ν ). Similarly, for a sequence ~ν = h ν i i ℓ p ≤ i Let p, q be two conditions in P . For m < ω we write p ≤ m q if p ≤ ∗ q and a pn = a qn , A pn = A qn for all n < m .Therefore, for each m < ω , ≤ m is κ m -closed and ≤ m +1 ⊆ ≤ m . Lemma 23. Let θ > λ + regular, ⊳ be a well-ordering of H θ , and M ≺ ( H θ ; ∈ , ⊳ ) satisfying P ∈ M and | M | = λ, V λ ⊆ M . Suppose that there exists an enumera-tion ~D = h D µ | µ < λ i of all dense open subsets of P in M , so that ~D ↾ ν ∈ M for every ν < λ . Then for every condition p ∈ P ∩ M and ℓ ∗ , ℓ p ≤ ℓ ∗ < ω , thereexists p ∗ ≥ ℓ ∗ p so that for each dense open set D ∈ M there are - • q ∈ M with p ≤ ℓ ∗ q ≤ ℓ ∗ p ∗ , • a finite ordinal n D < ω , and • a function N D : Q ℓ p ≤ i We note that the condition q ⌢ ~ν ⌢ ~ν in the statement of theLemma belongs to M as V λ ⊆ M . Therefore, Lemma 23 implies that p ∗ is ageneric condition for ( M, P ), namely, it forces the statement ˙ G ∩ ˇ M ∩ ˇ D = ∅ forevery dense open set D ∈ M . Proof. We assume for notational simplicity that ℓ p = 0. The proof for thegeneral case is similar. We fix for each n < ω a bijection ψ n : λ n → [ λ n ] n +1 × λ n in M . Our final condition p ∗ will be obtained as a limit of a carefully constructedsequence h p n | ℓ ∗ ≤ n < ω i , starting from p ℓ ∗ = p , and consisting of conditionsin M . Moreover, it will satisfy p n ≤ n +1 p n +1 for all n ≥ ℓ ∗ . Suppose that16 n has been defined for some n ≥ ℓ ∗ . Our goal is to construct an extension p n +1 ≥ n +1 p n , so that for every ordinal µ , λ n − ≤ µ < λ n there exists afunction N D µ : Q i ≤ n A p n +1 i → ω so that for every ~ν ∈ Y i ≤ n A p n +1 i and ~ν ∈ Y n +1 ≤ i 17e note that since dom( g i,k ) = dom( a p n k ), it is disjoint from dom( f i,k ), andwe can therefore take their unions f ∗ i,k = f i,k ∪ g i,k , k < n to define a sequenceof functions ~f ∗ i = h f ∗ i, , . . . , f ∗ i,n i . By concatenating the ( n + 1)-sequence ~f ∗ i withthe tail q i , we get a condition q i ∗ = ~f ∗ i ⌢ q i ∈ P with ℓ q i ∗ = n + 1, to which weapply the last clause of Lemma 21 (Strong Prikry Property) and find a directextension q i ∗∗ ≥ ∗ q i ∗ and an integer N so that for every ~ν ∈ Q k ≤ N A q i ∗∗ n +1+ k , q i ∗∗ ⌢ ~ν belongs to D µ i . Specifically, we choose q i ∗∗ ∈ M to be such a conditionwhich is minimal according to the fixed well-ordering ⊳ of H θ , and define • N D µi ( ~ν i ) = N , • ~f δ = h f δ,k | k ≤ n i with f δ,k = f q i ∗∗ k \ g i,k , and • q δ = h q δm | m ≥ n + 1 i with q δm = ( q i ∗∗ ) m for every m ≥ n + 1.Finally, given ~f λ n = h f λ n ,k | k ≤ n i and q λ n = h q δ n m | m ≥ n + 1 i . wedefine p n +1 ≥ ∗ p n by setting a p n +1 m = a p n m and A p n +1 m = A p n m and f p n +1 m = f λ n ,m for m ≤ n , and p n +1 m = q λ n m for m ≥ n + 1. Our use of the well-ordering ⊳ throughout the construction guarantees that p n +1 ∈ M .This concludes the construction of the sequence h p n | ℓ ∗ ≤ n < ω i . We nowdefine p ∗ ≥ ∗ p by p ∗ n = p nn . It is straightforward to verify from the constructionthat p ∗ ≥ ℓ ∗ satisfies the conclusion in the statement of the Lemma.We now show that there are plenty of models M , satisfying the conclusionof Lemma 23. Proposition 25. Let ~M = h M α | α < λ + i be an internally approachablesequence (i.e., ~M ↾ β ∈ M β +1 for every β < λ + ) ⊆ -increasing and continuoussequence of elementary substructures M α ≺ ( H θ ; ∈ , ⊳ ) of size | M α | = λ , andsatisfy M α ∩ λ + ∈ λ + . For every limit ordinal α < λ + of cof( α ) = ω , satisfying α = M α ∩ λ + , ℓ ∗ < ω , and p ∈ M α , there exists a direct extension p ∗ ≥ ℓ ∗ p satisfying the conclusion of Lemma 23 with respect to M = M α .Moreover, if the approachable ideal on λ + is trivial, i.e., I [ λ + ] = λ + , then therequirement of cof( α ) = ω can be removed.Proof. Suppose first that cof( α ) = ω and let h α n | n < ω i be a cofinal sequencein α . Then M α = ∪ n M α n , and for each n < ω since M α n ∈ M α , there exists anenumeration ~D n = h D nµ | µ < λ i ∈ M α of all dense open subsets of P in M α n .Using bijections from λ n × n to λ n , we can form a sequence ~D = h D µ | µ < λ i so that for every n < ω , ~D ↾ λ n enumerates ~D i ↾ λ n for each i < n . Therefore ~D enumerates all dense open sets of P in M α and satisfies ~D ↾ β ∈ M α for every β < λ . It follows from Lemma 23 that for every condition p ∈ M α and ℓ ∗ < ω there exists a direct extension p ∗ ≥ ℓ ∗ p as in the statement of the lemma. Thisconcludes the first part of the statement. thus, dom( f δ,k ) is disjoint from dom( g i,k ) = dom( a p n k ). I [ λ + ] = λ + . We proceed to prove by induction on limitordinals α < λ + with α = M α ∩ λ + , that for every ℓ ∗ < ω and p ∈ M α , thereis p ∗ ≥ ℓ ∗ p satisfying the desirable property for M α . Let α be such an ordinaland assume the statment holds for all β < α . If cof( α ) = ω we are done by thefirst case above. Therefore, suppose that cof( α ) = ρ is an uncountable regularcardinal. Since I [ λ + ] = λ + , there exists a closed and unbounded subset X ⊂ α of order-type otp( X ) = ρ so that X ∩ β ∈ M α ′ whenever β < α ′ , α ′ ∈ { α } ∪ X .Moreover, since ~M ↾ β belongs to M α ′ , so does ~M ↾ ( X ∩ β ) = h M γ | γ ∈ X ∩ β i .Given ℓ ∗ < ω as in the statement of the claim, we further increase it to assumethat κ ℓ ∗ > ρ . Let h β i | i ≤ ρ i be an increasing enumeration of the limit points β in X ∪ { α } which satisfy that M β ∩ λ + = β . Given p ∈ M α , we may assumethat p ∈ M β and denote it by p . Then, by applying the inductive assumptionand using the well ordering ⊳ , we form a sequence of conditions h p i | i ≤ ρ i which is increasing in ≤ ℓ ∗ , so that for each i < ρ p i ∈ M β i +1 and p i +1 ≥ ℓ ∗ p i is the ⊳ -minimal such extension, which is satisfies the conclusion of Lemma 23for M β i +1 . Suppose now that j ≤ ρ is limit. Then every initial segment of h p i | i < j i belongs to M β j , and the sequence has an upper bound in ≤ ℓ ∗ sincethis ordering is κ ℓ ∗ -closed and κ ℓ ∗ > ρ . Defining the upper bound by p j , itfollows from the continuity of the sequence ~M that p j satisfies the desirableproperty for M β j . In particular, for j = ρ , we obtain a suitable condition p ∗ = p ρ for M = M α .The following consequences of Lemma 23 and Proposition 25 will play a keyrole in our arguments concerning Approachable Bounded Subset Property in V [ G ]. Lemma 26. Let ˙ F be a P -name of a function from λ <ω to ordinals, and p ∈ P .There is a direct extension p ∗ ≥ ∗ p and a function f ∗ : [ λ ] <ω × [ λ ] <ω → On which provide the following recipe for deciding the P -names of ordinals ˙ F ( ~µ ) , ~µ ∈ [ λ ] <ω :For every ~µ ∈ [ λ ] <ω there are n ~µ < ω and a function N ~µ : [ λ ] <ω → ω suchthat for every ~ν ∈ Q ℓ p ≤ i Let M ≺ ( H θ ; ∈ , ⊳ ) be a model of size which satisfies the assumptionof Lemma 23 and has ˙ F , p ∈ M (the proof of Proposition 25 shows that suchstructures exist). Since λ ⊆ M then for every ~µ ∈ [ λ ] <ω , the dense open set E ~µ = { q ∈ P | ∃ ξ ∈ On , q (cid:13) ˙ F (ˇ ~µ ) = ˇ ξ } belongs to M . By taking p ∗ ≥ ∗ p as in the statement of Lemma 23 we obtainthe desired extension of p . Corollary 27. P preserves λ + .Proof. If ˙ F : λ → λ + is a P -name of a function, then by Lemma 26 for everycondition p there