On precipitousness of the nonstationary ideal over a supercompact

1986 ◽  
Vol 51 (3) ◽  
pp. 648-662 ◽  
Author(s):  
Moti Gitik
Keyword(s):  
General Theorem ◽  
Supercompact Cardinal ◽  
Generic Extension ◽  
Inner Model ◽  
Nonstationary Ideal

Namba [N] proved that the nonstationary ideal over a measurable (NSκ) cannot be κ+-saturated. Baumgartner, Taylor and Wagon [BTW] asked if it is possible for NSκ to be precipitous over a measurable κ. A model with this property was constructed by the author, and shortly after Foreman, Magidor and Shelah [FMS] proved a general theorem that after collapsing of a supercompact or even a superstrong to the successor of κ, NSκ became precipitous. This theorem implies that it is possible to have the nonstationary ideal precipitous over even a supercompact cardinal. Just start with a supercompact κ and a superstrong λ > κ. Make supercompactness of κ indistractible as in [L] and then collapse λ to be κ+.The aim of our paper is to show that the existence of a supercompact cardinal alone already implies the consistency of the nonstationary ideal precipitous over a supercompact. The proof gives also the following: if κ is a λ-supercompact for λ ≥ (2κ)+, then there exists a generic extension in which κ is λ-supercompact and NSκ is precipitous. Thus, for a model with NSκ precipitous over a measurable we need a (2κ)+-supercompact cardinal κ. Jech [J] proved that the precipitous of NSκ over a measurable κ implies the existence of an inner model with o(κ) = κ+ + 1. In §3 we improve this result a little by showing that the above assumption implies an inner model with a repeat point.The paper is organized as follows. In §1 some preliminary facts are proved. The model with NSκ precipitous over a supercompact is constructed in §2.

SUITABLE EXTENDER MODELS I

2010 ◽  
Vol 10 (01n02) ◽  
pp. 101-339 ◽  
Author(s):  
W. HUGH WOODIN
Keyword(s):  
Strong Evidence ◽  
Large Cardinal ◽  
Central Issue ◽  
Supercompact Cardinal ◽  
Model Program ◽  
Inner Model

We investigate both iteration hypotheses and extender models at the level of one supercompact cardinal. The HOD Conjecture is introduced and shown to be a key conjecture both for the Inner Model Program and for understanding the limits of the large cardinal hierarchy. We show that if the HOD Conjecture is true then this provides strong evidence for the existence of an ultimate version of Gödel's constructible universe L. Whether or not this "ultimate" L exists is now arguably the central issue for the Inner Model Program.

scholarly journals Jónsson cardinals, Erdős cardinals, and the core model

1999 ◽  
Vol 64 (3) ◽  
pp. 1065-1086 ◽  
Author(s):  
W. J. Mitchell
Keyword(s):  
Minimal Model ◽  
Independent Interest ◽  
Core Model ◽  
Generic Extension ◽  
The Core ◽  
Steel Core ◽  
Inner Model ◽  
Jonsson Cardinals ◽  
Woodin Cardinal

AbstractWe show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ5-Erdős in K.In the absence of the Steel core model K we prove the same conclusion for any model L[] such that either V = L[] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L[].The proof includes one lemma of independent interest: If V = L[A], where A ⊂ κ and κ is regular, then Lκ[A] is a Jónsson algebra. The proof of this result. Lemma 2.5, is very short and entirely elementary.

ON LÖWENHEIM–SKOLEM–TARSKI NUMBERS FOR EXTENSIONS OF FIRST ORDER LOGIC

2011 ◽  
Vol 11 (01) ◽  
pp. 87-113 ◽  
Author(s):  
MENACHEM MAGIDOR ◽  
JOUKO VÄÄNÄNEN
Keyword(s):  
Order Logic ◽  
Strong Form ◽  
The Other ◽  
First Order Logic ◽  
Supercompact Cardinal ◽  
Inaccessible Cardinal ◽  
First Order ◽  
Inner Model ◽  
Singular Cardinals ◽  
Second Order Logic

We show that, assuming the consistency of a supercompact cardinal, the first (weakly) inaccessible cardinal can satisfy a strong form of a Löwenheim–Skolem–Tarski theorem for the equicardinality logic L(I), a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim–Skolem–Tarski theorem for the equicardinality logic at κ implies the Singular Cardinals Hypothesis above κ as well as Projective Determinacy.

scholarly journals IN SEARCH OF ULTIMATE-LTHE 19TH MIDRASHA MATHEMATICAE LECTURES

2017 ◽  
Vol 23 (1) ◽  
pp. 1-109 ◽  
Author(s):  
W. HUGH WOODIN
Keyword(s):  
Model Theory ◽  
Model Problem ◽  
Supercompact Cardinal ◽  
Inner Model Theory ◽  
Inner Model ◽  
Complete Account

AbstractWe give a fairly complete account which first shows that the solution to the inner model problem for one supercompact cardinal will yield an ultimate version ofLand then shows that the various current approaches to inner model theory must be fundamentally altered to provide that solution.

