Large cardinal structures below ℵω

1986 ◽  
Vol 51 (3) ◽  
pp. 591-603 ◽  
Author(s):  
Arthur W. Apter ◽  
James M. Henle
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Axiom Of Choice ◽  
Weak Consistency ◽  
Consistency Strength ◽  
Supercompact Cardinals ◽  
Axiom Of Determinacy ◽  
Normal Measure ◽  
The Axiom Of Choice ◽  
Measurable Cardinals

The theory of large cardinals in the absence of the axiom of choice (AC) has been examined extensively by set theorists. A particular motivation has been the study of large cardinals and their interrelationships with the axiom of determinacy (AD). Many important and beautiful theorems have been proven in this area, especially by Woodin, who has shown how to obtain, from hypermeasurability, models for the theories “ZF + DC + ∀α < ℵ1(ℵ1 → (ℵ1)α)” and . Thus, consequences of AD whose consistency strength appeared to be beyond that of the more standard large cardinal hypotheses were shown to have suprisingly weak consistency strength.In this paper, we continue the study of large cardinals in the absence of AC and their interrelationships with AD by examining what large cardinal structures are possible on cardinals below ℵω in the absence of AC. Specifically, we prove the following theorems.Theorem 1. Con(ZFC + κ1 < κ2are supercompact cardinals) ⇒ Con(ZF + DC + The club filter on ℵ1is a normal measure + ℵ1and ℵ2are supercompact cardinals).Theorem 2. Con(ZF + AD) ⇒ Con(ZF + ℵ1, ℵ2and ℵ3are measurable cardinals which carry normal measures + μωis not a measure on any of these cardinals).

AD and the supercompactness of ℵ1

1981 ◽  
Vol 46 (4) ◽  
pp. 822-842 ◽  
Author(s):  
Howard Becker
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Axiom Of Choice ◽  
Axiom Of Determinacy ◽  
The Subject ◽  
The Axiom Of Choice ◽  
Dependent Choice

Since the discovery of forcing in the early sixties, it has been clear that many natural and interesting mathematical questions are not decidable from the classical axioms of set theory, ZFC. Therefore some mathematicians have been studying the consequences of stronger set theoretic assumptions. Two new types of axioms that have been the subject of much research are large cardinal axioms and axioms asserting the determinacy of definable games. The two appear at first glance to be unrelated; one of the most surprising discoveries of recent research is that this is not the case.In this paper we will be assuming the axiom of determinacy (AD) plus the axiom of dependent choice (DC). AD is false, since it contradicts the axiom of choice. However every set in L[R] is ordinal definable from a real. Our axiom that definable games are determined implies that every game in L[R] is determined (in V), and since a strategy is a real, it is determined in L[R]. That is, L[R] ⊨ AD. The axiom of choice implies L[R] ⊨ DC. So by embedding ourselves in L[R], we can assume AD + DC and begin proving theorems. These theorems true in L[R] imply corresponding theorems in V, by e.g. changing “every set” to “every set in L[R]”. For more information on AD as an axiom, and on some of the points touched on here, the reader should consult [14], particularly §§7D and 8I. In this paper L[R] will no longer even be mentioned. We just assume AD for the rest of the paper.

An AD-like model

1985 ◽  
Vol 50 (2) ◽  
pp. 531-543 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinals ◽  
Strong Sense ◽  
Supercompact Cardinals ◽  
Singular Cardinals ◽  
Partial Orderings ◽  
Measurable Cardinals ◽  
Jonsson Cardinals

A very fruitful line of research in recent years has been the application of techniques in large cardinals and forcing to the production of models in which certain consequences of the axiom of determinateness (AD) are true or in which certain “AD-like” consequences are true. Numerous results have been published on this subject, among them the papers of Bull and Kleinberg [4], Bull [3], Woodin [15], Mitchell [11], and [1], [2].Another such model will be constructed in this paper. Specifically, the following theorem is proven.Theorem 1. Con(ZFC + There are cardinals κ < δ < λ so that κ is a supercompact limit of supercompact cardinals, λ is a measurable cardinal, and δ is λ supercompact) ⇒ Con(ZF + ℵ1 and ℵ2 are Ramsey cardinals + The ℵn for 3 ≤ n ≤ ω are singular cardinals of cofinality ω each of which carries a Rowbottom filter + ℵω + 1 is a Ramsey cardinal + ℵω + 2 is a measurable cardinal).It is well known that under AD + DC, ℵ2 and ℵ2 are measurable cardinals, the ℵn for 3 ≤ n < ω are singular Jonsson cardinals of cofinality ℵ2, ℵω is a Rowbottom cardinal, and ℵω + 1 and ℵω + 2 are measurable cardinals.The proof of the above theorem will use the existence of normal ultrafilters which satisfy a certain property (*) (to be defined later) and an automorphism argument which draws upon the techniques developed in [9], [2], and [4] but which shows in addition that certain supercompact Prikry partial orderings are in a strong sense “homogeneous”. Before beginning the proof of the theorem, however, we briefly mention some preliminaries.

Ramsey-like cardinals

2011 ◽  
Vol 76 (2) ◽  
pp. 519-540 ◽  
Author(s):  
Victoria Gitman
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Large Fragment ◽  
Weakly Compact ◽  
Elementary Embeddings ◽  
Measurable Cardinals

AbstractOne of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V = L.

