Characterising subsets of ω1 constructible from a real

1994 ◽  
Vol 59 (4) ◽  
pp. 1420-1432 ◽  
Author(s):  
P. D. Welch
Keyword(s):  
Upper Bound ◽  
Large Cardinal ◽  
Core Model ◽  
The Core ◽  
Measurable Cardinals

AbstractA small large cardinal upper bound in V for proving when certain subsets of ω1 (including the universally Baire subsets) are precisely those constructible from a real is given. In the core model we find an exact equivalence in terms of the length of the mouse order; we show that ∀B ⊆ ω1 [B is universally Baire ⇔ B ϵ L[r] for some real r] is preserved under set-sized forcing extensions if and only if there are arbitrarily large “admissibly measurable” cardinals.

scholarly journals Ramsey-like cardinals II

2011 ◽  
Vol 76 (2) ◽  
pp. 541-560 ◽  
Author(s):  
Victoria Gitman ◽  
P. D. Welch
Keyword(s):  
Upper Bound ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Core Model ◽  
Consistency Strength ◽  
The Core ◽  
Countable Ordinal ◽  
Elementary Embeddings ◽  
Remarkable Cardinals

AbstractThis paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

scholarly journals GAMES AND RAMSEY-LIKE CARDINALS

2019 ◽  
Vol 84 (1) ◽  
pp. 408-437 ◽  
Author(s):  
DAN SAATTRUP NIELSEN ◽  
PHILIP WELCH
Keyword(s):  
Core Model ◽  
The Core ◽  
Game Theoretic ◽  
Measurable Cardinals ◽  
Remarkable Cardinals

AbstractWe generalise the α-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α to arbitrary ordinals α, and answer several questions posed in that paper. In particular, we show that α-Ramseys are downwards absolute to the core model K for all α of uncountable cofinality, that strategic ω-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α-Ramsey cardinals are equiconsistent with measurable cardinals for all α > ω. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990), and Sharpe and Welch (2011).

The complexity of the core model

1998 ◽  
Vol 63 (4) ◽  
pp. 1393-1398
Author(s):  
William J. Mitchell
Keyword(s):  
Core Model ◽  
Boolean Combination ◽  
The Core ◽  
Inner Model ◽  
Measurable Cardinals

AbstractIf there is no inner model with a cardinal κ such that ο(κ) = κ++ then the set K ∩ Hω1 is definable over Hω1 by a Δ4 formula, and the set of countable initial segments of the core model is definable over Hω1 by a Π3 formula. We show that if there is an inner model with infinitely many measurable cardinals then there is a model in which is not definable Σ3 by any Σ3 formula, and K ∩ Hω1 is not definable by any boolean combination of Σ3 formulas.

The reals in core models

1987 ◽  
Vol 52 (1) ◽  
pp. 64-67
Author(s):  
Philip Welch
Keyword(s):  
Core Model ◽  
The Core ◽  
Measurable Cardinals

AbstractWe set = ‹, ≤L, # ›, where is the set of degrees of nonconstructibility for countable sets of countable ordinals. We show how to define inductively over this structure the degrees of such sets of ordinals in Κ, the core model, and the next few core models thereafter, i.e. without reference to mice, premice or measurable cardinals

The consistency strength of the free-subset property for ωω

1984 ◽  
Vol 49 (4) ◽  
pp. 1198-1204 ◽  
Author(s):  
Peter Koepke
Keyword(s):  
Measurable Cardinal ◽  
Core Model ◽  
Reverse Direction ◽  
Ground Model ◽  
Consistency Strength ◽  
The Core ◽  
Inner Model ◽  
Free Subset ◽  
Coherent Sequence ◽  
Measurable Cardinals

A subset X of a structure S is called free in S if ∀x ∈ Xx ∉ S[X − {x}]; here, S[Y] is the substructure of S generated from Y by the functions of S. For κ, λ, μ cardinals, let Frμ(κ, λ) be the assertion:for every structure S with κ ⊂ S which has at most μ functions and relations there is a subset X ⊂ κ free in S of cardinality ≥ λ.We show that Frω(ωω, ω), the free-subset property for ωω, is equiconsistent with the existence of a measurable cardinal (2.2,4.4). This answers a question of Devlin [De].In the first section of this paper we prove some combinatorial facts about Frμ(κ, λ); in particular the first cardinal κ such that Frω(κ, ω) is weakly inaccessible or of cofinality ω (1.2). The second section shows that, under Frω(ωω, ω), ωω is measurable in an inner model. For the convenience of readers not acquainted with the core model κ, we first deduce the existence of 0# (2.1) using the inner model L. Then we adapt the proof to the core model and obtain that ωω is measurable in an inner model. For the reverse direction, we essentially apply a construction of Shelah [Sh] who forced Frω(ωω, ω) over a ground model which contains an ω-sequence of measurable cardinals. We show in §4 that indeed a coherent sequence of Ramsey cardinals suffices. In §3 we obtain such a sequence as an endsegment of a Prikry sequence.

