The well-foundedness of the Mitchell order

1993 ◽  
Vol 58 (3) ◽  
pp. 931-940 ◽  
Author(s):  
J. R. Steel
Keyword(s):  
Critical Point ◽  
Elementary Proof ◽  
Large Cardinal ◽  
Canonical Embedding ◽  
Inner Models ◽  
Normal Ultrafilter ◽  
Well Foundedness

Let E ⊲ F iff E and F are extenders and E ∈ Ult(V, F). Intuitively, E ⊲ F implies that E is weaker—embodies less reflection—than F. The relation ⊲ was first considered by W. Mitchell in [M74], where it arises naturally in connection with inner models and coherent sequences. Mitchell showed in [M74] that the restriction of ⊲ to normal ultrafilters is well-founded.The relation ⊲ is now known as the Mitchell order, although it is not actually an order. It is irreflexive, and its restriction to normal ultrafilters is transitive, but under mild large cardinal hypotheses, it is not transitive on all extenders. Here is a counterexample. Let κ be (λ + 2)-strong, where λ > κ and λ is measurable. Let E be an extender with critical point κ and let U be a normal ultrafilter with critical point λ such that U ∈ Ult(V, E). Let i: V → Ult(V, U) be the canonical embedding. Then i(E) ⊲ U and U ⊲ E, but by 3.11 of [MS2], it is not the case that i(E) ⊲ E. (The referee pointed out the following elementary proof of this fact. Notice that i ↾ Vλ+2 ∈ Ult(V, E) and X ∈ Ea ⇔ X ∈ i(E)i(a). Moreover, we may assume without loss of generality that = support(E). Thus, if i(E) ∈ Ult(V, E), then E ∈ Ult(V, E), a contradiction.)By going to much stronger extenders, one can show the Mitchell order is not well-founded. The following example is well known. Let j: V → M be elementary, with Vλ ⊆ M for λ = joω(crit(j)). (By Kunen, Vλ+1 ∉ M.) Let E0 be the (crit(j), λ) extender derived from j, and let En+1 = i(En), where i: V → Ult(V, En) is the canonical embedding. One can show inductively that En is an extender over V, and thereby, that En+1 ⊲ En for all n < ω. (There is a little work in showing that Ult(V, En+1) is well-founded.)

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.

Splitting number at uncountable cardinals

1997 ◽  
Vol 62 (1) ◽  
pp. 35-42 ◽  
Author(s):  
Jindřich Zapletal
Keyword(s):  
Large Cardinal ◽  
Inner Models ◽  
Splitting Number ◽  
Uncountable Cardinal ◽  
Uncountable Cardinals

AbstractWe study a generalization of the splitting number s to uncountable cardinals. We prove that 𝔰(κ) > κ+ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption 𝔰(ℵω) > ℵω+1 has a considerable large cardinal strength as well.

Forbidden intervals

2009 ◽  
Vol 74 (4) ◽  
pp. 1081-1099 ◽  
Author(s):  
Matthew Foreman
Keyword(s):  
Critical Point ◽  
Set Theory ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Elementary Embeddings ◽  
Measurable Cardinals ◽  
Precipitous Ideals ◽  
Strongly Compact Cardinals

Many classical statements of set theory are settled by the existence of generic elementary embeddings that are analogous the elementary embeddings posited by large cardinals. [2] The embeddings analogous to measurable cardinals are determined by uniform, κ-complete precipitous ideals on cardinals κ. Stronger embeddings, analogous to those originating from supercompact or huge cardinals are encoded by normal fine ideals on sets such as [κ]<λ or [κ]λ.The embeddings generated from these ideals are limited in ways analogous to conventional large cardinals. Explicitly, if j: V → M is a generic elementary embedding with critical point κ and λ supnЄωjn(κ) and the forcing yielding j is λ-saturated then j“λ+ ∉ M. (See [2].)Ideals that yield embeddings that are analogous to strongly compact cardinals have more puzzling behavior and the analogy is not as straightforward. Some natural ideal properties of this kind have been shown to be inconsistent:Theorem 1 (Kunen). There is no ω2-saturated, countably complete uniform ideal on any cardinal in the interval [ℵω, ℵω).Generic embeddings that arise from countably complete, ω2-saturated ideals have the property that sup . So the Kunen result is striking in that it apparently allows strong ideals to exist above the conventional large cardinal limitations. The main result of this paper is that it is consistent (relative to a huge cardinal) that such ideals exist.

