AXIOM I0 AND HIGHER DEGREE THEORY

2015 ◽  
Vol 80 (3) ◽  
pp. 970-1021 ◽  
Author(s):  
XIANGHUI SHI
Keyword(s):  
Critical Point ◽  
Degree Theory ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Generic Absoluteness ◽  
Perfect Set ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Measurable Cardinals

AbstractIn this paper, we analyze structures of Zermelo degrees via a list of four degree theoretic questions (see §2) in various fine structure extender models, or under large cardinal assumptions. In particular we give a detailed analysis of the structures of Zermelo degrees in the Mitchell model for ω many measurable cardinals. It turns out that there is a profound correlation between the complexity of the degree structures at countable cofinality singular cardinals and the large cardinal strength of the relevant cardinals. The analysis applies to general degree notions, Zermelo degree is merely the author’s choice for illustrating the idea.I0(λ) is the assertion that there is an elementary embedding j : L(Vλ+1) → L(Vλ+1) with critical point < λ. We show that under I0(λ), the structure of Zermelo degrees at λ is very complicated: it has incomparable degrees, is not dense, satisfies Posner–Robinson theorem etc. In addition, we show that I0 together with a mild condition on the critical point of the embedding implies that the degree determinacy for Zermelo degrees at λ is false in L(Vλ+1). The key tool in this paper is a generic absoluteness theorem in the theory of I0, from which we obtain an analogue of Perfect Set Theorem for “projective” subsets of Vλ+1, and the Posner–Robinson follows as a corollary. Perfect Set Theorem and Posner–Robinson provide evidences supporting the analogy between $$AD$$ over L(ℝ) and I0 over L(Vλ+1), while the failure of degree determinacy is one for disanalogy. Furthermore, we conjecture that the failure of degree determinacy for Zermelo degrees at any uncountable cardinal is a theorem of $$ZFC$$.

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.

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.

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.

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 a condition for Cohen extensions which preserve precipitous ideals

1981 ◽  
Vol 46 (2) ◽  
pp. 296-300 ◽  
Author(s):  
Yuzuru Kakuda
Keyword(s):  
Measurable Cardinal ◽  
Chain Condition ◽  
Elementary Embedding ◽  
Normal Ideal ◽  
Uncountable Cardinal ◽  
Thin Sets ◽  
Measurable Cardinals ◽  
Precipitous Ideal ◽  
Precipitous Ideals ◽  
The Ideal

T. Jech and K. Prikry introduced the concept of precipitous ideals as a counterpart of measurable cardinals for small cardinals (Jech and Prikry [1]). To get a precipitous ideal on ℵ1( W. Mitchell used the Cohen extension by the standard Levy collapse that makes κ = ℵ1, where κ is a measurable cardinal in the ground model (Jech, Magidor, Mitchell and Prikry [2]). His proof essentially used the fact that , where is the notion of forcing of Levy collapse and j is the elementary embedding obtained by a normal ultrafilter on κ.On the other hand, we know that has the κ-chain condition. In this paper, we show that the κ-chain condition for notions of forcing plays an essential role for preserving normal precipitous ideals. Namely,Theorem 1. Let κ be a regular uncountable cardinal and I a normal ideal on κ. Let be a notion of forcing with the κ-chain condition. Then, I is precipitous iff ⊩“the ideal on κ generated by I is precipitous”.As an application of Theorem 1, we have the following theorem.Theorem 2. Let κ be a regular uncountable cardinal, and Γ be a κ-saturated normal ideal on κ. Then, {a < κ; the ideal of thin sets on a is precipitous} has either Γ-measure one or Γ-measure zero, and the ideal of thin sets on κ is precipitous iff {α< κ; the ideal of thin sets on α is precipitous) has Γ-measure one.