are p ∗ ≥ ∗ p and a function f ∗ : [ λ ] <ω × [ λ ] <ω → λ + in V , sothat p ∗ forces rng( ˙ F ) is contained in rng( f ∗ ).19et G ⊆ P be a generic filter. By a standard density argument, for every α < λ + and n < ω there exists p ∈ G so that ℓ p > n and α ∈ dom( f pn ). Wedefine the generic scale h t α | α < λ + i by t α ( n ) = f pn ( α ) for any such a condition p ∈ G .Recalling that our setup includes that a n (0) = 0 and E n (0) is a normalmeasure on λ n , we get that the sequence h ρ n | n < ω i , given by ρ n = t ( n ),is generic over V for the diagonal Prikry forcing with the sequence of normalmeasures h E n (0) | n < ω i .Recall that for every n < ω , there exists a function g n : λ n → λ n so that j E n ( g n )( λ n ) is the supremum of the generators of E n , and is inaccessible in M E n .It follows from a standard density argument that the sequence h t α | α < λ + i isa scale in the product Q n g n ( ρ n ), and that g n ( ρ n ) < λ n is regular for almostall n < ω . Moreover, it is straightforward to verify that our assumption thatthe functions a n in conditions p ∈ P are continuous and have closed domains,implies that the scale h t α | α < λ + i is continuous. Notation 28. In V [ G ], we denote g n ( ρ n ) by τ n . Theorem 29. The Approachable Bounded Subset Property (ABSP) holds in V [ G ] with respect to the sequence h τ n | n < ω i .Proof. Suppose otherwise, then there exists a stationary set S ⊆ P λ ( H θ ) ofinternally approachable structures N ≺ ( H θ ; ∈ ) such that for every N ∈ S and n < ω there is a function F Nn : [ λ ] k Nn → λ in N , of a finite arity k Nn < ω , and afinite sequence of distinct numbers ~d N,n = h d N,n , . . . , d N,nk Nn i ⊆ ω \ n , satisfying χ N ( τ d N,n ) ≤ F Nn (cid:18) χ N ( τ d N,n ) , . . . , χ N ( τ d N,nkNn ) (cid:19) < τ d N,n . By Lemma 13, applied to the assignments N 7→ h F Nn | n < ω i and N 7→ h ~d N,n | n < ω i , there exists a stationary set S ∗ ⊆ λ + and two fixed sequences h F n | n <ω i , h ~d n | n < ω i , with F n : [ λ ] k n → λ and ~d n = h d n , . . . d nk n i , such that for every δ ∈ S ∗ there exists N ∈ S so that δ = χ N ( λ + ), h F Nn | n < ω i = h F n | n < ω i ,and h ~d N,n | n < ω i = h ~d n | n < ω i . For each δ ∈ S ∗ there are m δ < ω and N ∈ S ∗ such that for every n ≥ m δ , t δ ( n ) = χ N ( τ n ) and thus, t δ ( d n ) ≤ F n (cid:0) t δ ( d n ) , . . . , t δ ( d nk n ) (cid:1) < τ d n (1)We move back to V to contradict the above, and complete the proof. Let p be a condition forcing the statement of (1) with respect to the P -names ˙ S ∗ , h ˙ F n | n < ω i , and h ˙ ~d n | n < ω i . By taking a direct extension if needed, wemay assume p decides the integer values for ~d n ⊆ ω \ n , for all n < ω . ApplyLemma 26 repeatedly for each F n , n < ω , to form sequences, h p n | n < ω i of ≤ ∗ -extensions of p , and h f n | n < ω i of functions, f n : [ λ ] <ω × [ λ ] <ω → λ , sothat for each n < ω , p n ≥ ∗ p n − and f n are formed to satisfy the conclusion of20emma 26 with respect to ˙ F n . We define α = sup [ n,m (dom( a p n m ) ∪ dom( f p n m )) ! + 1and let p ∗ be a common direct extension of h p n | n < ω i with α = max(dom( a p ∗ m ))for all m ≥ ℓ p ∗ . Next, let q be an extension of p ∗ which forces ˇ δ ∈ ˙ S ∗ for someordinal δ > α . Since q extends p ∗ , it is a direct extension of p ∗ ⌢ ~ν ∗ for some ~ν ∗ ∈ Q ℓ p ∗ ≤ i<ℓ ∗ A p ∗ i . By taking a direct extension of q if needed, we may alsoassume that q decides the integer values m δ from above and that δ ∈ dom( a qn )for some n < ω .Next, we pick n < ω satisfying n ≥ m δ , ℓ q and δ ∈ dom( a qn ), and denote forease of notation, k n , h d n , . . . , d nk n i by k , h d , . . . , d k i respectively. Our choice of p ∗ ≥ ∗ p n , and function f n guarantee that for every ~µ = h µ d , . . . , µ d k i ∈ [ λ ] k there are n ~µ ∗ < ω and a function N ~µ ∗ : Y ℓ q ≤ i ABSP holds in V [ G ] with respect to h τ n | n < ω i and thus, byLemma 15, AFSP holds and there are no continuous scales on Q n τ n which areessentially tree-like. ℵ ω We define a variant ˆ P of the forcing P from the previous section, to obtain theresult of Theorem 29 in a model where h τ n | n < ω i form a subsequence of thefirst uncountable cardinals.Conditions q ∈ ˆ P are pairs q = h p, h i of sequences, p = h p n | n < ω i and h = h h high − i ⌢ h h n | n ∈ ω i satisfying the following conditions:22. p ∈ P , i.e., p satisfies Definition 19 above2. for every n < ℓ p , h n = h h low n , h high n i , is a pair of functions, which satisfythe following properties: • h low n ∈ Coll( ρ pn , < τ pn ) where ρ pn = f pn (0), and τ pn = g n ( ρ pn ), • h high n ∈ Coll(( τ pn ) + , < ρ pn +1 ) if n < ℓ p − 1, and h high ℓ p − ∈ Coll(( τ pℓ p − ) + , <λ ℓ p ).3. for every n ≥ ℓ p , h n = h h low n , h high n i , is a pair of functions, which satisfythe following properties: • dom( h low n ) = dom( h high n ) = A pn , • for every ν ∈ A np , h low n ( ν ) ∈ Coll( ρ νn , < τ νn )where ρ νn = π n mc( a pn ) , ( ν ) and τ νn = g n ( ρ νn ), and h high n ( ν ) ∈ Coll (cid:0) ( τ νn ) + , < λ n +1 (cid:1) h high − belongs to Coll( ω, < ρ p ) if ℓ p ≥ 1, and to Coll( ω, < λ ) otherwise5. h high ℓ p − ∈ V ρ νℓp for every ν ∈ A ℓ p p , and h high n − ( ν ′ ) ∈ V ρ νn for every n > ℓ p , ν ′ ∈ A pn − , and ν ∈ A np .A condition q ∗ = h p ∗ , h ∗ i is a direct extension of p = h p, h i if the followingconditions hold:1. p ∗ ≥ ∗ p in the sense of P ,2. for every n < ℓ p , h low n ⊆ ( h ∗ n ) low and h high n ⊆ ( h ∗ n ) high ,3. for every n ≥ ℓ p , and ν ∈ A p ∗ n , h low n ( π n mc( a p ∗ n ) , mc( a pn ) ( ν )) ⊆ ( h ∗ n ) low ( ν ), and h high n ( π n mc( a p ∗ n ) , mc( a pn ) ( ν )) ⊆ ( h ∗ n ) high ( ν ).Given a condition q = h p, h i and an ordinal ν ∈ A pℓ p , we define the one-pointend-extension of q by ν , denoted q ⌢ h ν i to be the condition h p ′ , h ′ i given asfollows: • p ′ = p ⌢ h ν i in the sense of P , in particular ℓ p ′ = ℓ p + 1, • h ′ n = h n for every n ≤ ℓ p , and in addition, ( h ′ ℓ p ) high is now considered asa condition of the restricted collapse poset Coll( ρ pℓ p − , < τ νℓ p ) (replacingColl( ρ pℓ p − , < λ ℓ p )). • ( h ′ ℓ p ) low = h low ℓ p ( ν ) and ( h ′ ℓ p ) high = h high ℓ p ( ν ), By Definition 19 ρ pn < τ pn < λ n are both inaccessible for n ≥ h ′ n = h n for every n ≥ ℓ p + 1.Given a condition q = h p, h i and a finite sequence ~ν = h ν ℓ p , . . . , ν n − i ∈ Q ℓ p ≤ i Suppose that D ⊆ ˆ P is a dense open set and q = h p, h i ∈ ˆ P a condition. Then there exists a direct extension q ∗ ≥ ∗ q , n < ω , such thatfor every ~ν ∈ Q ℓ p ≤ i Let G ⊆ ˆ P be a generic filter over V . The Approachable BoundedSubset Property (ABSP) holds in V [ G ] with respect to the sequence hℵ n +2 | n <ω i . Let M | = ZFC − be a premouse such that every countable hull of M has an( ω + 1) iteration strategy, λ ∈ M a limit cardinal (in M ) of V -cofinality ω (which need not agree with its cofinality in M ) such that λ + exists in M .Note if N is a premouse and α ∈ N is such that N | = α is the largest cardinal,then we let ( α + ) N = On ∩N .Let ~κ := h κ n : n < ω i be a sequence of M -cardinals cofinal in λ . We do noteasume ~κ is in M . Let τ n := ( κ + n ) M . We will define a sequence in Q n<ω τ n that