On P-points over a measurable cardinal

1981 ◽  
Vol 46 (1) ◽  
pp. 59-66
Author(s):  
A. Kanamori
Keyword(s):  
Measurable Cardinal ◽  
Focus Of Attention ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Inner Model ◽  
Measurable Cardinals ◽  
The Universe

This paper continues the study of κ-ultrafilters over a measurable cardinal κ, following the sequence of papers Ketonen [2], Kanamori [1] and Menas [4]. Much of the concern will be with p-point κ-ultrafilters, which have become a focus of attention because they epitomize situations of further complexity beyond the better understood cases, normal and product κ-ultrafilters.For any κ-ultrafilter D, let iD: V → MD ≃ Vκ/D be the elementary embedding of the universe into the transitization of the ultrapower by D. Situations of U < RKD will be exhibited when iU(κ) < iD(κ), and when iU(κ) = iD(κ). The main result will then be that if the latter case obtains, then there is an inner model with two measurable cardinals. (As will be pointed out, this formulation is due to Kunen, and improves on an earlier version of the author.) Incidentally, a similar conclusion will also follow from the assertion that there is an ascending Rudin-Keisler chain of κ-ultrafilters of length ω + 1. The interest in these results lies in the derivability of a substantial large cardinal assertion from plausible hypotheses on κ-ultrafilters.

On the ordering of certain large cardinals

1979 ◽  
Vol 44 (4) ◽  
pp. 563-565
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Inaccessible Cardinal ◽  
Elementary Embedding ◽  
Strongly Compact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
The Universe ◽  
Weakly Compact Cardinal

It is well known that the first strongly inaccessible cardinal is strictly less than the first weakly compact cardinal which in turn is strictly less than the first Ramsey cardinal, etc. However, once one passes the first measurable cardinal the inequalities are no longer strict. Magidor [3] has shown that the first strongly compact cardinal may be equal to the first measurable cardinal or equal to the first super-compact cardinal (the first supercompact cardinal is strictly larger than the first measurable cardinal). In this note we will indicate how Magidor's methods can be used to show that it is undecidable whether one cardinal (the first strongly compact) is greater than or less than another large cardinal (the first huge cardinal). We assume that the reader is familiar with the ultrapower construction of Scott, as presented in Drake [1] or Kanamori, Reinhardt and Solovay [2].Definition. A cardinal κ is huge (or 1-huge) if there is an elementary embedding j of the universe V into a transitive class M such that M contains the ordinals, is closed under j(κ) sequences, j(κ) > κ and j ↾ Rκ = id. Let κ denote the first huge cardinal, and let λ = j(κ).One can see from easy reflection arguments that κ and λ are inaccessible in V and, in fact, that κ is measurable in V.

Proper Forcing and Remarkable Cardinals

2000 ◽  
Vol 6 (2) ◽  
pp. 176-184 ◽  
Author(s):  
Ralf-Dieter Schindler
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Exact Formulation ◽  
Axiom Of Determinacy ◽  
Slow Pace ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Proper Forcing ◽  
Remarkable Cardinals ◽  
The Universe

The present paper investigates the power of proper forcings to change the shape of the universe, in a certain well-defined respect. It turns out that the ranking among large cardinals can be used as a measure for that power. However, in order to establish the final result I had to isolate a new large cardinal concept, which I dubbed “remarkability.” Let us approach the exact formulation of the problem—and of its solution—at a slow pace.Breathtaking developments in the mid 1980s found one of its culminations in the theorem, due to Martin, Steel, and Woodin, that the existence of infinitely many Woodin cardinals with a measurable cardinal above them all implies that AD, the axiom of determinacy, holds in the least inner model containing all the reals, L(ℝ) (cf. [6[, p. 91). One of the nice things about AD is that the theory ZF + AD + V = L(ℝ) appears as a choiceless “completion” of ZF in that any interesting question (in particular, about sets of reals) seems to find an at least attractive answer in that theory (cf., for example, [5] Chap. 6). (Compare with ZF + V = L!) Beyond that, AD is very canonical as may be illustrated as follows.Let us say that L(ℝ) is absolute for set-sized forcings if for all posets P ∈ V, for all formulae ϕ, and for all ∈ ℝ do we have thatwhere is a name for the set of reals in the extension.

