Huge Cardinals Research Articles

Model theoretic characterizations of large cardinals

We consider compactness characterizations of large cardinals. Based on results of Benda [Ben78], we study compactness for omitting types in various logics. In $$\mathbb{L}$$κ,κ, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vopĕnka’s Principle.

Superstrong and other large cardinals are never Laver indestructible

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals (all for n ? 3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if ? exhibits any of them, with corresponding target ?, then in any forcing extension arising from nontrivial strategically <?-closed forcing $${\mathbb{Q} \in V_\theta}$$Q?V?, the cardinal ? will exhibit none of the large cardinal properties with target ? or larger.

CHANG’S CONJECTURE, GENERIC ELEMENTARY EMBEDDINGS AND INNER MODELS FOR HUGE CARDINALS

AbstractWe introduce a natural principleStrong Chang Reflectionstrengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show thatdecisiveideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found in [10].1,2

Forbidden intervals

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]&lt;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.

Incompatible Ω-Complete Theories

AbstractIn 1985 the second author showed that if there is a proper class of measurable Woodin cardinals andandare generic extensions ofVsatisfying CH thenandagree on all Σ12-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ12. Moreover, CH is the unique Σ12-statement with this feature in the sense that any other Σ12-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axiomsAsuch that relative to large cardinal axioms ZFC +Ais Ω-complete for all of third-order arithmetic. Going further, for each specifiable segmentVλof the universe of sets (for example, one might takeVλto be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axiomsAsuch that relative to large cardinal axioms ZFC +Ais Ω-complete for the theory ofVλ. If such theories exist, extend one another, and are unique in the sense that any other such theoryBwith the same level of Ω-completeness asAis actually Ω-equivalent toAover ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies ¬-CH.

Smoke and mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals

Since the work of Godel and Cohen many questions in infinite combinatorics have been shown to be independent of the usual axioms for mathematics, Zermelo Frankel Set Theory with the Axiom of Choice (ZFC). Attempts to strengthen the axioms to settle these problems have converged on a system of principles collectively known as Large Cardinal Axioms. These principles are linearly ordered in terms of consistency strength. As far as is currently known, all natural independent combinatorial statements are equiconsistent with some large cardinal axiom. The standard techniques for showing this use forcing in one direction and inner model theory in the other direction. The conspicuous open problems that remain are suspected to involve combinatorial principles much stronger than the large cardinals for which there is a current fine-structural inner model theory for. The main results in this paper show that many standard constructions give objects with combinatorial properties that are, in turn, strong enough to show the existence of models with large cardinals are larger than any cardinal for which there is a standard inner model theory.

Large cardinals and locally defined well-orders of the universe

By forcing over a model of ZFC + GCH (above ℵ 0 ) with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H ( κ + ) ( κ ≥ ω 2 a regular cardinal) is a well-order of H ( κ + ) definable over the structure 〈 H ( κ + ) , ∈ 〉 by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in addition to forcing a locally defined well-order of the universe, preserve many of the n -huge cardinals from the ground model (for all n ).

The spectrum of elementary embeddings [formula omitted

In 1970, K. Kunen, working in the context of Kelley–Morse set theory, showed that the existence of a nontrivial elementary embedding j : V → V is inconsistent. In this paper, we give a finer analysis of the implications of his result for embeddings V → V relative to models of ZFC. We do this by working in the extended language { ∈ , j } , using as axioms all the usual axioms of ZFC (for ∈ -formulas), along with an axiom schema that asserts that j is a nontrivial elementary embedding. Without additional axiomatic assumptions on j, we show that that the resulting theory (denoted ZFC+BTEE) is weaker than an ω -Erdös cardinal, but stronger than n -ineffables. We show that natural models of ZFC+BTEE give rise to Schindler’s remarkable cardinals. The approach to inconsistency from ZFC+BTEE forks into two paths: extensions of ZFC+BTEE+Cofinal Axiom and ZFC+BTEE+¬Cofinal Axiom, where Cofinal Axiom asserts that the critical sequence κ , j ( κ ) , j 2 ( κ ) , … is cofinal in the ordinals. We describe near-minimal inconsistent extensions of each of these theories. The path toward inconsistency from ZFC+BTEE+¬Cofinal Axiom is paved with a sequence of theories of increasing large cardinal strength. Indeed, the extensions of the theory ZFC +“ j is a nontrivial elementary embedding” form a hierarchy of axioms, ranging in strength from Con(ZFC) to the existence of a cardinal that is super- n -huge for every n , to inconsistency. This hierarchy is parallel to the usual hierarchy of large cardinal axioms, and can be used in the same way. We also isolate several intermediate-strength axioms which, when added to ZFC+BTEE, produce theories having strengths in the vicinity of a measurable cardinal of high Mitchell order, a strong cardinal, ω Woodin cardinals, and n -huge cardinals. We also determine precisely which combinations of axioms, of the form ZFC + BTEE + Σ m - Separation j + Σ n - Replacement j result in inconsistency.