A WELLORDER OF THE REALS WITH SATURATED

2019 ◽  
Vol 84 (4) ◽  
pp. 1466-1483
Author(s):  
SY-DAVID FRIEDMAN ◽  
STEFAN HOFFELNER
Keyword(s):  
Inner Model ◽  
Nonstationary Ideal ◽  
Image Position ◽  
Woodin Cardinal

AbstractWe show that, assuming the existence of the canonical inner model with one Woodin cardinal $M_1 $ , there is a model of $ZFC$ in which the nonstationary ideal on $\omega _1 $ is $\aleph _2 $-saturated and whose reals admit a ${\rm{\Sigma }}_4^1 $-wellorder.

On generic elementary embeddings

1989 ◽  
Vol 54 (3) ◽  
pp. 700-707 ◽  
Author(s):  
Moti Gitik
Keyword(s):  
Measurable Cardinal ◽  
Generic Extension ◽  
Natural Question ◽  
The Core ◽  
Blowing Up ◽  
Elementary Embeddings ◽  
Nonstationary Ideal ◽  
Stationary Set ◽  
Precipitous Ideal ◽  
Precipitous Ideals

Suppose that I is a precipitous ideal over a cardinal κ and j is a generic embedding of I. What is the nature of j? If we assume the existence of a supercompact cardinal then, by Foreman, Magidor and Shelah [FMS], it is quite unclear where some of such j's are coming from. On the other hand, if ¬∃κ0(κ) = κ++, then, by Mitchell [Mi], the restriction of j to the core model is its iterated ultrapower by measures of it. A natural question arising here is if each iterated ultrapower of can be obtained as the restriction of a generic embedding of a precipitous ideal. Notice that there are obvious limitations. Thus the ultrapower of by a measure over λ cannot be obtained as a generic embedding by a precipitous ideal over κ ≠ λ. But if we fix κ and use iterated ultrapowers of which are based on κ, then the answer is positive. Namely a stronger statement is true:Theorem. Let τ be an ordinal and κ a measurable cardinal. There exists a generic extension V* of V so that NSℵ1 (the nonstationary ideal on ℵ1) is precipitous and, for every iterated ultrapower i of V of length ≤ τ by measures of V based on κ, there exists a stationary set forcing “the generic ultrapower restricted to V is i”.Our aim will be to prove this theorem. We assume that the reader is familiar with the paper [JMMiP] by Jech, Magidor, Mitchell and Prikry. We shall use the method of that paper for constructing precipitous ideals. Ideas of Levinski [L] for blowing up 2ℵ1 preserving precipitousness and of our own earlier paper [Gi] for linking together indiscernibles will be used also.

The genericity conjecture

1994 ◽  
Vol 59 (2) ◽  
pp. 606-614 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Generic Extension ◽  
Class Theory ◽  
First Order ◽  
Class A ◽  
Inner Model ◽  
Special Case

The Genericity Conjecture, as stated in Beller-Jensen-Welch [1], is the following:(*) If O# ∉ L[R], R ⊆ ω, then R is generic over L.We must be precise about what is meant by “generic”.Definition (Stated in Class Theory). A generic extension of an inner model M is an inner model M[G] such that for some forcing notion ⊆ M:(a) 〈M, 〉 is amenable and ⊩ is 〈M, 〉-definable for sentences.(b) G ⊆ is compatible, closed upwards, and intersects every 〈M, 〉-definable dense D ⊆ .A set x is generic over M if it is an element of a generic extension of M. And x is strictly generic over M if M[x] is a generic extension of M.Though the above definition quantifies over classes, in the special case where M = L and O# exists, these notions are in fact first order, as all L-amenable classes are definable over L[O#]. From now on assume that O# exists.Theorem A. The Genericity Conjecture is false.The proof is based upon the fact that every real generic over L obeys a certain definability property, expressed as follows.Fact. If R is generic over L, then for some L-amenable class A, Sat〈L, A〉 is not definable over 〈L[R],A〉, where Sat〈L,A〉 is the canonical satisfaction predicate for 〈L,A〉.

