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 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.

THE SOLIDITY AND NONSOLIDITY OF INITIAL SEGMENTS OF THE CORE MODEL

2018 ◽  
Vol 83 (3) ◽  
pp. 920-938
Author(s):  
GUNTER FUCHS ◽  
RALF SCHINDLER
Keyword(s):  
Core Model ◽  
Solid Core ◽  
The Core ◽  
Inner Model ◽  
Cohen Real ◽  
Image Position ◽  
Woodin Cardinal

AbstractIt is shown that $K|{\omega _1}$ need not be solid in the sense previously introduced by the authors: it is consistent that there is no inner model with a Woodin cardinal yet there is an inner model W and a Cohen real x over W such that $K|{\omega _1}\,\, \in \,\,W[x] \setminus W$. However, if ${0^{\rm{\P}}}$ does not exist and $\kappa \ge {\omega _2}$ is a cardinal, then $K|\kappa$ is solid. We draw the conclusion that solidity is not forcing absolute in general, and that under the assumption of $\neg {0^{\rm{\P}}}$, the core model is contained in the solid core, previously introduced by the authors.It is also shown, assuming ${0^{\rm{\P}}}$ does not exist, that if there is a forcing that preserves ${\omega _1}$, forces that every real has a sharp, and increases $\delta _2^1$, then ${\omega _1}$ is measurable in K.

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).

scholarly journals Jónsson cardinals, Erdős cardinals, and the core model

1999 ◽  
Vol 64 (3) ◽  
pp. 1065-1086 ◽  
Author(s):  
W. J. Mitchell
Keyword(s):  
Minimal Model ◽  
Independent Interest ◽  
Core Model ◽  
Generic Extension ◽  
The Core ◽  
Steel Core ◽  
Inner Model ◽  
Jonsson Cardinals ◽  
Woodin Cardinal

AbstractWe show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ5-Erdős in K.In the absence of the Steel core model K we prove the same conclusion for any model L[] such that either V = L[] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L[].The proof includes one lemma of independent interest: If V = L[A], where A ⊂ κ and κ is regular, then Lκ[A] is a Jónsson algebra. The proof of this result. Lemma 2.5, is very short and entirely elementary.

Successive weakly compact or singular cardinals

1999 ◽  
Vol 64 (1) ◽  
pp. 139-146 ◽  
Author(s):  
Ralf-Dieter Schindler
Keyword(s):  
Core Model ◽  
Weakly Compact ◽  
The Core ◽  
Inner Model ◽  
Singular Cardinals ◽  
Woodin Cardinal

AbstractIt is shown in ZF that if δ < δ+ < Ω are such that δ and δ+ are either both weakly compact or singular cardinals and Ω is large enough for putting the core model apparatus into action then there is an inner model with a Woodin cardinal.

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.

Countable unions of simple sets in the core model

1996 ◽  
Vol 61 (1) ◽  
pp. 293-312 ◽  
Author(s):  
P. D. Welch
Keyword(s):  
Core Model ◽  
Countable Union ◽  
Skolem Function ◽  
The Core ◽  
Inner Model ◽  
Type Property ◽  
Simple Sets

AbstractWe follow [8] in asking when a set of ordinals X ⊆ α is a countable union of sets in K, the core model. We show that, analogously to L, an X closed under the canonical Σ1 Skolem function for Kα can be so decomposed provided K is such that no ω-closed filters are put on its measure sequence, but not otherwise. This proviso holds if there is no inner model of a weak Erdős-type property.

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

Ramsey cardinals, α-Erdös cardinals, and the core model

1991 ◽  
Vol 56 (1) ◽  
pp. 108-114
Author(s):  
Dirk R. H. Schlingmann
Keyword(s):  
Measurable Cardinal ◽  
Core Model ◽  
The Core ◽  
Transitive Model ◽  
Normal Measure ◽  
Inner Model

The core model K was introduced by R. B. Jensen and A. J. Dodd [DoJ]. K is the union of Gödel's constructible universe L together with all mice, i.e., , and K is a transitive model of ZFC + (V = K) + GCH (see [DoJ]). V = K is consistent with the existence of Ramsey cardinals [M], and if cf(α) > ω, V = K is consistent with the existence of α-Erdös cardinals [J]. Let K be Ramsey. Then there is a smallest inner model Wκ of ZFC in which κ is Ramsey. We have Wκ ⊨ V = K and Wκ ⊆ K [M]. The existence of Wκ with is equivalent to the existence of a sharplike mouse on N ⊨ K with N ⊨ κ Ramsey. (A mouse N on is called sharplike provided .) We have , where is the mouse iteration of N. N is the oleast mouse not in Wκ (see [J] and [DJKo]). Here < denotes the mouse order. The context always clarifies whether the mouse order or the usual <-relation is meant.The main result of §1 is that Wκ ⊨ κ is the only Ramsey cardinal. A similar result has been found true in the smallest inner model L[U] of ZFC + “κ is measurable” if U is a normal measure on κ: L[U] ⊨ κ is the only measurable cardinal [Ku].

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