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.

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.

Indescribable cardinals and elementary embeddings

1991 ◽  
Vol 56 (2) ◽  
pp. 439-457 ◽  
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.

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.

Perfect trees and elementary embeddings

2008 ◽  
Vol 73 (3) ◽  
pp. 906-918 ◽  
Author(s):  
Sy-David Friedman ◽  
Katherine Thompson
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Generic Extension ◽  
Elementary Embeddings ◽  
Preliminary Version ◽  
Cohen Forcing ◽  
Difficult Cases ◽  
Generic Extensions ◽  
Reverse Easton Iteration

AbstractAn important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j* : M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), the generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d(κ) is less than 2κ for inaccessible κ, where d(κ) is the dominating number at κ) is internally consistent, given the existence of 0#.

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.

scholarly journals Large cardinals and definable well-orders on the universe

2009 ◽  
Vol 74 (2) ◽  
pp. 641-654 ◽  
Author(s):  
Andrew D. Brooke-Taylor
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Proper Class ◽  
The Universe

AbstractWe use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle , at a proper class of cardinals κ. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.

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.

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