Ramsey Cardinals Research Articles

A note on the Revised GCH

We give a proof of a general version of the Revised GCH, due originally to Shelah in [11]. Our proof eliminates the use of nice filters and their associated rank functions, and instead uses only the well-foundedness of reduced products of ordinals modulo countably complete filters. This removes the need to make assumptions about the existence of Ramsey cardinals in suitable core models.

Indestructibility properties of Ramsey and Ramsey-like cardinals

We develop new techniques, for use in indestructibility arguments, of lifting embeddings on transitive set models of ZFC− which lack closure and of lifting iterations of such embeddings to a forcing extension. We use these techniques to establish basic indestructibility results for Ramsey and the Ramsey-like cardinals – α-iterable, strongly Ramsey, and super Ramsey cardinals – introduced in [11]. We show that Ramsey, α-iterable, strongly Ramsey, and super Ramsey cardinals κ are indestructible by small forcing, the canonical forcing of the GCH, and the forcing to add a fast function on κ. We also show that if κ is one of these large cardinals, then there is a forcing extension in which the large cardinal property of κ becomes indestructible by Add(κ,θ) for any cardinal θ.The following are consequences of the indestructibility results. If κ is Ramsey, α-iterable, strongly Ramsey, or super Ramsey, then there is a forcing extension preserving this in which the GCH fails at κ. If κ is one of these large cardinals, then there is a forcing extension preserving the large cardinal property of κ in which κ is not even weakly compact in HOD. If κ is Ramsey, then there is a forcing extension in which κ remains virtually Ramsey, but is no longer Ramsey (this answers positively a question posed in [11]).

Ineffable limits of weakly compact cardinals and similar results

It is proved that if an uncountable cardinal κ has an ineffable subset of weakly compact cardinals, then κ is a weakly compact cardinal, and if κ has an ineffable subset of Ramsey (Rowbottom, Jónsson, ineffable or subtle) cardinals, then κ is a Ramsey (Rowbottom, J\'onsson, ineffable or subtle) cardinal.

A REFINEMENT OF THE RAMSEY HIERARCHY VIA INDESCRIBABILITY

AbstractWe study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of $\Pi ^1_n$ -indescribability where $n<\omega $ to that of $\Pi ^1_\xi $ -indescribability where $\xi \geq \omega $ . By iterating Feng’s Ramsey operator [12] on the various $\Pi ^1_\xi $ -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng’s original Ramsey hierarchy. We isolate Ramsey properties which provide strictly increasing hierarchies between Feng’s $\Pi _\alpha $ -Ramsey and $\Pi _{\alpha +1}$ -Ramsey cardinals for all odd $\alpha <\omega $ and for all $\omega \leq \alpha <\kappa $ . We also show that, given any ordinals $\beta _0,\beta _1<\kappa $ the increasing chains of ideals obtained by iterating the Ramsey operator on the $\Pi ^1_{\beta _0}$ -indescribability ideal and the $\Pi ^1_{\beta _1}$ -indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of $\Pi ^1_\xi $ -indescribability and Ramseyness.

Easton's theorem for Ramsey and strongly Ramsey cardinals

We show that, assuming GCH, if κ is a Ramsey or a strongly Ramsey cardinal and F is a class function on the regular cardinals having a closure point at κ and obeying the constraints of Easton's theorem, namely, F(α)≤F(β) for α≤β and α<cf(F(α)), then there is a cofinality-preserving forcing extension in which κ remains Ramsey or strongly Ramsey respectively and 2δ=F(δ) for every regular cardinal δ.

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ +, another for which 2κ = κ ++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that \({H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}\). Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.

Ramsey-like cardinals II

AbstractThis paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α ≤ ω1, that they are downward absolute to L for , and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.We show that the strongly Ramsey and super Ramsey cardinals from [5] are downward absolute to the core model K. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.

Making all cardinals almost Ramsey

We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + \({\neg {\rm AC}_\omega}\) in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost Ramsey cardinals”, and “ZF + DC + All infinite cardinals except possibly successors of singular cardinals are almost Ramsey”.