Chang's Conjecture and the Non-Stationary Ideal

2001 ◽  
Vol 66 (1) ◽  
pp. 144-170 ◽  
Author(s):  
Daniel Evan Seabold
Keyword(s):  
Partial Order ◽  
Order Type ◽  
Generic Extension ◽  
Forcing Axioms ◽  
Axiom Of Determinacy ◽  
Woodin Cardinals ◽  
Chang’S Conjecture ◽  
Nonstationary Ideal ◽  
Chang's Conjecture ◽  
Central Result

In The Axiom of Determinacy, Forcing Axioms and the Nonstationary Ideal [W1], Woodin constructs the partial order ℙmax, which in the presence of large cardinals yields a forcing extension of L(ℝ) where ZFC holds and the non-stationary ideal on ω1 (hereafter denoted NSω1) is ω2-saturated. The basic analysis of ℙmax forcing over L(ℝ) can be carried out assuming only the Axiom of Determinacy (AD). In the central result of this paper, we show that if one increases slightly the strength of the determinacy assumptions, then Chang's Conjecture—the assertion that every finitary algebra on ω2 has a subalgebra of order type ω1—holds in this extension as well. Specifically, we obtain:Corollary 4.6. Assume AD + V = L(ℝ, μ) + μ is a normal, fine measure on. Chang's Conjecture holds in any ℙmax-generic extension of L(ℝ).This technique for obtaining Chang's Conjecture is fairly general. We [Se] have adapted it to obtain Chang's Conjecture in the model presented by Steel and Van Wesep [SVW] and Woodin [Wl] has adapted it to his ℚmax forcing notion. In each of these models, as in the ℙmax extension, one forces over L(ℝ) assuming AD to obtain ZFC and NSω1 is ω2-saturated.By unpublished results of Woodin, the assumption for Corollary 4.6 is equiconsistent with the existence of ω2 many Woodin cardinals, and hence strictly stronger than ADL(ℝ). One would like to reduce this assumption to ADL(ℝ). Curiously, this reduction is not possible in the arguments for ℚmax or the Steel and Van Wesep model, and the following argument of Woodin suggests why it may not be possible for ℙmax either.

VARSOVIAN MODELS I

2018 ◽  
Vol 83 (2) ◽  
pp. 496-528 ◽  
Author(s):  
GRIGOR SARGSYAN ◽  
RALF SCHINDLER
Keyword(s):  
Core Model ◽  
Generic Extension ◽  
Strong Cardinal ◽  
Small Core ◽  
Woodin Cardinals ◽  
Inner Model ◽  
Specific Fragment ◽  
Image Position ◽  
Woodin Cardinal

AbstractLet Msw denote the least iterable inner model with a strong cardinal above a Woodin cardinal. By [11], Msw has a fully iterable core model, ${K^{{M_{{\rm{sw}}}}}}$, and Msw is thus the least iterable extender model which has an iterable core model with a Woodin cardinal. In V, ${K^{{M_{{\rm{sw}}}}}}$ is an iterate of Msw via its iteration strategy Σ.We here show that Msw has a bedrock which arises from ${K^{{M_{{\rm{sw}}}}}}$ by telling ${K^{{M_{{\rm{sw}}}}}}$ a specific fragment ${\rm{\bar{\Sigma }}}$ of its own iteration strategy, which in turn is a tail of Σ. Hence Msw is a generic extension of $L[{K^{{M_{{\rm{sw}}}}}},{\rm{\bar{\Sigma }}}]$, but the latter model is not a generic extension of any inner model properly contained in it.These results generalize to models of the form Ms (x) for a cone of reals x, where Ms (x) denotes the least iterable inner model with a strong cardinal containing x. In particular, the least iterable inner model with a strong cardinal above two (or seven, or boundedly many) Woodin cardinals has a 2-small core model K with a Woodin cardinal and its bedrock is again of the form $L[K,{\rm{\bar{\Sigma }}}]$.

scholarly journals Guessing models and the approachability ideal

2020 ◽  
pp. 2150003
Author(s):  
Rahman Mohammadpour ◽  
Boban Veličković
Keyword(s):  
Generic Extension ◽  
Singular Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Ideal Formula ◽  
Singular Cardinal Hypothesis ◽  
Nonstationary Ideal ◽  
The Universe ◽  
Weak Square

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call [Formula: see text] holds. This principle implies [Formula: see text] and [Formula: see text], and hence the tree property at [Formula: see text] and [Formula: see text], the Singular Cardinal Hypothesis, and the failure of the weak square principle [Formula: see text], for all regular [Formula: see text]. In addition, it implies that the restriction of the approachability ideal [Formula: see text] to the set of ordinals of cofinality [Formula: see text] is the nonstationary ideal on this set. The consistency of this last statement was previously shown by W. Mitchell.

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