scholarly journals Inner models with large cardinal features usually obtained by forcing

2011 ◽  
Vol 51 (3-4) ◽  
pp. 257-283 ◽  
Author(s):  
Arthur W. Apter ◽  
Victoria Gitman ◽  
Joel David Hamkins
Keyword(s):  
Large Cardinal ◽  
Inner Models

Many-times huge and superhuge cardinals

1984 ◽  
Vol 49 (1) ◽  
pp. 112-122 ◽  
Author(s):  
Julius B. Barbanel ◽  
Carlos A. Diprisco ◽  
It Beng Tan
Keyword(s):  
New York ◽  
Positive Integer ◽  
Critical Point ◽  
Order Type ◽  
Elementary Embedding ◽  
Rochester Institute Of Technology ◽  
Inner Model ◽  
Institute Of Technology ◽  
The Universe ◽  
Normal Ultrafilter

In this paper we consider various generalizations of the notion of hugeness. We remind the reader that a cardinal κ is huge if there exist a cardinal λ > κ, an inner model M which is closed under λ-sequences, and an elementary embedding i: V → M with critical point κ such that i(κ) = λ. We shall call λ a target for κ and shall write κ → (λ) to express this fact. Equivalently, κ is huge with target λ if and only if there exists a normal ultrafilter on P=κ(λ) = {X ⊆ λ:X has order type κ}. For the proof and additional facts on hugeness, see [3].We assume that the reader is familiar with the notions of measurability and supercompactness. If κ is γ-supercompact for each γ < λ, we shall say that κ is < λ-supercompact. We note that if κ → (λ), it follows immediately that κ is < λ-supercompact.Throughout the paper, n shall be used to denote a positive integer, the letters α, β, and δ shall denote ordinals, while κ, λ, γ, and η shall be reserved for cardinals. All addition is ordinal addition. V denotes the universe of all sets.All results except for Theorems 6b and 6c and Lemma 6d can be formalized in ZFC.This paper was written while the first named author was at Rochester Institute of Technology, Rochester, New York. We wish to thank the department of mathematics at R.I.T. for secretarial time and facilities.

Fine structure for tame inner models

1996 ◽  
Vol 61 (2) ◽  
pp. 621-639 ◽  
Author(s):  
E. Schimmerling ◽  
J. R. Steel
Keyword(s):  
Fine Structure ◽  
Large Cardinal ◽  
Structure Theory ◽  
Uniqueness Problem ◽  
Covering Property ◽  
Strong Cardinal ◽  
Woodin Cardinals ◽  
Inner Models ◽  
Normal Measure ◽  
Consistency Results

In this paper, we solve the strong uniqueness problem posed in [St2]. That is, we extend the full fine structure theory of [MiSt] to backgrounded models all of whose levels are tame (defined in [St2] and below). As a consequence, more powerful large cardinal properties reflect to fine structural inner models. For example, we get the following extension to [MiSt, Theorem 11.3] and [St2, Theorem 0.3].Suppose that there is a strong cardinal that is a limit of Woodin cardinals. Then there is a good extender sequence such that(1) every level of is a sound, tame mouse, and(2) ⊨ “There is a strong cardinal that is a limit of Woodin cardinals”.Recall that satisfies GCH if all its levels are sound. Another consequence of our work is the following covering property, an extension to [St1, Theorem 1.4] and [St3, Theorem 1.10].Suppose that fi is a normal measure on Ω and that all premice are tame. Then Kc, the background certified core model, exists and is a premouse of height Ω. Moreover, for μ-almost every α < Ω.Ideas similar to those introduced here allow us to extend the fine structure theory of [Sch] to the level of tame mice. The details of this extension shall appear elsewhere. From the extension of [Sch] and Theorem 0.2, new relative consistency results follow. For example, we have the following application.If there is a cardinal κ such that κ is κ+-strongly compact, then there is a premouse that is not tame.