THE CLUB GUESSING IDEAL: COMMENTARY ON A THEOREM OF GITIK AND SHELAH

It is shown in this paper that it is consistent (relative to almost huge cardinals) for various club guessing ideals to be saturated.

Gap Forcing: Generalizing the Lévy-Solovay Theorem

AbstractThe Lévy-Solovay Theorem[8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.

Flipping properties and huge cardinals

Flipping properties and huge cardinals

What’s new in logic?

One of the first delights of logic the average mathematics student encounters is the uncountability of the set of real numbers. There are more real numbers than natural numbers, more sets of real numbers than reals, more sets of sets of r e a l s . . , and so on. The student may or may not wonder how many different cardinalities there are and how large they can become. It is natural for him to give up wondering after a while: Either he decides the number of such cardinalities is infinite and, forgetting momentarily that he has just learned there to be different infinities, the student considers the matter settled, or he realises there to be a dazzling array of infinities and that he has no handle on the problem. Then again, he could go on to become a specialist and create a few handles. The genus of set theory, i.e., the number of the handles I spoke of, is itself almost a large cardinal. There are: weak and strong inaccessibles; Mahlo cardinals of various ranks; measurables; weakly and strongly compact cardinals; supercompact cardinals; indescribable and extendible cardinals; huge cardinals; and now two more due to Woodin. Despite the wide variety of such cardinal numbers and the variety of their names, experienced set theorists can linearly order them according to size and even announce which ones definitely exist, which ones probably exist, which ones might exist . . . . . As the cardinal gets bigger and bigger, the hypothesis that it exists (called a large cardinal hypothesis or an axiom of infinity) gets progressively more quest ionable as an addit ional axiom of set theory.

The Ultrafilter Characterization of Huge Cardinals

A huge cardinal can be characterized using ultrafilters. After an argument is made for a particular ultrafilter characterization, it is used to prove the existence of a measurable cardinal above the huge cardinal, and an ultrafilter over the set of all subsets of this measurable cardinal of size smaller than the huge cardinal. Finally, this last ultrafilter is disassembled intact by a process which often produces a different ultrafilter from the one started out with. An important point of this paper is given the existence of the particular ultrafilter characterization of a huge cardinal mentioned above these results are proved in Zermelo-Fraenkel set theory without the axiom of choice.

The ultrafilter characterization of huge cardinals

A huge cardinal can be characterized using ultrafilters. After an argument is made for a particular ultrafilter characterization, it is used to prove the existence of a measurable cardinal above the huge cardinal, and an ultrafilter over the set of all subsets of this measurable cardinal of size smaller than the huge cardinal. Finally, this last ultrafilter is disassembled intact by a process which often produces a different ultrafilter from the one started out with. An important point of this paper is given the existence of the particular ultrafilter characterization of a huge cardinal mentioned above these results are proved in Zermelo-Fraenkel set theory without the axiom of choice.

On the existence of nonregular ultrafilters and the cardinality of ultrapowers

Assuming the consistency of huge cardinals, we prove that ω 3 {\omega _3} can carry an ultrafilter D such that ω 1 ω 3 / D {\omega _1}^{{\omega _3}}/D has cardinality ω 3 {\omega _3} . (Hence D is not ( ω 3 , ω 1 ) ({\omega _3},\,{\omega _1}) regular.) Similarly ω 2 {\omega _2} can carry an ultrafilter D such that ω ω 2 / D {\omega ^{{\omega _2}}}/D has cardinality ω 2 {\omega _2} . (Hence D is not ( ω 2 , ω ) ({\omega _2},\,\omega ) regular.)