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.

A guide to “Coding the universe” by Beller, Jensen, Welch

1985 ◽  
Vol 50 (4) ◽  
pp. 1002-1019 ◽  
Author(s):  
Sy D. Friedman
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Affirmative Answer ◽  
Partial Ordering ◽  
Proper Class ◽  
Covering Lemma ◽  
Singular Cardinals ◽  
The Universe

In the wake of Silver's breakthrough on the Singular Cardinals Problem (Silver [74]) followed one of the landmark results in set theory, Jensen's Covering Lemma (Devlin-Jensen [74]): If 0# does not exist then for every uncountable x ⊆ ORD there exists a constructible Y ⊇ X, card(Y) = card(X). Thus it is fair to say that in the absence of large cardinals, V is “close to L”.It is natural to ask, as did Solovay, if we can fairly interpret the phrase “close to L” to mean “generic over L”. For example, if V = L[a], a ⊆ ω and if 0# does not exist then is V-generic over L for some partial ordering ∈ L? Notice that an affirmative answer implies that in the absence of 0#, no real can “code” a proper class of information.Jensen's Coding Theorem provides a negative answer to Solovay's question, in a striking way: Any class can be “coded” by a real without introducing 0#. More precisely, if A ⊆ ORD then there is a forcing definable over 〈L[A], A〉 such that ⊩ V = L[a], a ⊆ ω, A is definable from a. Moreover if 0# ∉ L[A] then ⊩ 0# does not exist. Now as any M ⊨ ZFC can be generically extended to a model of the form L[A] (without introducing 0#) we obtain: For any 〈M, A〉 ⊨ ZFC (that is, M ⊨ ZFC and M obeys Replacement for formulas mentioning A as a predicate) there is an 〈M, A〉-definable forcing such that ⊩ V = L[a], a ⊆ ω, 〈M, A〉 is definable from a. Moreover if 0# ∉ M then ⊩ 0# does not exist.

Homogeneous changes in cofinalities with applications to HOD

2017 ◽  
Vol 17 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Omer Ben-Neria ◽  
Spencer Unger
Keyword(s):  
Large Cardinals ◽  
Related Question ◽  
New Technique ◽  
Singular Cardinals ◽  
Uncountable Cardinal ◽  
A New Technique

We present a new technique for changing the cofinality of large cardinals using homogeneous forcing. As an application we show that many singular cardinals in [Formula: see text] can be measurable in HOD. We also answer a related question of Cummings, Friedman and Golshani by producing a model in which every regular uncountable cardinal [Formula: see text] in [Formula: see text] is [Formula: see text]-supercompact in HOD.

An AD-like model

1985 ◽  
Vol 50 (2) ◽  
pp. 531-543 ◽  
Author(s):  
Arthur W. Apter
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinals ◽  
Strong Sense ◽  
Supercompact Cardinals ◽  
Singular Cardinals ◽  
Partial Orderings ◽  
Measurable Cardinals ◽  
Jonsson Cardinals

A very fruitful line of research in recent years has been the application of techniques in large cardinals and forcing to the production of models in which certain consequences of the axiom of determinateness (AD) are true or in which certain “AD-like” consequences are true. Numerous results have been published on this subject, among them the papers of Bull and Kleinberg [4], Bull [3], Woodin [15], Mitchell [11], and [1], [2].Another such model will be constructed in this paper. Specifically, the following theorem is proven.Theorem 1. Con(ZFC + There are cardinals κ < δ < λ so that κ is a supercompact limit of supercompact cardinals, λ is a measurable cardinal, and δ is λ supercompact) ⇒ Con(ZF + ℵ1 and ℵ2 are Ramsey cardinals + The ℵn for 3 ≤ n ≤ ω are singular cardinals of cofinality ω each of which carries a Rowbottom filter + ℵω + 1 is a Ramsey cardinal + ℵω + 2 is a measurable cardinal).It is well known that under AD + DC, ℵ2 and ℵ2 are measurable cardinals, the ℵn for 3 ≤ n < ω are singular Jonsson cardinals of cofinality ℵ2, ℵω is a Rowbottom cardinal, and ℵω + 1 and ℵω + 2 are measurable cardinals.The proof of the above theorem will use the existence of normal ultrafilters which satisfy a certain property (*) (to be defined later) and an automorphism argument which draws upon the techniques developed in [9], [2], and [4] but which shows in addition that certain supercompact Prikry partial orderings are in a strong sense “homogeneous”. Before beginning the proof of the theorem, however, we briefly mention some preliminaries.