Co-critical points of elementary embeddings

1985 ◽  
Vol 50 (1) ◽  
pp. 220-226
Author(s):  
Michael Sheard
Keyword(s):  
Critical Point ◽  
Measurable Cardinal ◽  
Elementary Embedding ◽  
Proper Class ◽  
Ground Model ◽  
Target Model ◽  
Inner Models ◽  
Inner Model ◽  
Elementary Embeddings

Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universeLinto itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model” (and in the latter case they are equal). It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: ifis a generic ultrafilter arising from forcing with a precipitous ideal on a successor cardinalκ, then the ultraproduct of the ground model viacollapsesκ. Such considerations suggest a classification of how close the target model comes to “fitting inside” the ground model.Definition 1.1. LetMandNbe inner models (transitive, proper class models) of ZFC, and letj:M→Nbe an elementary embedding. Theco-critical pointofjis the least ordinalλ, if any exist, such that there isX⊆λ, X∈NbutX∉M. Such anXis called anew subsetofλ.It is easy to see that the co-critical point ofj:M→Nis a cardinal inN.

Between strong and superstrong

1986 ◽  
Vol 51 (3) ◽  
pp. 547-559 ◽  
Author(s):  
Stewart Baldwin
Keyword(s):  
Critical Point ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Proper Class ◽  
Consistency Strength ◽  
Strong Cardinal ◽  
Definition Of ◽  
Strong Cardinals

Definition. A cardinal κ is strong iff for every x there is an elementary embedding j:V → M with critical point κ such that x ∈ M.κ is superstrong iff ∃j:V → M with critical point κ such that Vj(κ) ∈ M.These definitions are natural weakenings of supercompactness and hugeness respectively and display some of the same relations. For example, if κ is superstrong then Vκ ⊨ “∃ proper class of strong cardinals”, but the smallest superstrong cardinal is less than the smallest strong cardinal (if both types exist). (See [SRK] and [Mo] for the arguments involving supercompact and huge, which translate routinely to strong and superstrong.)Given any two types of large cardinals, a typical vague question which is often asked is “How large is the gap in consistency strength?” In one sense the gap might be considered relatively small, since the “higher degree” strong cardinals described below (a standard trick that is nearly always available) and the Shelah and Woodin hierarchies of cardinals (see [St] for a definition of these) seem to be (at least at this point in time) the only “natural” large cardinal properties lying between strong cardinals and superstrong cardinals in consistency strength.

Deconstructing inner model theory

2002 ◽  
Vol 67 (2) ◽  
pp. 721-736 ◽  
Author(s):  
Ralf-Dieter Schindler ◽  
John Steel ◽  
Martin Zeman
Keyword(s):  
Model Theory ◽  
Initial Segment ◽  
Large Cardinal ◽  
Inner Model Theory ◽  
Inner Models ◽  
Inner Model

In this paper we shall repair some errors and fill some gaps in the inner model theory of [2]. The problems we shall address affect some quite basic definitions and proofs.We shall be concerned with condensation properties of canonical inner models constructed from coherent sequences of extenders as in [2]. Condensation results have the general form: if x is definable in a certain way over a level , then either x ∈ , or else from x we can reconstruct in a simple way.The first condensation property considered in [2] is the initial segment condition, or ISC. In section 1 we show that the version of this condition described in [2] is too strong, in that no coherent in which the extenders are indexed in the manner of [2], and which is such that L[] satisfies the mild large cardinal hypothesis that there is a cardinal which is strong past a measurable, can satisfy the full ISC of [2]. It follows that the coherent sequences constructed in [2] do not satisfy the ISC of [2]. We shall describe the weaker ISC which these sequences do satisfy, and indicate the small changes in the arguments of [2] this new condition requires.

Projective well-orderings and bounded forcing axioms

2005 ◽  
Vol 70 (2) ◽  
pp. 557-572
Author(s):  
Andrés Eduardo Caicedo
Keyword(s):  
Large Cardinal ◽  
Forcing Axioms ◽  
Woodin Cardinals ◽  
Inner Models

AbstractIn the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

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