isincreasing, tree-like, and continuous.Let C λ, M := { α < ( λ + ) M |M|| α ≺ M|| ( λ + ) M } . For α ∈ C λ, M let M α bethe collapsing level for α . Let n α be minimal such that ρ M α n +1 = λ , p α := p M α n α +1 ,and w α := w M α n α +1 . Let also F α be the top predicate of M α .By Lemma 17 there exists some M nα E M such that C ( M nα ) is isomorphicto Hull M α n α +1 ( κ n ∪ { p α } ). f ~κ, M α ( n ) = ( ( κ + n ) M nα { w α , λ } ∈ Hull M α n α +1 ( κ n ∪ { p α } )0 otherwiseNote that the above function is non-zero almost everywhere, that is if λ ∈C ( M α ). This can fail if (and only if) M α is active and ν M α = λ . Such α wewill call anomalous. For such α we define: f ~κ, M α ( n ) = ( ( κ + n ) Ult( M ; F α ↾ κ n ) κ n > µ M α γ < λ such that the trivialcompletion of F α ↾ κ n is indexed at γ . We that it is impossible to have thealternative case as κ n is a cardinal and hence not an index It also cannot be type Z. Type Z extenders have a largest generator. M nα is the least level of M over which a surjectionfrom κ n on to the ordinal f ~κ, M α ( n ) is definable. Hence the ordinal defines thelevel and vice versa. In cases where it is clear which mouse and which sequence of cardinals we aretalking about, e.g. for the rest of this subsection, we will omit the superscripts. Lemma 33. Let α < β both in C . If m is such that f α ( m ) = f β ( m ) then f α ( n ) = f β ( n ) for all n ≤ m .Proof. Note first that if f β ( m ) = 0, then f β ( n ) = 0 and the same holds for α .Let us then consider f β ( m ) = 0, it follows that M mα = M mβ . We will start withthe assumption that neither α nor β are anomalous. In that situation we musthave that w α ∈ Hull M α n α +1 ( κ n ∪ { p α } ). This implies that p α collapses down to p n α +1 ( M mα ). The same, of course, holds for β . Note we must have n α = n β . Itfollows that C ( M nα ) ∼ = Hull M mα n α +1 ( κ n ∪ { p n α +1 ( M mα ) } )= Hull M mβ n β +1 ( κ n ∪ { p n β +1 ( M mβ ) } ) ∼ = C ( M nβ ) . This implies f β ( n ) = f α ( n ). Note that f β ( n ) = 0 if and only if w n β +1 ( M mβ ) / ∈ Hull M mβ n β +1 ( κ n ∪ { p n β +1 ( M mβ ) } ) and similarly for α .Assume then that at least one of α and β is anomalous. Let us assume that α is anomalous, the proof for β is only notationally different. We will realizethat, in fact, both must be anomalous. As types are preserved by taking hullswe must have that both are active type III. As at least one is anomalous wedo know that the top extender of M mα has no generators above κ m . If thenthe other were not to be anomalous we must have that λ is an element of theappropriate hull. This implies that C ( M mβ ) has ordinals and hence generatorsabove κ n . Contradiction!As then both are anomalous and M mα = M mβ , we have F α ↾ κ m = F β ↾ κ m .From this follows µ M α = µ M β and F α ↾ κ n = F β ↾ κ n . Therefore f α ( n ) = f β ( n ). Lemma 34. Let α < β both in C . Then f α ( n ) < f β ( n ) for all but finitely many n .Proof. Note that since α < β are in C then M α = M β and so M α ∈ M β . Letus first assume that β is not anomalous.Let n ∗ be such that M α ∈ Hull M β n β +1 ( κ n ∗ ∪ { p β } ). The pre-image of M α in M nβ ( n ≥ n ∗ ) can then compute M nα and hence f α ( n ) correctly.If on the other hand β were anomalous, let n ∗ be such that M α is generatedby some a ∈ [ κ n ∗ ] <ω , i.e. M α = ι F β ( h )( a ) for some h ∈ ( µ M β M|| µ M β ). Then f α ( n ) ( n ≥ n ∗ ) can be computed from ι F β ↾ κ n ( h )( a ) inside Ult( M ; F β ↾ κ n ) by Lo´s’s Theorem. 26 emma 35. Let β ∈ C be of uncountable cofinality. Then β is a continuitypoint of the sequence ( i.e. f β is the exact upper bound of h f α : α ∈ C ∩ β i ).Proof. Let α n < f β ( n ). We shall find some α < β such that f α dominates h α n : n < ω i almost everywhere. Towards that end, we deal first with the casewhere β is not anomalous.For almost all n < ω we have some surjection from κ n onto α n in M nβ ,given by some parameter a n ∈ [ κ n ] <ω and term τ n . Let ξ n < ρ n β ( M β ) besuch that the image of such a surjection is (Σ )-definable over M|| ξ n withTh M β n β ( ξ n , p n β ( M β )) as an additional predicate.By Lemma 16, ρ n β ( M β ) has uncountable cofinality. So ξ := sup n<ω ξ n <ρ n β ( M β ). Take then some A that codes the Σ theory of M|| ξ with Th M β n β ( ξ, p n β ( M β ))as an additional predicate. Such an A exists in M β .Pick some α < β such that A ∈ M α . Let n < ω be such that A has apre-image ¯ A in M nα . M nα can then compute α n as the ordertype of { ( γ, δ ) | ( k, a n a h γ, δ i ) ∈ ¯ A } where k is the G¨odel number of “ τ n ( a n )( γ ) < τ n ( a n )( δ ) ′′ . Hence α n < f α ( n ).Similarly, if α were to be anomalous, we can pick n such that A = ι F α ( h )( a ) forsome h ∈ µ M α M|| µ M α and a ∈ [ κ n ] <ω . The rest of the argument remains thesame.Let us then assume that β is anomalous. Pick h n ∈ µ M β M|| µ M β such that ι F β ( h n )( a n ) is a surjection from κ n onto α n for some a n ∈ [ κ n ] <ω . We havethat cof(( µ M β , + ) M ) > ω .Pick then some ξ < ( µ M β , + ) M such that h h n : n < ω i ⊂ M|| ξ . By weakamenability the extender fragment ¯ F := { ( a, X ) ∈ F β | X ∈ M|| ξ, a ∈ [ λ ] <ω } in M β . Pick then α < β with ¯ F ∈ M α . Any M nα containing ¯¯ F a pre-image of ¯ F can then compute α n as the ordertype of { ( γ, δ ) | B γ,δn ∈ ¯¯ F } where B γ,δn = { ¯ a ∈ (cid:2) µ M β (cid:3) | b γ,δn | | h a n ,b γ,δn n (¯ a )(id γ,b γ,δn (¯ a )) < h a n ,b γ,δn n (¯ a )(id δ,b γ,δn (¯ a )) } , and b γ,δn := a ∪ { γ, δ } . Hence α n < f α ( n ). Lemma 36. Assume h κ n : n < ω i ∈ M , then h f α : α ∈ C i is a scale in Q n<ω τ n ∩ M .Proof. Let f ∈ Q n<ω ( τ n /J bd ) ∩ M . Pick α ∈ C such that f ∈ M α . Then f ( n ) < f α ( n ) for all but finitely many n . Remark 37. We note that it is possible to associate a sequence in Q n<ω τ n to anyinitial segment of M projecting to λ and it would obey the established rules.In certain situations we will want to consider a variant construction. Let usconsider an additional set of parameters ~α := h α n : n < ω i ∈ Q n<ω τ n . Let β ∈ C. 27y the condensation lemma there exists some M n,α n β such that C ( M n,α n β ) isisomorphic to Hull M β n β +1 ( κ n ∪ { p β a h α n i} ). We then define: f ~κ,~α, M β ( n ) = ( ( κ + n ) M n,αnβ { λ, w β } ∈ Hull M β n β +1 ( κ n ∪ { p β a h α n i} )0 otherwiseIf β is anomalous, then we use F β ↾ ( α n + 1) (instead of F β ↾ κ n ) to define thesequence.This sequence will behave just like the previously defined sequence. Theproofs are mostly the same. The only minor problem adapting these argumentslie in the preservation of standard parameters. Let p nβ be the image of p β underthe collapse map in M n,α n β . Then p nβ might fail to be the standard parameterof M n,α n β as it can fail to be a good parameter.Though certainly we do know that p nβ a h α n i is a parameter so the standardparameter is below it in the lexicographic order. As we do have a preimage ofthe solidity witness in M n,α n β , its standard parameter can only be lesser on thatlast component, i.e. p n β +1 ( M n,α n β ) = p nβ a α ′ with α ′ ≤ α n .Then M m,α m β can always compute M n,α n β from its standard parameter andthe ordinal α n in a consistent