Abstract We provide a model theoretical and tree property-like characterization of $\lambda $ - $\Pi ^1_1$ -subcompactness and supercompactness. We explore the behavior of these combinatorial principles at accessible cardinals.

Abstract In this paper we study the notion of $C^{(n)}$ -supercompactness introduced by Bagaria in [3] and prove the identity crises phenomenon for such class. Specifically, we show that consistently the least supercompact is strictly below the least $C^{(1)}$ -supercompact but also that the least supercompact is $C^{(1)}$ -supercompact (and even $C^{(n)}$ -supercompact). Furthermore, we prove that under suitable hypothesis the ultimate identity crises is also possible. These results solve several questions posed by Bagaria and Tsaprounis.

Abstract We present an alternative proof that from large cardinals, we can force the tree property at $\kappa ^+$ and $\kappa ^{++}$ simultaneously for a singular strong limit cardinal $\kappa $ . The advantage of our method is that the proof of the tree property at the double successor is simpler than in the existing literature. This new approach also works to establish the result for $\kappa =\aleph _{\omega ^2}$ .

Abstract We construct a model in which the tree property holds in ${\aleph _{\omega + 1}}$ and it is destructible under $Col\left( {\omega ,{\omega _1}} \right)$ . On the other hand we discuss some cases in which the tree property is indestructible under small or closed forcings.

Abstract Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at ${\kappa ^{ + + }}$ , assuming that $\kappa = {\kappa ^{ < \kappa }}$ and there is a weakly compact cardinal aboveκ. If in additionκis supercompact then we can forceκto be ${\aleph _\omega }$ in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a ${\kappa ^{ + + }}$ -Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into ${\aleph _\omega }$ .

Abstract We show that from infinitely many supercompact cardinals one can force a model of ZFC where both the tree property and the stationary reflection hold at אω2+1.

Abstract An uncountable cardinal κ is called ${\omega _1}$ -strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$ -complete ultrafilter on I. We show that the first ${\omega _1}$ -strongly compact cardinal, ${\kappa _0}$ , cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$ . We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$ . Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$ .

Abstract There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH. where α is a cardinal at most κ++. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = α++, the maximum possible) and [1] (for α = κ+, after collapsing κ++). In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of Mitchell order α), [2] (as in [12], but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and [11] (as in [12], but where κ is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing. The proof uses κ-Sacks forcing and the “tuning fork” technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails.

This note proves two theorems. The first is that it is consistent to have for every n, but not have . This is done by carefully collapsing a supercompact cardinal and adding square sequences to each ωn. The crux of the proof is that in the resulting model every stationary subset of ℵω+1 ⋂ cof(ω) reflects to an ordinal of cofinality ω1, that is to say it has stationary intersection with such an ordinal. This result contrasts with compactness properties of square shown in [3]. In that paper it is shown that if one has square at every ωn, then there is a square type sequence on the points of cofinality ωk, k > 1 in ℵω+1. In particular at points of cofinality greater than ω1 there is a strongly non-reflecting stationary set of points of countable cofinality. The second result answers a question of Džamonja, by showing that there can be no squarelike sequence above a supercompact cardinal, where “squarelike” means that one replaces the requirement that the cofinal sets be closed and unbounded by the requirement that they be stationary at all points of uncountable cofinality.

Abstract. If ωn has the tree property for all 2 ≤ n < ω and , then for all and n < ω. Mnt(X) exists.

Abstract A vast and interesting family of natural semantics lor belief revision is defined. Suppose one is given a distance d between any two models. One may then define the revision of a theory K by a formula α as the theory defined by the set of all those models of α that are closest, by d. to the set of models of K. This family is characterized by a set of rationality postulates that extends the AGM postulates. The new postulates describe properties of iterated revisions.

In this paper we propose a new form of polymorphism for object-oriented languages, called correspondence polymorphism. It lies in a different dimension than either parametric or subtype polymorphism. In correspondence polymorphism, some methods are declared to correspond to other methods, via a correspondence relation. With this relation, it is possible to reuse non-generic code in various type contexts—not necessarily subtyping or matching contexts—without having to plan ahead for this reuse. Correspondence polymorphism has advantages over other expressive object type systems in that programmer-declared types still may be simple, first-order types that are easily understood. We define a simple language LCP that reflects these new ideas, illustrating its behavior with multiple examples. We present formal type rules and an operational semantics for LCP, and establish soundness of the type system with respect to reduction.

Abstract In this paper we prove the independence of for n ≥ 3. We show that can be forced to be above any ordinal of L using set forcing. For we prove that it can be forced, using set forcing, to be above any L cardinal κ such that κ is Π1 definable without parameters in L. We then show that cannot be forced by a set forcing to be above every cardinal of L Finally we present a class forcing construction to make greater than any given L cardinal.

In this paper we explicate a very weak version of the principle □ discovered by Jensen who proved it holds in the constructible universe L. This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals. In this section we show that this principle is equivalent to a common combinatorial device, which we call a Jensen matrix. In the second section we show that our principle is consistent with a supercompact cardinal. In the third section of this paper we show that this principle is exactly equivalent to the statement that every torsion free Abelian group has a filtration into σ-balanced subgroups. In the fourth section of this paper we show that this principle fails if you assume the Chang's Conjecture: In the fifth section of the paper we review the proofs that the various weak squares we consider are strictly decreasing in strength. Section 6 was added in an ad hoc manner after the rest of the paper was written, because the subject matter of Theorem 6.1 fit well with the rest of the paper. It deals with a principle dubbed “Not So Very Weak Square”, which appears close to Very Weak Square but turns out not to be equivalent.

