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.

Some pathological examples of precipitous ideals

2008 ◽  
Vol 73 (2) ◽  
pp. 492-511 ◽  
Author(s):  
Moti Gitik
Keyword(s):  
Measurable Cardinal ◽  
Normal Ideal ◽  
Precipitous Ideal ◽  
Precipitous Ideals

AbstractWe construct a model with an indecisive precipitous ideal and a model with a precipitous ideal with a non precipitous normal ideal below it. Such kind of examples were previously given by M. Foreman [2] and R. Laver [4] respectively. The present examples differ in two ways: first- they use only a measurable cardinal and second- the ideals are over a cardinal. Also a precipitous ideal without a normal ideal below it is constructed. It is shown in addition that if there is a precipitous ideal over a cardinal κ such that• after the forcing with its positive sets the cardinality of κ remains above ℵ1• there is no a normal precipitous ideal then there is 0†.

Precipitous ideals

1980 ◽  
Vol 45 (1) ◽  
pp. 1-8 ◽  
Author(s):  
T. Jech ◽  
M. Magidor ◽  
W. Mitchell ◽  
K. Prikry
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Ground Model ◽  
Generic Set ◽  
Precipitous Ideal ◽  
Precipitous Ideals ◽  
The Ideal

The properties of small cardinals such as ℵ1 tend to be much more complex than those of large cardinals, so that properties of ℵ1 may often be better understood by viewing them as large cardinal properties. In this paper we show that the existence of a precipitous ideal on ℵ1 is essentially the same as measurability.If I is an ideal on P(κ) then R(I) is the notion of forcing whose conditions are sets x ∈ P(κ)/I, with x ≤ x′ if x ⊆ x′. Thus a set D R(I)-generic over the ground model V is an ultrafilter on P(κ) ⋂ V extending the filter dual to I. The ideal I is said to be precipitous if κ ⊨R(I)(Vκ/D is wellfounded).One example of a precipitous ideal is the ideal dual to a κ-complete ultrafilter U on κ. This example is trivial since the generic ultrafilter D is equal to U and is already in the ground model. A generic set may be viewed as one that can be worked with in the ground model even though it is not actually in the ground model, so we might expect that cardinals such as ℵ1 that cannot be measurable still might have precipitous ideals, and such ideals might correspond closely to measures.

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.

Nonabsoluteness of elementary embeddings

1989 ◽  
Vol 54 (3) ◽  
pp. 774-778
Author(s):  
Friedrich Wehrung
Keyword(s):  
Measurable Cardinal ◽  
Elementary Embedding ◽  
Minimal Support ◽  
Elementary Embeddings ◽  
Measurable Cardinals

Ifκis a measurable cardinal, let us say that a measure onκis aκ-complete nonprincipal ultrafilter onκ. IfUis a measure onκ, letjUbe the canonical elementary embedding ofVinto its Ultrapower UltU(V). Ifxis a set, say thatUmovesxwhenjU(x)≠x; say thatκmovesxwhen some measure onκmovesx. Recall Kunen's lemma (see [K]): “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof (see [K]) and Fleissner's proof (see [KM, III, §10]) are essentially nonconstructive.The following proposition can be proved by using elementary facts about iterated ultrapowers.Proposition.Let ‹Un: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n= Um(m, n ∈ ω). Then, for each θ inUltU(V),if E is the (minimal) support of θ inUltU(V),then, for all m ∈ ω, Ummoves θ iff E ∩ [ωm, ω(m + 1))≠ ∅.

A MODEL WITH A PRECIPITOUS IDEAL, BUT NO NORMAL PRECIPITOUS IDEAL

2013 ◽  
Vol 13 (01) ◽  
pp. 1250008
Author(s):  
MOTI GITIK
Keyword(s):  
Measurable Cardinal ◽  
Precipitous Ideal ◽  
Precipitous Ideals

Starting with a measurable cardinal κ of the Mitchell order κ++ we construct a model with a precipitous ideal on ℵ1 but without normal precipitous ideals. This answers a question by T. Jech and K. Prikry. In the constructed model there are no Q-point precipitous filters on ℵ1, i. e. those isomorphic to extensions of Cubℵ1.

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.

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.

Seminormal λ-generated ideals on Pκλ

1988 ◽  
Vol 53 (1) ◽  
pp. 92-102
Author(s):  
C. A. Johnson
Keyword(s):  
Normal Ideal ◽  
Selective Property ◽  
Weakly Compact ◽  
Uncountable Cardinal ◽  
The Ideal

In this paper we consider the problem of lifting properties of the Fréchet ideal Iκ = {X ⊆ κ: ∣X∣ < κ} on a regular uncountable cardinal κ, to an analogue about Iκλ, the ideal of not unbounded subsets of Pκλ. With this in mind, in §1 we introduce and study the class of seminormal λ-generated ideals on Pκλ. We shall see that ideals belonging to this class display properties which are clearly analogous to those of the Fréchet ideal on κ (for instance, with regard to saturation, normality and weak selectivity) and yet are closely related to Iκλ. Our results here show that if λ<λ = λ, then many restrictions of Iκλ are weakly selective, nowhere precipitous and, quite suprisingly, seminormal (but nowhere normal). These latter two results suggest the question of whether any restriction of Iκλ can ever be normal. In §2 we prove that if κ is strongly inaccessible, λ<κ = 2λ and NSκλ, the ideal of nonstationary subsets of Pκλ, has a mild selective property, then NSκλ∣A = Iκλ∣A for some stationary A ⊆ Pκλ.In [1] Baumgartner showed that if κ is weakly compact and P is the collection of indescribable subsets of κ, then P → (P, κ)2. As a Pκλ analogue of indescribability, Carr (see [3]–[5]) introduced the λ-Shelah property, but was unable to derive the natural Pκλ analogue of Baumgartner's result, (where NShκλ is the normal ideal on Pκλ induced by the λ-Shelah property). In §3 we show that the problem lies in the fact that, as far as we know, NShκλ is not sufficiently distributive, and derive conditions which are sufficient and, in a sense, necessary to yield partitions related to .

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 $$ZFC$$.

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