matter, guaranteeing tree-likeness of the sequence.Everything else goes through with minor changes. Let now each of the κ n be an inaccessible cardinal in M . We want to extractfrom M β , β ∈ C , a sequence of structures that singularize some g β ( n ) < κ n .For this we need a vector of parameters ~α = h α n : n < ω i where α n < κ n . Wealso do require that sup n<ω α n = λ . When do these parameters give rise to theright structure? This will depend on whether β is anomalous or not. Whenbegin with listing three key factors for the case β is not anomalous:(1) βn sup(Hull M β n β +1 ( α n ∪ { p β } ) ∩ κ n ) > α n ;(2) βn κ n ∈ Hull M β n β +1 ( α n ∪ { p β } )(3) βn Hull M β n β +1 ( α n ∪ { p β } ) is cofinal in ρ n β ( M β ).If β is anomalous, we have the following two considerations:(4) βn sup( { ι F β ( h )( a ) | h ∈ µ M β ( µ M β ) , a ∈ [ α n ] <ω } ∩ κ n ) > α n ;(5) βn κ n = ι F β ( h )( a ) for some h ∈ µ M β ( µ M β ) and a ∈ [ α n ] <ω .We say β is adequate iff (1) βn + (2) βn + (3) βn or (4) βn + (5) βn (depending ontype) are met for all but finitely many n . If β is adequate and not anomalous28hen g ~κ,~α, M β ( n ) := ( sup(Hull M β n β +1 ( α n ∪ { p β } ) ∩ κ n ) (1) βm + (2) βm + (3) βm ∀ m ≥ n M nβ be the unique initial segment of M such that C ( M nβ )is isomorphic to Hull M β n β +1 ( g β ( n ) ∪ { p β } ). (Note that the second case in thecondensation lemma cannot hold as g β ( n ) is a limit of cardinals and hence acardinal itself. This follows by elementarity, trivially so when n β > βn .)If on the other hand β is anomalous then g ~κ,~α, M β ( n ) := ( sup( { ι F β ( h )( a ) | h ∈ µ M β µ M β , a ∈ [ α n ] <ω } ∩ κ n ) (4) βm + (5) βm ∀ m ≥ n M nβ will be the unique initial segment of M with the trivial completionof F β ↾ g β ( n ) as its top extender. As in the previous section, we will omitsuperscripts for the remainder of this section.To ensure tree-likeness for this sequence we need a strong interdependencebetween the ordinal g β ( n ) and structure M nβ . Towards that end notice that g β ( n ) is definably singularized over M nβ . The next lemma will show that M nβ is the least level of M with this property. Lemma 38. g β ( n ) is regular in M nα for all n such that (1) βn + (2) βn + (3) βn or (4) βn + (5) βn holds.Proof. First we will consider β that is not anomalous. Since κ n is regular in M β ,it will then be enough to show that sup(Hull M β n β +1 ( g β ( n ) ∪ { p β } ) ∩ κ n ) = g β ( n ).Let ξ < κ n be such that ξ ∈ Hull M β n β +1 ( g β ( n ) ∪ { p β } ). We can then take γ < g β ( n ) and δ < ρ n β ( M β ) such that ξ ∈ Hull N δ ( γ ) where N δ is M|| δ togetherwith Th M β n β ( δ, p n β ( M β )) as an additional predicate. We can take γ and δ to bein Hull M β n β +1 ( α n ∪ { p β } ) (by definition of g β ( n ) and (3) n respectively).Then η := sup(Hull N δ ( γ ) ∩ κ n ) is also in that hull (uses (2) n ) and thus ξ < η < g β ( n ).Now consider an anomalous β . We will show that g β ( n ) is regular inUlt( M ; F β ↾ g β ( n )). We have some h ∈ µ M β µ M β and a ∈ [ α n ] <ω such that κ n = ι F β ( h )( a ). We will show that g β ( n ) = ι F β ↾ g α ( n ) ( h )( a ). As this pair repre-sents a regular cardinal in the larger ultrapower this will suffice.Pick then some h and b such that b ∈ [ g β ( n )] <ω and ι F α ↾ g β ( n ) ( h )( b ) <ι F β ↾ g β ( n ) ( h )( a ). Pick some c ∈ [ α n ] <ω (w.l.o.g. a ⊂ c ) and h such that b ⊂ ι F β ( h )( c ). Define h : (cid:2) µ M β (cid:3) | c | → µ M β by d sup { h ( e ) | e ∈ [ h ( d )] | b | , h ( e ) Let α ∈ C be such that g β ( n ) is defined for all but finitely many n . Let ~α ∗ := h α ∗ n : n < ω i be such that α n ≤ α ∗ n < g α ( n ) for all but finitelymany n . Then g ~κ, M ,~α ∗ β is defined and agrees with g β almost everywhere. Lemma 40. Say β ∗ is adequate, then every β > β ∗ in C of uncountable cofi-nality is also adequate.Proof. Let us assume for simplicity’s sake that (1) β ∗ n + (2) β ∗ n + (3) β ∗ n holds forall n . Let then n ∗ be such that β ∗ ∈ Hull M β n β +1 ( α m ∪ { p β } ) for all m ≥ n ∗ . Thenthat hull can compute Hull M β ∗ n β ∗ +1 ( α m ∪ { p β ∗ } ) for all m ≥ n ∗ . (1) βm + (2) βm thenfollows.That is if β ∗ is not anomalous. If it is anomalous note that Hull M β n β +1 ( α m ∪{ p β } ) has access to the extender F β ∗ and can compute κ m from it assuming α m > µ M β ∗ . (1) βm follows for similar reasons.(3) βm almost everywhere follows for cofinality reasons.If β is anomalous then we take some h and a ∈ [ α m ] <ω such that β ∗ = ι F β ( h )( a ) and let τ be some r Σ n β ∗ +1 -term such that κ m = τ M β ∗ m ( b m , p β ∗ ) for b m ∈ [ α m ] <ω . Define then h m : (cid:2) µ M β (cid:3) | a ∪ b m | → µ M β by c τ M ha,a ∪ bm ( c ) m (id b m ,a ∪ b m ( c ) , p h a,a ∪ bm ( c ) ) . We then have ι F β ( h )( a ∪ b m ) = κ m . (4) βm follows for similar reasons.The idea is similar if β ∗ and β are both anomalous. (Pick h, a representing F α ∗ etc.) We skip further details.Assuming the existence of an adequate ordinal β ∗ we can then show that h g β : β ∈ C \ β ∗ ∩ cof( >ω ) i is increasing (mod finite), tree-like, and continuousas before. Lemma 41. Let h κ n : n < ω i and h α n : n < ω i both in M then there exists anadequate β ∗ and h g β : β ∈ C \ β ∗ ∩ cof( >ω ) i is a scale in Q n<ω κ n ∩ M .Proof. Any β ∗ of uncountable cofinality (in M ) such that M|| β ∗ contains bothsequences will be adequate. The rest is then as before.Note that while we have only considered sequences of a “pure” type, wecould easily deal with sequences h κ n : n < ω i of regular cardinals with bothsuccessor cardinals and inaccessible cardinals by mixing both constructions us-ing parameters where needed. With this we finish the proof of Theorem 6. Remark 42. Assuming that λ is not subcompact in M the sequences we definedshould be very good, but we have yet to check this in detail. The proof wouldpresumably proceed along similar lines as in [7].30 Core models and the tree-like scale We now want to consider the question when the sequences constructed in theprevious section are scales in V . For this we need to consider the right mouse.The natural candidate is, of course, the core model. But even core modelsequences are not always scales.To keep the following as accessible as possible we are going to operate undera smallness assumption. This will allow us to cover all known anti tree-like scaleresults while greatly simplify the following arguments.This assumption is:There is no inner model W and F ∈ W a total extendersuch that gen( F ) ≥ (crit( F ) ++ ) W (2) Corollary 43. There is no ω -iterable premouse ( M , ∈ , ~E, F ) such that gen( F ) > (crit( F ) ++ ) M .Proof. Assume towards a contradiction that ( M ; ∈ , ~E, F ) is a counterexample.Then we can generate an inner model W by iterating the top extender out of theuniverse. Note that by a standard reflection argument, ω -iterability is enoughto ensure that this model is wellfounded. By the initial segment condition F ↾ (crit( F ) ++ ) W is then in W contradicting (2).The reader should be aware, though, that our main results will hold undermuch weaker anti-Large Cardinals assumptions (up to one Woodin cardinaland beyond). Neither should the choice of indexing scheme affect