Abstract The paper is a continuation of [The SCH revisited], In § 1 we define a forcing with countably many nice systems. It is used, for example, to construct a model “GCH below κ, c f κ = ℵ0, and 2κ > κ+ω” from 0(κ) = κ+ω. In §2 we define a triangle iteration and use it to construct a model satisfying “{μ ≤ λ∣c f μ = ℵ0 and pp(μ) > λ} is countable for some λ”. The question of whether this is possible was asked by S. Shelah. In §3 a forcing for blowing the power of a singular cardinal without collapsing cardinals or adding new bounded subsets is presented. Answering a question of H. Woodin, we show that it is consistent to have “c f κ = ℵ0. GCH below κ, 2κ > κ+, and ”. In §4 a variation of the forcing of [The SCH revisited, §1] is defined. It behaves nicely in iteration processes. As an application, we sketch a construction of a model satisfying: “κ is a measurable and 2κ ≥ κ+α for some α, κ < c f α < α” starting with 0(κ) = κ+α. This answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.

Abstract We show that relative to the consistency of a supercompact cardinal does not imply . The model-theoretic transfer property ⟨ℵ1, ℵ0⟩ → ⟨ℵω + 1, ℵω⟩ does not imply , and it is consistent to have an ultrafilter on ℵω + 1 which is λ-indecomposible for all ω < λ < ℵω.

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 either a) there is a real number r ∈ V [G] with r ∉ V, or b) 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.

Abstract It is proven that the following statement: “there exists a club C ⊆ κ such that every α ∈ C is an inaccessible cardinal in L and, for every δ a limit point of C, C ∩ δ is almost contained in every club of δ of L” is equiconsistent with a weakly compact cardinal if δ = ℵ1, and with a weakly compact cardinal of order 1 if δ = ℵ2.

Abstract Assume ZFC + “There is a weakly compact cardinal” is consistent. Then: (i) For every S ⊆ ω, ZFC + “S and the monadic theory of ω2 are recursive each in the other” is consistent; and (ii) ZFC + “The full second-order theory of ω2 is interpretable in the monadic theory of ω2” is consistent.

Abstract We characterize the ordinals α of uncountable cofinality such that α is the standard part of a nonstandard model of ZFC (or equivalently KP).

Abstract We prove that the statement “For every pair A, B, stationary subsets of ω2, composed of points of cofinality ω, there exists an ordinal α such that both A ∩ α and B ∩ α are stationary subsets of α is equiconsistent with the existence of weakly compact cardinal. (This completes results of Baumgartner and Harrington and Shelah.) We also prove, assuming the existence of infinitely many supercompact cardinals, the statement “Every stationary subset of ωω+1 has a stationary initial segment.”

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.

Let us consider the following infinite game between two players, Empty and Nonempty. We are given a large set S. Empty opens the game by choosing a large subset S0 of S; then Nonempty chooses a large set S1 ⊆ S0; then Empty chooses large S2 ⊆ S, etc. The game is over after ω moves. If ⋂n=0xSn is empty then Empty wins, and if ⋂n=0∞Sn is nonempty then Nonempty wins. If “large” means “infinite”, then Empty can beat Nonempty rather easily: he chooses So countable, S0 = {a0, a1,…, an,…}, and then he chooses S2 such that a0 ∉ S2, S4 such that a1, ∉ S4 and so on. Next we assume that S is a set of uncountable cardinality, and that “large” means “of cardinality ∣S∣”. Then still Empty can win, but his winning strategy is somewhat more sophisticated: Let us identify S with a cardinal number κ. Thus each subset of S of size κ is a set of ordinals below κ. For each X ⊆ κ of size κ, let fx be the unique order-preserving mapping of X onto κ, and let F(X) = {x ϵ X: f(x) is a successor ordinal}. Empty's strategy is to play S0 = F(K), and when Nonempty plays S2k − 1, let S2k = F(S2k − 1).

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].

The language Ln is obtained from the first order predicate calculus by adjoining the quantifier Qn which binds n variables. The formula Qnυ1 … υnΨ is given a κ-interpretation for each infinite cardinal κ, namely, “there is a set X of power κ such that Ψx1 … xn holds for all distinct x1 … xn ϵ X”. L<ω is the result of adjoining all the Qn quantifiers for each n ϵ ω to the first order predicate calculus. In [4] we showed that under the assumption (cf. [3]) L<ω is countably compact under the ω1-interpretation, and that any sentence σ ϵ L<ω that has a model in some κ-interpretation where κ is a regular infinite cardinal has a model in the ω1 interpretation. However, compactness for L<ω in the κ-interpretation for κ an infinite successor cardinal other than ω1 and the transfer of satisfiability from ω1 to any higher power remain open questions under any set theoretic assumptions. Here we restrict our attention to a small fragment L2− of L2 consisting of universal first order formulas along with formulas of the kind Q2υ1υ2∀υ3 … υnΨ and ¬Q2υ1υ2∀υ3 … υnφ where Ψ and φ are open and no function symbol of arity > 1 occurs in any formula. Assuming the existence of a κ-Souslin tree, this language is λ compact in the κ-interpretation when λ < κ.