On elementary embeddings from an inner model to the universe

2001 ◽  
Vol 66 (3) ◽  
pp. 1090-1116 ◽  
Author(s):  
J. Vickers ◽  
P. D. Welch
Keyword(s):  
Measurable Cardinal ◽  
Negative Answer ◽  
Core Model ◽  
Elementary Embedding ◽  
Proper Class ◽  
The Core ◽  
Inner Model ◽  
Elementary Embeddings ◽  
The Universe ◽  
Closure Assumption

AbstractWe consider the following question of Kunen:Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M → V)imply Con(ZFC + ∃ a measurable cardinal)?We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of “there exists a proper class of almost Ramsey cardinals”. Conversely, if On is Ramsey, then such a j. M are definable.We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.

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.

Elementary embeddings and infinitary combinatorics

1971 ◽  
Vol 36 (3) ◽  
pp. 407-413 ◽  
Author(s):  
Kenneth Kunen
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Elementary Embeddings ◽  
The Universe

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings,j, from the universe,V, into some transitive submodel,M. See Reinhardt–Solovay [7] for more details. Ifjis not the identity, andκis the first ordinal moved byj, thenκis a measurable cardinal. Conversely, Scott [8] showed that wheneverκis measurable, there is suchjandM. If we had assumed, in addition, that, thenκwould be theκth measurable cardinal; in general, the wider we assumeMto be, the largerκmust be.

Nonabsoluteness of elementary embeddings

1989 ◽  
Vol 54 (3) ◽  
pp. 774-778
Author(s):  
Friedrich Wehrung
Keyword(s):  
Measurable Cardinal ◽  
Elementary Embedding ◽  
Minimal Support ◽  
Elementary Embeddings ◽  
Measurable Cardinals

Ifκis a measurable cardinal, let us say that a measure onκis aκ-complete nonprincipal ultrafilter onκ. IfUis a measure onκ, letjUbe the canonical elementary embedding ofVinto its Ultrapower UltU(V). Ifxis a set, say thatUmovesxwhenjU(x)≠x; say thatκmovesxwhen some measure onκmovesx. Recall Kunen's lemma (see [K]): “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof (see [K]) and Fleissner's proof (see [KM, III, §10]) are essentially nonconstructive.The following proposition can be proved by using elementary facts about iterated ultrapowers.Proposition.Let ‹Un: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n= Um(m, n ∈ ω). Then, for each θ inUltU(V),if E is the (minimal) support of θ inUltU(V),then, for all m ∈ ω, Ummoves θ iff E ∩ [ωm, ω(m + 1))≠ ∅.

Weak partition relations and measurability

1986 ◽  
Vol 51 (1) ◽  
pp. 33-38
Author(s):  
Mitchell Spector
Keyword(s):  
Measurable Cardinal ◽  
Order Type ◽  
Large Cardinal ◽  
Partition Relation ◽  
Consistency Strength ◽  
Partition Relations ◽  
Inner Model ◽  
Increasing Function ◽  
Natural Way

The concept of "partition relation" has proven to be extremely important in the development of the theory of large cardinals. This is due in good part to the fact that the ordinal numbers which appear as parameters in partition relations provide a natural way to define a detailed hierarchy of the corresponding large cardinal axioms. In particular, the study of cardinals satisfying Ramsey-Erdös-style partition relations has yielded a great number of very interesting large cardinal axioms which lie in strength strictly between inaccessibility and measurability. It is the purpose of this paper to show that this phenomenon does not occur if we use infinite exponent partition relations; no such partition relation has consistency strength strictly between inaccessibility and measurability. We also give a complete determination of which infinite exponent partition relations hold, assuming that there is no inner model of set theory with a measurable cardinal.Our notation is standard. If F is a function and x is a set, then F″x denotes the range of F on x. If X is a set of ordinals and α is an ordinal, then [X]α is the collection of all subsets of X of order type α. We identify a member of [X]α with a strictly increasing function from α to X. If p ∈ [X]α and q ∈ [α]β, then the composition of p with q, which we denote pq, is a member of [X]β.