their validity(though we have yet to check this in detail).The most immediate payoff of (2) will be that all iterations we are going toconsider are linear (This is one instance in which ms-indexing will make thingssimpler for us). Proposition 44. Let M be a ω -iterable premouse, and T a normal iterationtree on M . Then no α < β ≤ lh( T ) is such that crit( E T β ) < gen( E T α ) .Proof. Let α < β such that crit( E T β ) < gen( E T α ). There are three cases: Case 1: crit( E T α ) < crit( E T β )By agreement between models in an iteration we have that crit( E T β ) is inac-cessible in M T α || lh( E T α ) and thus (crit( E T α ) ++ ) M T α || lh( E T α ) < crit( E T β ). As E T α has generators above crit( E T β ), M T α | lh( E T α ) is a counterexample to Corollary43. Case 2: crit( E T α ) > crit( E T β )In M T β due to the agreement between models in an iteration, lh( E T α ) > crit( E T β ) is a cardinal in M T β so by strong acceptability crit( E T α ) is inaccessiblein M T β and thus above (crit( E T β ) ++ ) M β ). As T is a normal iteration lh( E T β ) > lh( E T α ) > (crit( E T α ) + ) M T β and so gen( E T β ) > crit( E T α ) but then M T β | lh( E T β ) isa counterexample to Corollary 43. 31 ase 3: crit( E T α ) = crit( E T β )Because T is normal we have lh( E T α ) < lh( E T β ) but this means that E T β musthave generators cofinal in (crit( E T β ) ++ ) M T β . Now, let γ be the last drop in theinterval ( α, β ] if it exists or α + 1 otherwise. We can assume that crit( E T γ − ) ≥ crit( E T β ). ι T γ − ,β (crit( E T γ − )) is then the critical point of an extender on the M T β sequence and greater than crit( E T β ). As E T β must be total over M T β . Thus wecan produce a class size model W containing E T β and agreeing with M T β past(crit( E T β ) ++ ) W which contradicts (2).Another consequence of (2) is that the Jensen-Steel core model K exists by[18]. Note that by the smallness assumption there can be no anomalous ordinalsin K . For the following results we will follow the general framework of the proofof weak covering for that model. Before going into the proofs we shall take quicknote of the involved objects.Let λ be a singular cardinal of countable cofinality. Let ~κ := h κ n : n < ω i be a sequence cofinal in λ . Let τ n := ( κ + n ) K . Consider some X ≺ H θ ( θ >> λ )and let σ X : H X → X be the inverse of the transitive collapse map. X will needto satisfy certain properties: • certain phalanxes “lift” through σ X • card( X ) < λ , • X is tight on ~κ (and h τ n : n < ω i ), i.e. X ∩ Q n<ω κ n is cofinal in Q n<ω ( X ∩ κ n ) /J bd , • the collection of X ≺ H θ with the above three properties is stationary.The first point is quite vague, and we will provide more details where neededin the course of the argument. By [21] ω -closed X do satisfy the first property,but it seems possible that there are not enough, i.e. stationary many, X withall properties available. In such cases, by [22] we do know that for every inter-nally approachable chain ~Y := h Y i : i < κ i in H θ there exists some i < κ ofuncountable cofinality such that Y i satisfies the first property. That it satisfiesthe other properties should be easy to see.Let then from now on X be some such set with the required properties.Let σ X : H X → X be an isomorphism where H X is transitive. Write K X := σ − X ” [ K ], λ X := σ − X ( λ ), κ Xn := σ − X ( κ n ), etc.As is standard we will compare K X with K , we should have (for our choiceof X ) that the iteration tree on K X is trivial (this is (1) α from [21] or (1) iα from [22] respectively). Let then I X be the iteration tree on K that arises fromthe co-iteration. We will simplify notation by writing M Xα for M I X α etc. Let ζ X := lh( I X ) be the length of the iteration. Lemma 45. (crit( σ X ) + ) K X < (crit( σ X ) + ) K and if E X is defined then it is nottotal over K . K X and K agree up to (crit( σ X ) + ) K X as a result of the conden-sation lemma. Proof. Assume towards a contradiction that (crit( σ X ) + ) K X = crit( σ X ) + ) K .Then E σ X the (crit( σ X ) , σ X (crit( σ X )))-extender derived from σ X measures allsubsets of its critical point that are in K . It also coheres with K by the ele-mentarity of σ X . (This is a little bit of a lie. We would actually need to knowthat all Mitchell-Steel initial segments of E σ X are on the K -sequence. But ifthis fails we could simply apply the argument we are about to give to the leastmissing segment instead.)We do know that the phalanx hh K, Ult( K ; E σ X ) i , σ X (crit( σ X )) i is iterable.This is (2) α from [21] or [22] where crit( σ X ) = ( ℵ α ) K X . (Once again this issomething of a lie. We actually have to replace K with an appropriate soundnesswitness in the above statement, but we can choose W such that it agrees with K past the level we actually care about. Thus this will not make a differencehere.)But then by [28, 8.6] we have that E σ X is on the K -sequence. It shouldbe obvious that gen( E σ X ) = σ X (crit( σ X )) and thus K | lh( E σ X ) contradictsProposition 43.As for the second part, assume E X is applied to K . By the first part, ifcrit( E X ) ≥ crit( σ X ), then we must truncate. If (crit( E X ) + ) K X = crit( σ X ),then by elementarity (crit( E X ) + ) K = σ X (crit( σ X ) so we must truncate.If crit( σ X ) ≥ (crit( E X ) ++ ) K X then its generators must be cofinal in crit( σ X ).So, if the strict inequality holds then E X contradicts Corollary 43. A similarargument applies if lh( E X ) > (crit( σ X ) + ) K X .So, we must have that crit( σ X ) = (crit( E X ) ++ ) K X . Consider M X . It agreeswith K X up to (crit( σ X ) + ) K X and that ordinal is a cardinal there. Thus wecan apply the extender E σ X to it. The properties of X will guarantee that˜ K := Ult( M X ; E σ X ) is iterable (similar to the proof of [21, Lemma 3.13]).We have that K and ˜ K agree up to sup( σ X ” (cid:2) (crit( σ X ) + ) K X (cid:3) ) which liespast σ X (crit( σ X )) their common crit( E X ) ++ , but on the other hand(crit( E X ) +++ ) ˜ K = sup( σ X ” (cid:2) (crit( σ X ) + ) K X (cid:3) ) < σ X ((crit( σ X ) + ) = (crit( E X ) +++ ) K as a result of weak covering.Consider then ˜ E the first extender applied on the K side in the co-iterationof K and ˜ K . Its index must be above σ X (crit( σ X )) but its critical point cannotbe larger than crit( E X ). ˜ E on the K -sequence then contradicts (2).Remember now the sequence h κ n : n < ω i and the sequence of successors h τ n : n < ω i . The general idea for the following proofs is to find some ordinal α X < λ + such that the natural scales of the core model at α X align with thecharacteristic function of X .From now on we shall assume that κ n is a cutpoint of (the extender sequenceof) K and hence κ Xn is a cutpoint of K X . ( α ∈ ( M ; ∈ , ~E ) is a cutpoint (of ~E )iff whenever crit( E β ) < α , then lh( E α ) < α for all β ∈ dom( ~E ).)33 emma 46. There exist some n X , k X < ω , a sequence of models hN Xn : n X ≤ n < ω i , and maps h υ Xn,m : n X ≤ n ≤ m < ω i such that: • (( κ Xn ) + ) N Xn = τ Xn and N Xn agrees with K X up to τ Xn for all n ≥ n X ; • N Xn is ( k X + 1) -sound above κ Xn for all n ≥ n X ; • υ Xn,m : C ( N Xn ) → C ( N Xm ) is r Σ k X +1 -elementary for all m ≥ n ≥ n X ; • crit( υ Xn,m ) ≥ κ Xn for all m ≥ n ≥ n X . For our purposes the critical point of the identity will be defined as theordinals of its domain. Proof. There are a couple of cases. Case 1: I X has no indices below λ .In that case, we have K X | ( λ + ) K X E K . By Lemma 45 we do know that some N ′ E K exists with (crit( σ X ) + ) N ′ = (crit( σ X ) + ) K X and ρ ω ( N ′ ) ≤ crit( σ X ).By assumption we must have K X | ( λ + ) K X E N ′ .Take then N ∗ to be the smallest initial segment of K that end-extends K X | ( λ + ) K X such that ρ ω ( N ∗ ) < λ X .Let k X be minimal such that ρ k X +1 ( N ∗ ) < λ X . Let n X be minimal suchthat κ Xn X ≥ ρ k X +1 ( N ∗ ). We then let N Xn := N ∗ for all n ≥ n X , and let υ Xn,m be the identity for all m ≥ n ≥ n X . As an initial segment of K , N ∗ is sound sothis works. Case 2: The set { lh( E Xβ ) | β < ζ X } is bounded below λ X .Let η X < ζ X be minimal such that E Xη X has length >λ X , if it exists. If thereis no such ordinal, let η X = ζ X . We must then have that M Xη X agrees with K X past λ X . If M Xη X has some proper initial segment of length greater than λ X projecting below λ X , then this is no different from the previous case.So let us assume that this is not the case. Let n X be minimal such that κ Xn X the set { lh( E Xβ ) | β < η X } . By Lemma 45, M Xη X is not a weasel and is( k X + 1)-sound above κ Xn X for some unique k X .We then let N Xn := M Xη X for all n ≥ n X , and υ Xn,m the identity for all m ≥ n ≥ n X . Case 3: The set { lh( E Xβ ) | β < ζ X } is cofinal below λ X .Let η X := sup( { β < ζ X | lh( E Xβ ) < λ X } ). By assumption and Lemma 45there is some drop in the interval (0 , η X ). Let then γ + 1 be the last such.Let k X be minimal such that ρ k X +1 (( M Xγ +1 ) ∗ ) ≤ crit( E Xγ ). Let n X be min-imal such that κ Xn X ≥ lh( E Xγ ). Let η Xn < η X be minimal such that crit( E Xη Xn ) ≥ κ Xn for n ≥ n X .Let then N Xn := M Xη Xn and υ Xn,m := ι Xη Xn ,η Xm for m ≥ n ≥ n X . It is easy to seethat the maps are as wanted, but it remains to check that N Xn is ( k X + 1)-soundabove κ Xn . This is going to be the one critical use of the assumption that κ Xn isa cutpoint.We have to show that the generators of the iteration up to η Xn are boundedby κ Xn . If η Xn is a limit this is obvious as by choice of η Xn all previous critical34oints are less than κ Xn . So assume that η Xn = δ + 1 and E Xδ has a generator ≥ κ Xn . By the initial segment condition we then have that the trivial completion G of E Xδ ↾ κ Xn is on the sequence of K X . But we have crit( G ) = crit( E Xδ ) < κ Xn and lh( G ) > κ Xn , contradicting that κ Xn is a cutpoint.The covering argument goes through three cases. Thanks to Lemma 45 wecan eliminate one of these cases, we will now see that we can also eliminate theother less than convenient case. Lemma 47. If N Xn for n ≥ n X has a top extender, then µ Xn , its critical point,is ≥ κ Xn .Proof. Let us first assume that N Xn has been constructed according to Case 1or Case 2. Then λ X is a limit cardinal in N Xn and thus by (2) µ Xn cannot besmaller than λ X .If N Xn is constructed according to Case 3, then some ordinal ≥ κ Xn has to bethe critical point of an extender on the N Xn -sequence. As no overlaps can existon the N Xn -sequence, µ Xn ≥ κ Xn follows. Remark 48. Note that N Xn in the notation of [21] is the mouse P γ where κ n = ℵ K X γ . Recall that P γ is the least initial segment (if it exists) of M Xδ thatdefines a subset of κ n not in K X where δ < ζ X is least such that gen( E Xδ ) > κ n .In addition, by the preceding lemma P γ = Q γ , i.e. we are avoiding protomicein this construction.Let then N X := dirlim( hN Xn , υ Xn,m : n X ≤ n ≤ m < ω i ) and υ Xn : C ( N Xn ) →C ( N X ) the direct limit map. It should be easy to see that N X is wellfoundedand that the direct limit maps are r Σ k X +1 -elementary as they are generated byan iteration on K . But more is true: Lemma 49. The phalanx (( K X , N X ) , λ X ) is iterable.Proof. We cannot quote [21] here as it seems a priori possible that the mouse(or weasel) P β , where λ X = ℵ K X β , from that proof is not equal to N X . (Thiswould happen if ( λ + X ) K X is not equal to ( λ + X ) N X .)Nevertheless, the proof presented in [21] works just as well with N X substi-tuted for P β .For those readers not content with this answer, we want to point out thatthere is an easy cheat available to us in this case as (2) implies that λ X mustbe a cutpoint in N X , and hence the iterability of the phalanx reduces to theiterability of N X . The latter holds as N X is an iterate of the core model. Theorem 50. Let λ be a singular cardinal of countable cofinality. Let ~κ := h κ n : n < ω i be a sequence of K -cut points cofinal in λ . Let τ n := ( κ + n ) K , then Q n<ω τ n carries a continuous, tree-like scale.Proof. We will show that ~f := h f ~κ,Kα : α ∈ C λ,K i as defined in the last section isthat scale. Towards that purpose we need to show that this sequence is cofinal35n Q n<ω τ n /J bd . Let g ∈ Q n<ω τ n /J bd be arbitrary. Let X ≺ ( H θ ; ∈ , K || θ, ~f ) be ofgood type, as explained at the beginning of this section, with g ∈ X . It willsuffice to show that there is some α X such that f ~κ,Kα X ( n ) = sup( X ∩ τ n ) for allbut finitely many n .Let hN Xn , υ Xn,m : n X ≤ n ≤ m < ω i and hN X , υ Xn : n X ≤ n < ω i be aspreviously discussed appropriate to our choice of ~κ and X .The first step will be to show that we can realize the least level of K to definea surjection onto sup( X ∩ τ n ) by taking an ultrapower of N Xn for n ≥ n X . Let O Xn := Ult k X ( N Xn ; σ X ↾ K X | τ Xn ) and ˜ σ Xn be the ultrapower map for n ≥ n X . (This ultrapower is formed using equivalence classes [ f, a ] σ X where a ∈ [ κ n ] <ω and f is a function with domain (cid:2) κ Xn (cid:3) | a | that is r Σ k X -definable over N Xn .)We do know that these models are wellfounded, in fact, the phalanx (( K, O Xn ) , κ n )must be iterable. (This is (2) β from [21] or [22], where κ Xn = ℵ K X β .) This meansthat O Xn is an inital segment of K . Furthermore, O Xn is sound above κ n , and˜ σ Xn ( τ Xn ) = sup( X ∩ τ n ) is a cardinal there by the choice of N Xn . This meansthat O Xn is the level of K we are looking for.The next step must be to tie the sequence hO Xn : n X ≤ n < ω i to some levelof K projecting to λ . Our candidate is O X := Ult k X ( N X ; σ X ↾ K X | λ X ). Let˜ σ X be the ultrapower map. By Lemma 49 and the lifting properties of our X not only is O X wellfounded, but it is an initial segment of the core model. Let α X := ( λ + ) O X .The last thing we need are appropriate embeddings from O Xn into O X for n ≥ n X . Define π Xn : C ( O Xn ) → C ( O X ): let π Xn ([ f, a ] σ X ) = h υ Xn ( f ) ↾ (cid:2) κ Xn (cid:3) <ω , a i σ X .It is to be understood here that if f is not an element of C ( N Xn ) but merely de-finable over it, then υ Xn ( f ) is the function over C ( N X ) with the same definitionand parameters moved according to υ Xn .Let now f an r Σ k X definable function over C ( N Xn ), φ an r Σ k X -formula, and a ∈ [ κ n ] <ω . C ( O Xn ) | = φ ([ f, a ] σ X ) ⇔ a ∈ σ X ( { b ∈ (cid:2) κ Xn (cid:3) <ω |N Xn | = φ ( f ( b )) } ) ⇔ a ∈ σ X ( { b ∈ (cid:2) κ Xn (cid:3) <ω |N X | = φ ( υ Xn ( f )( b )) } ) ⇔ C ( O X ) | = φ ( h υ Xn ( f ) ↾ (cid:2) κ Xn (cid:3) <ω , a i σ X )This shows that π Xn is r Σ k X -elementary. Consider then the following dia-gram: 36 ( O X ) C ( N X ) ˜ σ X ♠♠♠♠♠♠♠♠♠♠♠♠♠ C ( O Xn ) π Xn O O C ( N Xn ) υ Xn O O ˜ σ Xn ♠♠♠♠♠♠♠♠♠♠♠♠♠ The diagram commutes, and all of υ Xn , ˜ σ X , ˜ σ Xn are cofinal (in ρ k X ( · )). Thusso is π Xn which shows that it is r Σ k X +1 -elementary. Also note that the criticalpoint of π Xn is ≥ κ n .It then follows that C ( O Xn ) is isomorphic to Hull O X k X +1 ( κ Xn ∪ { p k X +1 ( O X ) } ),so sup( X ∩ τ n ) = f ~κ,Kα X ( n ) for n ≥ n X . Remark 51. The last line is inaccurate, as it seems possible that α X / ∈ C λ,K meaning f ~κ,Kα X might not be defined. Nevertheless the structure O Xn is definablefrom α X and κ n in K which implies that the sequence h sup( X ∩ τ n ) : n X ≤ n <ω i is dominated by some f ~κ,Kβ for β ∈ C λ,K . Corollary 52. In the above situation α X = sup( X ∩ λ + ) .Proof. By continuity f ~κ,K sup( X ∩ λ + ) is the exact upper bound of h f ~κ,Kβ : β < sup( X ∩ λ + ) i . On the other hand, as we know that ~f is a scale, by the tightnessof X we also know that h sup( X ∩ τ n ) : n < ω i is also an exact upper bound forthis sequence. This implies that both agree almost everywhere, but the latterequals f ~κ,Kα X almost everywhere. The desired equality then follows.Let us now move on to the second theorem. This one concerns scales onproducts that concentrate on ordinals that are inaccessible in K . We will seethat scales on such ordinals are significantly more restricted. Theorem 53. Let λ be a singular cardinal of countable cofinality. Let h κ n : n <ω i be a cofinal sequence such that each κ n is an inaccessible limit of cutpointsof K . Assume there is some δ < λ such that ordinals β with o K ( β ) ≥ δ arebounded in each of the κ n . Then Q n<ω κ n admits a continuous, tree-like scale. Let from now on ~κ := h κ n : n < ω i and δ < λ be as in the statement of thetheorem. As this theorem deals with scales on ordinals which are inaccessible in K we will have need of a theorem that provides information about the possiblecofinalites of such ordinals. In general, we cannot expect these cofinalities to behigh because of the existence of Prikry forcing. The next theorem essentiallystates that this is the only real obstacle. Versions of this theorem for different37orms of the core model have existed for some time, but its newest form appro-priate for the Jensen-Steel core model is due to Mitchell and Schimmerling. Theorem 54 (Mitchell-Schimmerling) . Assume there is no inner model witha Woodin cardinal, and let K be the Jensen-Steel core model. Let α ≥ ℵ besuch that α is regular in K , but cof( α ) < card( α ) . Then o K ( α ) ≥ ν where cof( α ) = ω · ν . See [20]. Alternatively, as we only deal with linear iterations here it shouldbe plausible that the results from [4] even though not directly applicable can bemimicked here to achieve a similar end.Let us now again consider some X ≺ ( H θ ; ∈ , . . . ) containing relevant objects.In addition to its previous properties we will require that cof(sup( X ∩ κ n )) >δ . Note then that by our assumption and 54, and this fact will be crucial,sup( X ∩ κ n ) is a singular cardinal in K .We will once again have need of the directed system hN Xn,m , υ Xn,m : n X ≤ n ≤ m < ω i and its limit hN X , υ Xn : n X ≤ n < ω i , but we will require someadditional properties. Lemma 55. There exist ˜ α Xn < κ Xn such that Hull N Xn k X +1 (˜ α Xn ∪ { p k X +1 ( N Xn ) } ) iscofinal in κ Xn .Proof. If the system is constructed as in Case 1 and 2 then there is a single α such that N Xn (which is independent of n ) is sound above α so the conclusionfollows.Consider then that the system is constructed as in Case 3. Pick n ≥ n X .Recall that N Xn = M Xη Xn and γ + 1 is the last drop below η Xn . Note that bydefinition of η Xn all critical points before that point are less than κ Xn . There aretwo cases. Case 3.1: η Xn = ¯ η + 1.In that case as κ Xn is a limit cardinal we must have that lh( E X ¯ η ) < κ Xn . Let˜ α Xn < κ Xn be such that M Xη Xn is sound above ˜ α Xn . Case 3.2: η Xn is a limit ordinal.Let γ < β < η Xn be such that ι Xβ,η Xn (¯ κ ) = κ Xn for some ¯ κ ∈ M Xβ . We musthave that ι Xβ,ξ (¯ κ ) ≥ crit( ι Xξ,η Xn ) for all ξ ∈ (cid:2) β, η Xn (cid:1) . The key is to consider whenequality holds in the above equation.Let us assume towards a contradiction that ι Xβ,ξ (¯ κ ) = crit( ι Xξ,η Xn ) for an un-bounded in η Xn set A . For ν ∈ lim( A ) ∩ η Xn we havecrit( ι Xν,η Xn ) ≥ sup ξ ∈ A ∩ ν crit( ι Xξ,η Xn ) = sup ξ ∈ A ∩ ν ι Xβ,ξ (¯ κ ) = ι Xβ,ν (¯ κ )and hence ν ∈ A . But then B := { ι Xβ,ξ (¯ κ ) | ξ ∈ A } is a club of indiscerniblesin κ Xn . As σ X is continuous at points of cofinality ω , C := σ X ” [ B ] is an ω -club in sup( X ∩ κ n ). As the latter was singular there must exist a club D ⊂ sup( X ∩ κ n ) consisting of K -singulars. But C ∩ D = ∅ , and C consists of K -regulars. Contradiction! 38e conclude that crit( ι Xξ,η Xn ) < ι Xβ,ξ (¯ κ ) for all ξ ≥ ν for some ν ∈ (cid:2) β, η Xn (cid:1) .This means that ι Xν,η Xn is continuous at ι Xβ,ν (¯ κ ). We then finish the argument bynoticing that M Xν is ( k X +1)-sound above crit( ι Xν,η Xn ), and thus Hull M XηXn k X +1 (crit( ι Xν,η Xn ) ∪{ p k X +1 ( M Xη xn ) } ) is cofinal in κ Xn . Proof of Theorem 53. We want to show that for some ~α X := h α Xn : n X ≤ n <ω i ∈ X the sequence h sup( X ∩ κ n ) : n < ω i agrees almost everywhere with g ~κ,K,~α X α X . (Implicit here is that α X will be adequate.)Recall the structures O Xn from the proof of the preceding theorem. Wewill need a slightly different structure here. Let ( O Xn ) ∗ := Ult k X ( N Xn ; σ X ↾ K X | κ Xn ). (This ultrapower is formed using equivalence classes [ f, a ] σ X where a ∈ [sup( X ∩ κ n )] <ω and f is a function with domain [ γ ] | a | where γ < κ n is acardinal with a ⊂ σ X ( γ ) and f is r Σ k X -definable over N Xn . Note that functionswith different domains can be compared by adding dummy values.)Let ¯ σ Xn be the ultrapower map. Note that ¯ σ Xn maps κ Xn cofinally into sup( X ∩ κ n ) so we have Hull ( O Xn ) ∗ n X +1 ( α Xn ∪{ p k X +1 (( O Xn ) ∗ ) } ) is cofinal in sup( X ∩ κ n ) where α Xn := ¯ σ Xn (˜ α Xn ).The phalanx (( K, ( O Xn ) ∗ ) , sup( X ∩ κ n )) is iterable as C (( O Xn ) ∗ ) can bemapped into C ( O Xn ) by a map with critical point sup( X ∩ κ n ), so ( O Xn ) ∗ isan initial segment of K , in fact, the least one to define a witness to the singu-larity of sup( X ∩ κ Xn ).Just as before we can map C (( O Xn ) ∗ ) into C ( O X ), so g ~κ,K,~α X α X ( n ) = sup( X ∩ κ n ) for all n ≥ n X . We would like to have ~α X ∈ X . This is obvious if X is ω -closed. If X is merely internally approachable then we can still find some ~α ′ ∈ X ∩ Q n<ω sup( X ∩ κ n ) that dominates ~α X almost everywhere. Then g ~κ,K,~α X α X and g ~κ,K,~α ′ α X agree almost everywhere by Corollary 39, so we can replace ~α X with ~α ′ . By Fodor’s Lemma we then have a stationary set of X and a single ~α suchthat g ~κ,K,~αα X agrees with h sup( X ∩ κ n : n < ω i almost everywhere. This thenshows that h g ~κ,K,~αα : α ∈ C ∩ cof( >ω ) i