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ1 satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω1 satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω1, and specifically if this and a stronger property hold under MA + ¬CH. Towards this we show that if MA + ¬CH holds, then for every family ? of ℵ1 many infinite subsets of ω1, one can find a family ? of finite subsets of ω1 which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?.

On elementary embeddings from an inner model to the universe

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.

Ramsey cardinals, α-Erdös cardinals, and the core model

The core model K was introduced by R. B. Jensen and A. J. Dodd [DoJ]. K is the union of Gödel's constructible universe L together with all mice, i.e., , and K is a transitive model of ZFC + (V = K) + GCH (see [DoJ]). V = K is consistent with the existence of Ramsey cardinals [M], and if cf(α) &gt; ω, V = K is consistent with the existence of α-Erdös cardinals [J]. Let K be Ramsey. Then there is a smallest inner model Wκ of ZFC in which κ is Ramsey. We have Wκ ⊨ V = K and Wκ ⊆ K [M]. The existence of Wκ with is equivalent to the existence of a sharplike mouse on N ⊨ K with N ⊨ κ Ramsey. (A mouse N on is called sharplike provided .) We have , where is the mouse iteration of N. N is the oleast mouse not in Wκ (see [J] and [DJKo]). Here &lt; denotes the mouse order. The context always clarifies whether the mouse order or the usual &lt;-relation is meant.The main result of §1 is that Wκ ⊨ κ is the only Ramsey cardinal. A similar result has been found true in the smallest inner model L[U] of ZFC + “κ is measurable” if U is a normal measure on κ: L[U] ⊨ κ is the only measurable cardinal [Ku].

A hierarchy of ramsey cardinals

Assuming the existence of a measurable cardinal, we define a hierarchy of Ramsey cardinals and a hierarchy of normal filters. We study some combinatorial properties of this hierarchy. We show that this hierarchy is absolute with respect to the Dodd-Jensen core model, extending a result of Mitchell which says that being Ramsey is absolute with respect to the core model.

An AD-like model

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 &lt; δ &lt; 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 &lt; 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.

The consistency strength of the free-subset property for ωω

A subset X of a structure S is called free in S if ∀x ∈ Xx ∉ S[X − {x}]; here, S[Y] is the substructure of S generated from Y by the functions of S. For κ, λ, μ cardinals, let Frμ(κ, λ) be the assertion:for every structure S with κ ⊂ S which has at most μ functions and relations there is a subset X ⊂ κ free in S of cardinality ≥ λ.We show that Frω(ωω, ω), the free-subset property for ωω, is equiconsistent with the existence of a measurable cardinal (2.2,4.4). This answers a question of Devlin [De].In the first section of this paper we prove some combinatorial facts about Frμ(κ, λ); in particular the first cardinal κ such that Frω(κ, ω) is weakly inaccessible or of cofinality ω (1.2). The second section shows that, under Frω(ωω, ω), ωω is measurable in an inner model. For the convenience of readers not acquainted with the core model κ, we first deduce the existence of 0# (2.1) using the inner model L. Then we adapt the proof to the core model and obtain that ωω is measurable in an inner model. For the reverse direction, we essentially apply a construction of Shelah [Sh] who forced Frω(ωω, ω) over a ground model which contains an ω-sequence of measurable cardinals. We show in §4 that indeed a coherent sequence of Ramsey cardinals suffices. In §3 we obtain such a sequence as an endsegment of a Prikry sequence.