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.

Believing the axioms. II

1988 ◽  
Vol 53 (3) ◽  
pp. 736-764 ◽  
Author(s):  
Penelope Maddy
Keyword(s):  
Continuum Hypothesis ◽  
Winning Strategy ◽  
Regularity Property ◽  
Large Cardinals ◽  
Natural Numbers ◽  
Regularity Properties ◽  
Finite Sequences ◽  
The Axiom Of Choice ◽  
Measurable Cardinals ◽  
Descriptive Set

This is a continuation of Believing the axioms. I, in which nondemonstrative arguments for and against the axioms of ZFC, the continuum hypothesis, small large cardinals and measurable cardinals were discussed. I turn now to determinacy hypotheses and large large cardinals, and conclude with some philosophical remarks.Determinacy is a property of sets of reals. If A is such a set, we imagine an infinite game G(A) between two players I and II. The players take turns choosing natural numbers. In the end, they have generated a real number r (actually a member of the Baire space ωω). If r is in A, I wins; otherwise, II wins. The set A is said to be determined if one player or the other has a winning strategy (that is, a function from finite sequences of natural numbers to natural numbers that guarantees the player a win if he uses it to decide his moves).Determinacy is a “regularity” property (see Martin [1977, p. 807]), a property of well-behaved sets, that implies the more familiar regularity properties like Lebesgue measurability, the Baire property (see Mycielski [1964] and [1966], and Mycielski and Swierczkowski [1964]), and the perfect subset property (Davis [1964]). Infinitary games were first considered by the Polish descriptive set theorists Mazur and Banach in the mid-30s; Gale and Stewart [1953] introduced them into the literature, proving that open sets are determined and that the axiom of choice can be used to construct an undetermined set.

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.

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.

scholarly journals Possible behaviours of the reflection ordering of stationary sets

1995 ◽  
Vol 60 (2) ◽  
pp. 534-547 ◽  
Author(s):  
Jiří Witzany
Keyword(s):  
Partial Ordering ◽  
Generic Extension ◽  
Maximal Antichain ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Stationary Subsets ◽  
Almost All

AbstractIf S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so thatWe prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

The large cardinal strength of weak Vopenka’s principle

2021 ◽  
pp. 2150024
Author(s):  
Trevor M. Wilson
Keyword(s):  
Lower Bound ◽  
Large Cardinal ◽  
Proper Class ◽  
Strong Cardinal ◽  
Elementary Embeddings ◽  
Vopěnka’S Principle ◽  
Measurable Cardinals

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal.

0# and some forcing principles

1986 ◽  
Vol 51 (1) ◽  
pp. 39-46 ◽  
Author(s):  
Matthew Foreman ◽  
Menachem Magidor ◽  
Saharon Shelah
Keyword(s):  
Real Number ◽  
Large Cardinals ◽  
Singular Limit ◽  
Partial Ordering ◽  
The Other ◽  
List Type ◽  
Consistency Strength ◽  
Axiom Of Determinacy ◽  
Partial Orderings ◽  
The Universe

It has been considered desirable by many set theorists to find maximality properties which state that the universe has in some sense “many sets”. The properties isolated thus far have tended to be consistent with each other (as far as we know). For example it is a widely held view that the existence of a supercompact cardinal is consistent with the axiom of determinacy holding in L(R). This consistency has been held to be evidence for the truth of these properties. It is with this in mind that the first author suggested the following:Maximality Principle If P is a partial ordering and G ⊆ P is a V-generic ultrafilter then eithera) there is a real number r ∈ V [G] with r ∉ V, orb) there is an ordinal α such that α is a cardinal in V but not in V[G].This maximality principle applied to garden variety partial orderings has startling results for the structure of V.For example, if for some , then P = 〈{p: p ⊆ κ, ∣p∣ < κ}, ⊆〉 neither adds a real nor collapses a cardinal. Thus from the maximality principle we can deduce that the G. C. H. fails everywhere and there are no inaccessible cardinals. (Hence this principle contradicts large cardinals.) Similarly one can show that there are no Suslin trees on any cardinal κ. These consequences help justify the title “maximality principle”.Since the maximality principle implies that the G. C. H. fails at strong singular limit cardinals it has consistency strength at least that of “many large cardinals”. (See [M].) On the other hand it is not known to be consistent, relative to any assumptions.

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