is a scale.We are going to finish by showing how to weaken the assumption of Theorem50 yet achieving the same result. It is here that we will make use of the sequence h f ~κ,K,~αα : α ∈ C λ,K i .We say a cardinal κ ∈ K is a weak cutpoint if crit( E ) < κ implies lh( E ) < ( κ + ) K for all extenders E on the K -sequence. Theorem 56. Let λ be a singular cardinal of countable cofinality. Let h κ n : n < ω i be a sequence of weak cutpoints cofinal in λ . Let τ n := ( κ + n ) K . Then Q n<ω τ n carries a continuous, tree-like scale. Lemma 57. There exist some n X , k X < ω , a sequence of ordinals h ˜ α Xn : n X ≤ n < ω i , a sequence of models hN Xn : n X ≤ n < ω i , and maps h υ Xn,m : n X ≤ n ≤ m < ω i such that: (( κ Xn ) + ) N Xn = τ Xn and N Xn agrees with K X up to τ Xn for all n ≥ n X ; • N Xn is ( k X + 1) -sound above κ Xn relative to p k X +1 ( N Xn ) a ˜ α Xn for all n ≥ n X ; • υ Xn,m : C ( N Xn ) → C ( N Xm ) is r Σ k X +1 -elementary for all m ≥ n ≥ n X ; • crit( υ Xn,m ) ≥ max { κ Xn , ˜ α Xn + 1 } for all m ≥ n ≥ n X .Proof. This proof goes through the same cases as the proof of Lemma 46, infact, many of the cases will be the same. (In those cases we can take ˜ α Xn to be0.) In the interest of time we shall only deal with the case that is unique to thissituation.Let us assume that η X := sup( { β < ζ X | lh( E Xβ ) < λ } ) is a limit ordinal.Let γ + 1 be the last drop in the interval (0 , η X ). Let k X be minimal such that ρ k X +1 (( M Xγ ) ∗ ) ≤ crit( E Xγ ). Let n X be minimal such that κ Xn X ≥ lh( E Xγ ). Let η Xn < η X be minimal such that crit( E Xη Xn ) ≥ κ Xn for n ≥ n X .Let then N Xn := M Xη Xn and υ Xn,m := ι Xη Xn ,η Xm for m ≥ n ≥ n X . Let n ≥ n X be such that η Xn = ˜ η Xn + 1 and gen( E X ˜ η Xn ) ≥ κ n . Otherwise the argument willproceed just as in the proof of Lemma 46 (and ˜ α Xn = 0).First note then gen( E X ˜ η Xn ) < τ Xn as otherwise κ Xn could not be a weak cutpointby the initial segment condition. Moreover, it then follows that E X ˜ η Xn has a largestgenerator as otherwise gen( E X ˜ η Xn ) ∈ (cid:0) κ Xn , τ Xn (cid:1) must be a cardinal in N Xn .Let ˜ α Xn be that largest generator. We will be done if we can show that κ Xn ∪ { ˜ α Xn } generates the whole ultrapower. Let then ˜ M := Ult( M X ˜ η Xn ; E X ˜ η Xn ↾ κ Xn ∪ { ˜ α Xn } ) and ˜ ι : C ( ˜ N ) → C ( N Xn ) be the canonical embedding.We have that ˜ α Xn ∈ (cid:0) κ Xn , τ Xn (cid:1) is in the range of ˜ ι , thus so is κ Xn = card N Xn (˜ α Xn )and some surjection from κ Xn on to ˜ α Xn . Then ˜ α Xn ⊂ ran ˜ ι and thus so are all ofthe other generators of E X ˜ η Xn . Remark 58. Note that in the “special” case of Lemma 57 unlike Remark 48 N Xn is not equal to the mouse P γ (where κ n = ℵ K X γ ) from [21], but it is equalto Q γ . To see this we must first realize that P γ must be an initial segment of M X ˜ η Xn as in this case E X ˜ η Xn has generators ≥ κ n .As the Dodd projectum of E X ˜ η Xn is below κ n we have, in fact, P γ = M X ˜ η Xn | lh( E X ˜ η Xn ).Note though that lifting this mouse by σ X would create a proto mouse. Hencewe must move to the mouse Q γ which is formed by applying the extender E X ˜ η Xn using the usual iteration tree rules. Hence the resulting mouse must be equalto M X ˜ η Xn +1 = N Xn . Proof of Theorem 56. Let hN Xn : n X ≤ n < ω i , h υ Xn,m : n X ≤ n ≤ m < ω i and h ˜ α Xn : n X ≤ n < ω i as in the lemma. We will find some α X and ~α X := h α Xn : n X ≤ n < ω i such that sup( X ∩ τ n ) = f ~κ,K,~αα X ( n ) for all n ≥ n X . In fact, α Xn = σ X (˜ α Xn ) will do. A priori ~α X will depend on X but we will be able todeal with that by pressing down just as in the proof of Theorem 53.40e can mostly proceed as in the proof of Theorem 50. We will form O Xn and O X as before, and generate embeddings between them by lifting υ Xn . Notethat υ Xn will not move ˜ α Xn as iteration maps do not move generators. So nei-ther will its lift move α Xn . Thus C ( O Xn ) will be isomorphic to Hull O X k X +1 ( κ n ∪{ p k X +1 ( O X ) a α Xn } ) as required.As before the fact that the phalanx hh K, O Xn i , κ n i is iterable follows from thecovering lemma, noticing that in this case we might have to consider the mouse Q β not P β as explained above. Fortunately, this does not change anythingabout the rest of the argument. We skip further detail. Proof of Theorem 5. We have different cases depending on if the κ i are limitcardinals or successor cardinals in the core model. Let us first assume that all κ i share a type. If that shared type is limit cardinals, then we can use Theorem53 to finish. If that type is successor cardinals we have two cases: if ¯ κ i is the K -predecessor of κ i is measurable, then it must be a cutpoint by the smallnessassumption therefore we can use Theorem 50 to finish; if it is not, then it mustbe a weak cutpoint thus we can use Theorem 56 to finish.In cases of mixed type, divide the sequence into three parts of pure type.Each of these parts do have a scale by the above. These individual scales canthen be integrated. This works as individual elements of the different scales canbe tied to some common ordinal <λ + . We conclude this work with a discussion on further possible developments, andopen questions.1. Consider the following natural strengthening of the ABSP with respectsequence of regular cardinals ~τ = h τ n | n < ω i with λ = ∪ n τ n : For everysufficiently large regular cardinal θ and internally approachable structure N ≺ ( H θ , ∈ , ~τ ), there is some m < ω , so that for every strictly increasingsequence d , . . . , d k ∈ ω \ m and F ∈ N , F : [ λ ] k → λ , if F ( χ N ( τ d ) , . . . , χ N ( τ d k )) < τ d then F ( χ N ( τ d ) , . . . , χ N ( τ d k )) ∈ N. Is it consistent?2. We saw in Section 2.1 that from the same large cardinal assumptions ofTheorem 29, it is consistent that ABSP holds with respect to a sequence h τ n | n < ω so that τ n = ℵ n for all n < ω . Is ABSP consistent withrespect to cofinite sequence of the ℵ n ’s?3. The definitions of Tree-like scales, Essentially Tree-like scales, ASFP, andABFP naturally extend to uncountable sequences of cardinals h τ i | i < ρ i ,41 > ℵ regular. Are those principles consistent? If so, what is theirconsistency strength?4. Another natural extension of the principles AFSP and ABFP, is to requirethe appropriate principle to hold for any elementary substrucute N ≺ ( H θ ∈ ~τ ). Is it consistent?5. Is there a version of Theorem 6 for Neeman-Steel long extender mice?6. Pereira showed in [25] that it consistent relative to the existence of asupercompact cardinal that there exist products Q n<ω τ n carrying a contin-uous tree-like scale of length greater than sup( h τ n i n ) + . Can the same beachieved from a weaker large cardinal assumptions at the level of strongcardinals? References [1] Omer Ben-Neria. On singular stationarity ii. Journal of Symbolic Logic ,84(1):320–342, 2019.[2] Omer Ben-Neria, Moti Gitik, Itay Neeman, and Spencer Unger. On thepowersets of singular cardinals in hod. Proceedings of the American Math-ematical Society , 148:1777 – 1789, 2020.[3] Omer Ben-Neria and Spencer Unger. Homogeneous changes in cofinalitieswith applications to hod. 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