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.

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.

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.

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.

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.

Some results concerning strongly compact cardinals

1985 ◽  
Vol 50 (4) ◽  
pp. 874-880
Author(s):  
Yoshihiro Abe
Keyword(s):  
Order Type ◽  
Supercompact Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
Identity Function ◽  
Power Set ◽  
Elementary Embeddings ◽  
Strongly Compact Cardinals

This paper consists of two parts. In §1 we mention the first strongly compact cardinal. Magidor proved in [6] that it can be the first measurable and it can be also the first supercompact. In [2], Apter proved that Con(ZFC + there is a supercompact limit of supercompact cardinals) implies Con(ZFC + the first strongly compact cardinal κ is ϕ(κ)-supercompact + no α < κ is ϕ(α)-supercompact) for a formula ϕ which satisfies certain conditions.We shall get almost the same conclusion as Apter's theorem assuming only one supercompact cardinal. Our notion of forcing is the same as in [2] and a trick makes it possible.In §2 we study a kind of fine ultrafilter on Pκλ investigated by Menas in [7], where κ is a measurable limit of strongly compact cardinals. He showed that such an ultrafilter is not normal in some case (Theorems 2.21 and 2.22 in [7]). We shall show that it is not normal in any case (even if κ is supercompact). We also prove that it is weakly normal in some case.We work in ZFC and much of our notation is standard. But we mention the following: the letters α,β,γ… denote ordinals, whereas κ,λ,μ,… are reserved for cardinals. R(α) is the collection of sets rank <α. φM denotes the realization of a formula φ to a class M. Except when it is necessary, we drop “M”. For example, M ⊩ “κ is φ(κ)-supercompact” means “κ is φM(κ)-supercompact in M”. If x is a set, |x| is its cardinality, Px is its power set, and . If also x ⊆ OR, denotes its order type in the natural ordering. The identity function with the domain appropriate to the context is denoted by id. For the notation concerning ultrapowers and elementary embeddings, see [11]. When we talk about forcing, “⊩” will mean “weakly forces” and “p < q” means “p is stronger than q”.

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.

Supercompact cardinals, elementary embeddings and fixed points

1982 ◽  
Vol 47 (1) ◽  
pp. 84-88
Author(s):  
Julius B. Barbanel
Keyword(s):  
Fixed Points ◽  
Supercompact Cardinals ◽  
Inner Models ◽  
Elementary Embeddings ◽  
Universe Of Sets ◽  
The Universe

Supercompactness is usually defined in terms of the existence of certain ultrafilters. By the well-known procedure of taking ultrapowers of V (the universe of sets) and transitive collapses, one obtains transitive inner models of V and corresponding elementary embeddings from V into these inner models. These embeddings have been studied extensively (see, e.g. [3] or [4]). We investigate the action of these embeddings on cardinals. In particular, we establish a characterization, based upon cofinality, of which cardinals are fixed by these embeddings.

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.

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