Strongly compact cardinals, elementary embeddings and fixed points

1984 ◽  
Vol 49 (3) ◽  
pp. 808-812
Author(s):  
Yoshihiro Abe
Keyword(s):  
Fixed Points ◽  
Elementary Embedding ◽  
Inner Model ◽  
Power Set ◽  
Elementary Embeddings ◽  
Basic Facts ◽  
The Universe ◽  
Normal Ultrafilter ◽  
Strongly Compact Cardinals

J. Barbanel [1] characterized the class of cardinals fixed by an elementary embedding induced by a normal ultrafilter on Pκλ assuming that κ is supercompact. In this paper we shall prove the same results from the weaker hypothesis that κ is strongly compact and the ultrafilter is fine.We work in ZFC throughout. Our set-theoretic notation is quite standard. In particular, if X is a set, ∣X∣ denotes the cardinality of X and P(X) denotes the power set of X. Greek letters will denote ordinals. In particular γ, κ, η and γ will denote cardinals. If κ and λ are cardinals, then λ<κ is defined to be supγ<κγγ. Cardinal exponentiation is always associated from the top. Thus, for example, 2λ<κ means 2(λ<κ). V denotes the universe of all sets. If M is an inner model of ZFC, ∣X∣M and P(X)M denote the cardinality of X in M and the power set of X in M respectively.We review the basic facts on fine ultrafilters and the corresponding elementary embeddings. (For detail, see [2].)Definition. Assume κ and λ are cardinals with κ ≤ λ. Then, Pκλ = {X ⊂ λ∣∣X∣ < κ}.It is important to note that ∣Pκλ∣ = λ< κ.

Complexity of κ-ultrafilters and inner models with measurable cardinals

1984 ◽  
Vol 49 (3) ◽  
pp. 833-841 ◽  
Author(s):  
Claude Sureson
Keyword(s):  
Regular Cardinal ◽  
Measurable Cardinal ◽  
Order Type ◽  
Inaccessible Cardinal ◽  
Inner Models ◽  
Inner Model ◽  
Measurable Cardinals

The purpose of this paper is to establish a connection between the complexity of κ-ultrafilters over a measurable cardinal κ, and the existence of ascending Rudin-Keisler chains of κ-ultrafilters and of inner models with several measurable cardinals.If V is a model of ZFC + “There exists a measurable cardinal κ”, then V satisfies “There exists a normal κ-ultrafilter”, that is to say a “simple” κ-ultrafilter. The only known examples of “complex” κ-ultrafilters have been constructed by Kanamori [2], Ketonen [4] and Kunen (cf. [2]) with stronger hypotheses than measurability: compactness or supercompactness. Using the notions of skies and constellations defined by Kanamori [2] for the measurable case, and which witness the complexity of a κ-ultrafilter, we shall show the necessity of such assumptions, namely:Theorem 1. If λ < κ is a strongly inaccessible cardinal, the existence of a κ-ultrafilter with more than λ constellations implies that there is an inner model with two measurable cardinals if λ = ω and λ + 1 measurable cardinals otherwise.Theorem 2. Let θ < κ be an arbitrary ordinal. If there is a κ-ultrafilter such that the order-type of its skies is greater than ωθ, then there exists an inner model with θ + 1 measurable cardinals.And as a corollary, we obtain:Theorem 3. Let μ < κ be a regular cardinal. If there exists a κ-ultrafilter containing the closed-unbounded subsets of κ and {α < κ: cf(α) = μ}, then there is an inner model with two measurable cardinals if μ = ω, and μ + 1 measurable cardinals otherwise.

The measure quantifier

1979 ◽  
Vol 44 (1) ◽  
pp. 103-108 ◽  
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Measurable Cardinal ◽  
Order Logic ◽  
First Order Logic ◽  
Order Structure ◽  
Proper Class ◽  
First Order ◽  
Recursive Enumerability ◽  
Almost All ◽  
Measurable Cardinals ◽  
The Universe

Originally generalized quantifiers were introduced to specify that a given formula was true for “many x's” e.g. ⊨ Qxφ(x) iff card{x ∈ ∣∣ ∣ ⊨ φ[x]} ≥ ℵ0, ℵ1, or some fixed cardinal κ. In this paper we formalize the notion that φ{x) is true “for almost all x”. This is accomplished by referring to structures = (′, μ) where ′ is a first-order structure and μ is a measure of a suitable type on the universe of ′. We will prove that the language Lμ obtained from first-order logic by adjoining a quantifier Qμ, which ranges over the measure μ, is fully compact if we assume the existence of a proper class of measurable cardinals. As a corollary to the compactness theorem we obtain the recursive enumerability of the validities of Lμ. Finally, the Magidor-Malitz quantifiers Qkn (n ∈ ω) will be added to Lμ together with analogous quantifiers Qμm (m ∈ ω) to form Lκμ<ω,<ω, which is compact for sets of sentences of cardinality < κ, where κ is a measurable cardinal > ℵ0.An alternate approach to formalizing “for almost all” has been recently developed by Barwise, Kaufmann and Makkai [1] who follow a suggestion of Shelah [5].

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