Ramsey cardinals and constructibility

Large cardinal properties divide rather strikingly into two groups. “Small” large cardinal properties such as weak compactness always relativize to L, while in contrast “large” large cardinal properties such as Ramsey are incompatible with L. These properties seem to be similar otherwise and this sense of similarity is reinforced by the fact that many of the large cardinals do exist in the L-like model L(μ). This paper will show that the division is caused by an artificially restrictive class of “constructible” sets rather than an essential difference in the properties themselves. Specifically, we consider the class K of sets constructible from mice as defined by Dodd and Jensen [3] and proveTheorem 1. If ρ is Ramsey then ρ is Ramsey in K.A modification of the proof will showTheorem 2. If ρ is Jonson, then ρ is Ramsey in K.Since Ramsey implies Jonson this shows that the notions are equiconsistent. Theorem 2 was proved by Kunen [5] under the assumption that V = L(μ).The proof of Theorems 1 and 2 depends heavily on results of Dodd and Jensen [3] about K and mice. These results are stated without proof in §2. §2 also contains elementary (to a reader familiar with the theory of iterated ultrapowers) proofs of special cases of some of these lemmas sufficient to give a self contained proof of the following corollary of Theorem 1:Corollary. If ρ ≤ κ,ρ is Ramsey and L(μ) ⊨ μ is a measure on κ then L(μ) ⊨ ρ is Ramsey.

A Flipping Characterization of Ramsey Cardinals

Mathematical Logic QuarterlyVolume 24, Issue 1-6 p. 31-36 Article A Flipping Characterization of Ramsey Cardinals J. M. Henle, J. M. Henle Northampton and Cambridge, Mass. (U.S.A.)Search for more papers by this authorE. M. Kleinberg, E. M. Kleinberg Northampton and Cambridge, Mass. (U.S.A.)Search for more papers by this author J. M. Henle, J. M. Henle Northampton and Cambridge, Mass. (U.S.A.)Search for more papers by this authorE. M. Kleinberg, E. M. Kleinberg Northampton and Cambridge, Mass. (U.S.A.)Search for more papers by this author First published: 1978 https://doi.org/10.1002/malq.19780240105Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume24, Issue1-61978Pages 31-36 RelatedInformation

Some consequences of an infinite-exponent partition relation

Beginning with Ramsey's theorem of 1930, combinatorists have been intrigued with the notion of large cardinals satisfying partition relations. Years of research have established the smaller ones, weakly ineffable, Erdös, Jonsson, Rowbottom and Ramsey cardinals to be among the most interesting and important large cardinals in set theory. Recently, cardinals satisfying more powerful infinite-exponent partition relations have been examined with growing interest. This is due not only to their inherent qualities and the fact that they imply the existence of other large cardinals (Kleinberg [2], [3]), but also to the fact that the Axiom of Determinacy (AD) implies the existence of great numbers of such cardinals (Martin [5]).That these properties are more often than not inconsistent with the full Axiom of Choice (Kleinberg [4]) somewhat increases their charm, for the theorems concerning them tend to be a little odd, and their proofs, circumforaneous. The properties are, as far as anyone knows, however, consistent with Dependent Choice (DC).Our basic theorem will be the following: If k &gt; ω and k satisfies k→(k)k then the least cardinal δ such that has a δ-additive, uniform ultrafilter. In addition, if ACω is assumed, we will show that δ is greater than ω, and hence a measurable cardinal. This result will be strengthened somewhat when we prove that for any k, δ, if then .

Chang's conjecture and powers of singular cardinals

In [2] Galvin and Hajnal showed, as a corollary to a more general result, that if , is a strong limit cardinal, then . They established similar bounds for powers of singular cardinals of cofinality greater than ω. Jech and Prikry in [3] showed that the Galvin-Hajnal bound can be improved if we assume that ω1 carries an ω2 saturated ω1 complete, nontrivial ideal. (See [7] for definitions), namely: under the given assumption provided is a strong limit cardinal.In this paper we show that the same conclusion can be derived from Chang's Conjecture (see below) which is, at least consistencywise, a weaker assumption than the existence of an ω2 saturated ideal on ω1. We do not know if assumptions like these are necessary for obtaining the result.Our notations and terminology should be understood by any reader acquainted with set theory. Chang's Conjecture is the following model theoretic assumption introduced by C. C. Chang:which is deciphered as follows: Every structure 〈A, R,…〉 in a countable type where ∣A∣ = ω2, R ⊆ A, ∣R∣ = ω1 has an elementary substructure: 〈A′,R′,…〉 where ∣A′∣ = ω1 and ∣R′∣ = ω0. The consistency of Chang's Conjecture modulo the existence of Ramsey cardinals is claimed in [5].