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 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.

scholarly journals THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

2019 ◽  
Vol 85 (1) ◽  
pp. 338-366 ◽  
Author(s):  
JUAN P. AGUILERA ◽  
SANDRA MÜLLER
Keyword(s):  
Set Theory ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Image Position ◽  
Woodin Cardinal

AbstractWe determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

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.

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

2017 ◽  
Vol 82 (3) ◽  
pp. 1106-1131 ◽  
Author(s):  
PHILIPP LÜCKE ◽  
RALF SCHINDLER ◽  
PHILIPP SCHLICHT
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Woodin Cardinals ◽  
Perfect Set ◽  
Nonstationary Ideal ◽  
Definable Sets ◽  
Image Position ◽  
Stationary Subsets ◽  
Woodin Cardinal ◽  
Generic Ultrapowers

AbstractWe study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

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.

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.

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.

On the ultrafilters and ultrapowers of strong partition cardinals

1984 ◽  
Vol 49 (4) ◽  
pp. 1268-1272
Author(s):  
J.M. Henle ◽  
E.M. Kleinberg ◽  
R.J. Watro
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Partition Property ◽  
Axiom Of Determinacy ◽  
Almost Everywhere ◽  
Normal Measure ◽  
Constructible Sets ◽  
Inner Model ◽  
The Axiom Of Choice ◽  
Strong Partition Cardinals

A strong partition cardinal is an uncountable well-ordered cardinal κ such that every partition of [κ]κ (the size κ subsets of κ) into less than κ many pieces has a homogeneous set of size κ. The existence of such cardinals is inconsistent with the axiom of choice, and our work concerning them is carried out in ZF set theory with just dependent choice (DC). The consistency of strong partition cardinals with this weaker theory remains an open question. The axiom of determinacy (AD) implies that a large number of cardinals including ℵ1 have the strong partition property. The hypothesis that AD holds in the inner model of constructible sets built over the real numbers as urelements has important consequences for descriptive set theory, and results concerning strong partition cardinals are often applied in this context. Kechris [4] and Kechris et al. [5] contain further information concerning the relationship between AD and strong partition cardinals.We assume familiarity with the basic results on strong partition cardinals as developed in Kleinberg [6], [7], [8] and Henle [2]. Recall that a strong partition cardinal κ is measurable; in fact every stationary subset of κ is measure one under some normal measure on κ. If μ is a countably additive ultrafilter extending the closed unbounded filter on κ, then the length of the ultrapower [κ]κ under the less than almost everywhere μ ordering is again a measurable cardinal. In §1 below we establish a polarized partition property on these measurable cardinals.

Proper forcing and remarkable cardinals II

2001 ◽  
Vol 66 (3) ◽  
pp. 1481-1492 ◽  
Author(s):  
Ralf-Dieter Schindler
Keyword(s):  
Large Cardinal ◽  
Current Paper ◽  
Proper Forcing ◽  
Remarkable Cardinals

AbstractThe current paper proves the results announced in [5].We isolate a new large cardinal concept, “remarkability.” Consistencywise, remarkable cardinals are between ineffable and ω-Erdös cardinals. They are characterized by the existence of “0#-like” embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(ℝ) absoluteness for proper forcings. In particular, said absoluteness does not imply determinacy.

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.

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