Believing the axioms. II

1988 ◽  
Vol 53 (3) ◽  
pp. 736-764 ◽  
Author(s):  
Penelope Maddy
Keyword(s):  
Continuum Hypothesis ◽  
Winning Strategy ◽  
Regularity Property ◽  
Large Cardinals ◽  
Natural Numbers ◽  
Regularity Properties ◽  
Finite Sequences ◽  
The Axiom Of Choice ◽  
Measurable Cardinals ◽  
Descriptive Set

This is a continuation of Believing the axioms. I, in which nondemonstrative arguments for and against the axioms of ZFC, the continuum hypothesis, small large cardinals and measurable cardinals were discussed. I turn now to determinacy hypotheses and large large cardinals, and conclude with some philosophical remarks.Determinacy is a property of sets of reals. If A is such a set, we imagine an infinite game G(A) between two players I and II. The players take turns choosing natural numbers. In the end, they have generated a real number r (actually a member of the Baire space ωω). If r is in A, I wins; otherwise, II wins. The set A is said to be determined if one player or the other has a winning strategy (that is, a function from finite sequences of natural numbers to natural numbers that guarantees the player a win if he uses it to decide his moves).Determinacy is a “regularity” property (see Martin [1977, p. 807]), a property of well-behaved sets, that implies the more familiar regularity properties like Lebesgue measurability, the Baire property (see Mycielski [1964] and [1966], and Mycielski and Swierczkowski [1964]), and the perfect subset property (Davis [1964]). Infinitary games were first considered by the Polish descriptive set theorists Mazur and Banach in the mid-30s; Gale and Stewart [1953] introduced them into the literature, proving that open sets are determined and that the axiom of choice can be used to construct an undetermined set.

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.

Some consequences of an infinite-exponent partition relation

1977 ◽  
Vol 42 (4) ◽  
pp. 523-526 ◽  
Author(s):  
J. M. Henle
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Large Cardinals ◽  
Axiom Of Choice ◽  
Partition Relation ◽  
Basic Theorem ◽  
Axiom Of Determinacy ◽  
Ramsey’S Theorem ◽  
Partition Relations ◽  
Dependent Choice

Beginning with Ramsey's theorem of 1930, combinatorists have been intrigued with the notion of large cardinals satisfying partition relations. Years of research have established the smaller ones, weakly ineffable, Erdös, Jonsson, Rowbottom and Ramsey cardinals to be among the most interesting and important large cardinals in set theory. Recently, cardinals satisfying more powerful infinite-exponent partition relations have been examined with growing interest. This is due not only to their inherent qualities and the fact that they imply the existence of other large cardinals (Kleinberg [2], [3]), but also to the fact that the Axiom of Determinacy (AD) implies the existence of great numbers of such cardinals (Martin [5]).That these properties are more often than not inconsistent with the full Axiom of Choice (Kleinberg [4]) somewhat increases their charm, for the theorems concerning them tend to be a little odd, and their proofs, circumforaneous. The properties are, as far as anyone knows, however, consistent with Dependent Choice (DC).Our basic theorem will be the following: If k > ω and k satisfies k→(k)k then the least cardinal δ such that has a δ-additive, uniform ultrafilter. In addition, if ACω is assumed, we will show that δ is greater than ω, and hence a measurable cardinal. This result will be strengthened somewhat when we prove that for any k, δ, if then .

0# and some forcing principles

1986 ◽  
Vol 51 (1) ◽  
pp. 39-46 ◽  
Author(s):  
Matthew Foreman ◽  
Menachem Magidor ◽  
Saharon Shelah
Keyword(s):  
Real Number ◽  
Large Cardinals ◽  
Singular Limit ◽  
Partial Ordering ◽  
The Other ◽  
List Type ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Partial Orderings ◽  
The Universe

It has been considered desirable by many set theorists to find maximality properties which state that the universe has in some sense “many sets”. The properties isolated thus far have tended to be consistent with each other (as far as we know). For example it is a widely held view that the existence of a supercompact cardinal is consistent with the axiom of determinacy holding in L(R). This consistency has been held to be evidence for the truth of these properties. It is with this in mind that the first author suggested the following:Maximality Principle If P is a partial ordering and G ⊆ P is a V-generic ultrafilter then eithera) there is a real number r ∈ V [G] with r ∉ V, orb) there is an ordinal α such that α is a cardinal in V but not in V[G].This maximality principle applied to garden variety partial orderings has startling results for the structure of V.For example, if for some , then P = 〈{p: p ⊆ κ, ∣p∣ < κ}, ⊆〉 neither adds a real nor collapses a cardinal. Thus from the maximality principle we can deduce that the G. C. H. fails everywhere and there are no inaccessible cardinals. (Hence this principle contradicts large cardinals.) Similarly one can show that there are no Suslin trees on any cardinal κ. These consequences help justify the title “maximality principle”.Since the maximality principle implies that the G. C. H. fails at strong singular limit cardinals it has consistency strength at least that of “many large cardinals”. (See [M].) On the other hand it is not known to be consistent, relative to any assumptions.

scholarly journals Combined Maximality Principles up to large cardinals

2009 ◽  
Vol 74 (3) ◽  
pp. 1015-1046 ◽  
Author(s):  
Gunter Fuchs
Keyword(s):  
Regular Cardinal ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Consistency Strength ◽  
Regular Cardinals ◽  
Maximality Principles

AbstractThe motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of ∞), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.

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.

Proper forcing and L(ℝ)

2001 ◽  
Vol 66 (2) ◽  
pp. 801-810 ◽  
Author(s):  
Itay Neeman ◽  
Jindřich Zapletal
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Consistency Strength ◽  
Woodin Cardinals ◽  
Proper Forcing

AbstractWe present two ways in which the model L(ℝ) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing: we show further that a set of ordinals in V cannot be added to L(ℝ) by small forcing. The large cardinal needed corresponds to the consistency strength of ADL(ℝ): roughly ω Woodin cardinals.

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