Between strong and superstrong

1986 ◽  
Vol 51 (3) ◽  
pp. 547-559 ◽  
Author(s):  
Stewart Baldwin
Keyword(s):  
Critical Point ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Proper Class ◽  
Consistency Strength ◽  
Strong Cardinal ◽  
Definition Of ◽  
Strong Cardinals

Definition. A cardinal κ is strong iff for every x there is an elementary embedding j:V → M with critical point κ such that x ∈ M.κ is superstrong iff ∃j:V → M with critical point κ such that Vj(κ) ∈ M.These definitions are natural weakenings of supercompactness and hugeness respectively and display some of the same relations. For example, if κ is superstrong then Vκ ⊨ “∃ proper class of strong cardinals”, but the smallest superstrong cardinal is less than the smallest strong cardinal (if both types exist). (See [SRK] and [Mo] for the arguments involving supercompact and huge, which translate routinely to strong and superstrong.)Given any two types of large cardinals, a typical vague question which is often asked is “How large is the gap in consistency strength?” In one sense the gap might be considered relatively small, since the “higher degree” strong cardinals described below (a standard trick that is nearly always available) and the Shelah and Woodin hierarchies of cardinals (see [St] for a definition of these) seem to be (at least at this point in time) the only “natural” large cardinal properties lying between strong cardinals and superstrong cardinals in consistency strength.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

Co-stationarity of the ground model

2006 ◽  
Vol 71 (3) ◽  
pp. 1029-1043 ◽  
Author(s):  
Natasha Dobrinen ◽  
Sy-David Friedman
Keyword(s):  
Large Cardinals ◽  
Partial Ordering ◽  
Proper Class ◽  
Ground Model ◽  
Covering Theorem ◽  
Uncountable Cardinal ◽  
Cohen Forcing ◽  
Measurable Cardinals

AbstractThis paper investigates when it is possible for a partial ordering ℙ to force Pk(Λ)\V to be stationary in Vℙ. It follows from a result of Gitik that whenever ℙ adds a new real, then Pk(Λ)\V is stationary in Vℙ for each regular uncountable cardinal κ in Vℙ and all cardinals λ ≥ κ in Vℙ [4], However, a covering theorem of Magidor implies that when no new ω-sequences are added, large cardinals become necessary [7]. The following is equiconsistent with a proper class of ω1-Erdős cardinals: If ℙ is ℵ1-Cohen forcing, then Pk(Λ)\V is stationary in Vℙ, for all regular κ ≥ ℵ2and all λ ≩ κ. The following is equiconsistent with an ω1-Erdős cardinal: If ℙ is ℵ1-Cohen forcing, then is stationary in Vℙ. The following is equiconsistent with κ measurable cardinals: If ℙ is κ-Cohen forcing, then is stationary in Vℙ.

scholarly journals Laver sequences for extendible and super-almost-huge cardinals

1999 ◽  
Vol 64 (3) ◽  
pp. 963-983 ◽  
Author(s):  
Paul Corazza
Keyword(s):  
Critical Point ◽  
Upper Bound ◽  
Large Cardinal ◽  
Strong Cardinal ◽  
Regular Class ◽  
Strong Cardinals ◽  
Extendible Cardinals

AbstractVersions of Laver sequences are known to exist for supercompact and strong cardinals. Assuming very strong axioms of infinity, Laver sequences can be constructed for virtually any globally defined large cardinal not weaker than a strong cardinal; indeed, under strong hypotheses. Laver sequences can be constructed for virtually any regular class of embeddings. We show here that if there is a regular class of embeddings with critical point κ, and there is an inaccessible above κ, then it is consistent for there to be a regular class that admits no Laver sequence. We also show that extendible cardinals are Laver-generating, i.e., that assuming only that κ is extendible, there is an extendible Laver sequence at κ. We use the method of proof to answer a question about Laver-closure of extendible cardinals at inaccessibles. Finally, we consider Laver sequences for super-almost-huge cardinals. Assuming slightly more than super-almost-hugeness, we show that there are super-almost-huge Laver sequences, improving the previously known upper bound for such Laver sequences. We also describe conditions under which the canonical construction of a Laver sequence fails for super-almost-huge cardinals.

HAPPY AND MAD FAMILIES INL(ℝ)

2018 ◽  
Vol 83 (2) ◽  
pp. 572-597 ◽  
Author(s):  
ITAY NEEMAN ◽  
ZACH NORWOOD
Keyword(s):  
Generic Absoluteness ◽  
Mad Families ◽  
Recent Theorem ◽  
Image Position ◽  
Solovay Model ◽  
Happy Family

AbstractWe prove that, in the choiceless Solovay model, every set of reals isH-Ramsey for every happy familyHthat also belongs to the Solovay model. This gives a new proof of Törnquist’s recent theorem that there are no infinite mad families in the Solovay model. We also investigate happy families and mad families under determinacy, applying a generic absoluteness result to prove that there are no infinite mad families under$A{D^ + }$.

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