Thomas J. Jech. Two remarks on elementary embeddings of the universe. Pacific journal of mathematics, vol. 39 (1971), pp. 395–400.

Richard Mansfield

doi:10.2307/2272096

Elementary embeddings and infinitary combinatorics

Journal of Symbolic Logic ◽

10.2307/2269948 ◽

1971 ◽

Vol 36 (3) ◽

pp. 407-413 ◽

Cited By ~ 70

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.

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Strongly compact cardinals, elementary embeddings and fixed points

Journal of Symbolic Logic ◽

10.2307/2274133 ◽

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.

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Indescribable cardinals and elementary embeddings

Journal of Symbolic Logic ◽

10.2307/2274692 ◽

1991 ◽

Vol 56 (2) ◽

pp. 439-457 ◽

Cited By ~ 13

Author(s):

Kai Hauser

Keyword(s):

Set Theory ◽

Predicate Symbol ◽

Large Cardinal ◽

Unary Predicate ◽

Finite Sequences ◽

Elementary Embeddings ◽

Reflection Principles ◽

The One ◽

Image Position ◽

The Universe

Indescribability is closely related to the reflection principles of Zermelo-Fränkel set theory. In this axiomatic setting the universe of all sets stratifies into a natural cumulative hierarchy (Vα: α ϵ On) such that any formula of the language for set theory that holds in the universe already holds in the restricted universe of all sets obtained by some stage.The axioms of ZF prove the existence of many ordinals α such that this reflection scheme holds in the world Vα. Hanf and Scott noticed that one arrives at a large cardinal notion if the reflecting formulas are allowed to contain second order free variables to which one assigns subsets of Vα. For a given collection Ω of formulas in the ϵ language of set theory with higher type variables and a unary predicate symbol they define an ordinal α to be Ω indescribable if for all sentences Φ in Ω and A ⊆ VαSince a sufficient coding apparatus is available, this definition is (for the classes of formulas that we are going to consider) equivalent to the one that one obtains by allowing finite sequences of relations over Vα, some of which are possibly k-ary. We will be interested mainly in certain standardized classes of formulas: Let (, respectively) denote the class of all formulas in the language introduced above whose prenex normal form has n alternating blocks of quantifiers of type m (i.e. (m + 1)th order) starting with ∃ (∀, respectively) and no quantifiers of type greater than m. In Hanf and Scott [1961] it is shown that in ZFC, indescribability is equivalent to inaccessibility and indescribability coincides with weak compactness.

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On elementary embeddings from an inner model to the universe

Journal of Symbolic Logic ◽

10.2307/2695094 ◽

2001 ◽

Vol 66 (3) ◽

pp. 1090-1116 ◽

Cited By ~ 4

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.

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Supercompact cardinals, elementary embeddings and fixed points

Journal of Symbolic Logic ◽

10.2307/2273383 ◽

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.

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