The core model for sequences of measures. I

1984 ◽  
Vol 95 (2) ◽  
pp. 229-260 ◽  
Author(s):  
William J. Mitchell
Keyword(s):  
High Order ◽  
Core Model ◽  
Basic Properties ◽  
The Core ◽  
Inner Model ◽  
Measurable Cardinals

The model K() presented in this paper is a new inner model of ZFC which can contain measurable cardinals of high order. Like the model L() of [14], from which it is derived, K() is constructed from a sequence of filters such that K() satisfies for each (α, β) ε domain () that (α,β) is a measure of order β on α and the only measures in K() are the measures (α,β). Furthermore K(), like L(), has many of the basic properties of L: the GCH and ⃟ hold and there is a definable well ordering which is on the reals. The model K() is derived from L() by using techniques of Dodd and Jensen [2–5] to build in absoluteness for measurability and related properties.

scholarly journals The Jensen Covering Property

2001 ◽  
Vol 66 (4) ◽  
pp. 1505-1523 ◽  
Author(s):  
E. Schimmerling ◽  
W. H. Woodin
Keyword(s):  
Large Cardinal ◽  
Core Model ◽  
Covering Property ◽  
Prikry Forcing ◽  
Covering Lemma ◽  
Woodin Cardinals ◽  
Inner Models ◽  
Coherent Sequence ◽  
Measurable Cardinals

The Jensen covering lemma says that either L has a club class of indiscernibles, or else, for every uncountable set A of ordinals, there is a set B ∈ L with A ⊆ B and card (B) = card(A). One might hope to extend Jensen's covering lemma to richer core models, which for us will mean to inner models of the form L[] where is a coherent sequence of extenders of the kind studied in Mitchell-Steel [8], The papers [8], [12], [10] and [1] show how to construct core models with Woodin cardinals and more. But, as Prikry forcing shows, one cannot expect too direct a generalization of Jensen's covering lemma to core models with measurable cardinals.Recall from [8] that if L[] is a core model and α is an ordinal, then either Eα = ∅, or else Eα is an extender over As in [8], we assume here that if Eα is an extender, then Eα is below superstrong type in the sense that the set of generators of Eα is bounded in (crit(Eα)). Let us say that L[] is a lower-part core model iff for every ordinal α, Eα is not a total extender over L[]. In other words, if L[] is a lower-part core model, then no cardinal in L[] is measurable as witnessed by an extender on . Other than the “below superstrong” hypothesis, we impose no bounds on the large cardinal axioms true in the levels of a lower-part core model.

Scales in K(ℝ) at the end of a weak gap

2008 ◽  
Vol 73 (2) ◽  
pp. 369-390 ◽  
Author(s):  
J. R. Steel
Keyword(s):  
Induction Hypothesis ◽  
The Other ◽  
Core Model ◽  
The Core ◽  
Woodin Cardinals ◽  
Other Hand

In this note we shall proveTheorem 0.1. Letbe a countably ω-iterable-mouse which satisfies AD, and [α, β] a weak gap of. Supposeis captured by mice with iteration strategies in ∣α. Let n be least such that ; then we have that believes that has the Scale Property.This complements the work of [5] on the construction of scales of minimal complexity on sets of reals in K(ℝ). Theorem 0.1 was proved there under the stronger hypothesis that all sets definable over are determined, although without the capturing hypothesis. (See [5, Theorem 4.14].) Unfortunately, this is more determinacy than would be available as an induction hypothesis in a core model induction. The capturing hypothesis, on the other hand, is available in such a situation. Since core model inductions are one of the principal applications of the construction of optimal scales, it is important to prove 0.1 as stated.Our proof will incorporate a number of ideas due to Woodin which figure prominently in the weak gap case of the core model induction. It relies also on the connection between scales and iteration strategies with the Dodd-Jensen property first discovered in [3]. Let be the pointclass at the beginning of the weak gap referred to in 0.1. In section 1, we use Woodin's ideas to construct a Γ-full a mouse having ω Woodin cardinals cofinal in its ordinals, together with an iteration strategy Σ which condenses well in the sense of [4, Def. 1.13]. In section 2, we construct the desired scale from and Σ.

scholarly journals Laver sequences for extendible and super-almost-huge cardinals

1999 ◽  
Vol 64 (3) ◽  
pp. 963-983 ◽  
Author(s):  
Paul Corazza
Keyword(s):  
Critical Point ◽  
Upper Bound ◽  
Large Cardinal ◽  
Strong Cardinal ◽  
Regular Class ◽  
Strong Cardinals ◽  
Extendible Cardinals

AbstractVersions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses. Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that κ is extendible, there is an extendible Laver sequence at κ. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals.

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