The paper is concerned with methods for blowing power of singular cardinals using short extenders. Thus, for example, starting with κ of cofinality ω with {α<κ oα+n} cofinal in κ for every n<ω we construct a cardinal preserving extension having the same bounded subsets of κ and satisfying 2κ=κ+δ+1 for any δ<1.

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

Given a supercompact cardinal κ and a regular cardinal Λ < κ, we describe a type of forcing such that in the generic extension the cofinality of κ is Λ, there is a very good scale at κ, a bad scale at κ, and SCH at κ fails. When creating our model we have great freedom in assigning the value of 2κ, and so we can make SCH hold or fail arbitrarily badly.

The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove (...) from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ. (shrink)

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 $2^\kappa > \kappa^{+\omega}$" 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 $pp(\mu) > \lambda\}$ 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 $\kappa, 2^\kappa > \kappa^+$, and ¬□* κ". 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 $\alpha, \kappa < c f \alpha < \alpha$" starting with 0(κ) = κ+α. This answers the question from Gitik's On measurable cardinals violating the continuum hypothesis. (shrink)

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to prove (...) some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals. (shrink)

We present a survey of combinatorial set theory relevant to the study of singular cardinals and their successors. The topics covered include diamonds, squares, club guessing, forcing axioms, and PCF theory.

The rank-into-rank and stronger large cardinal axioms assert the existence of certain elementary embeddings. By the preservation of the large cardinal properties of the embeddings under certain operations, strong implications between various of these axioms are derived.

This paper attempts to present and organize several problems in the theory of Singular Cardinals. The most famous problems in the area (bounds for the ℶ-function at singular cardinals) are well known to all mathematicians with even a rudimentary interest in set theory. However, it is less well known that the combinatorics of singular cardinals is a thriving area with results and problems that do not depend on a solution of the Singular Cardinals Hypothesis. We present here an annotated collection (...) of representative problems with some references. Where the problems are novel, attribution is attempted and it is noted where money is attached to particular problems. Three closely related themes are represented in these problems: stationary sets and stationary set reflection, variations of square and approachability, and the singular cardinals hypothesis. Underlying many of them are ideas from Shelah's PCF theory. Important subthemes were mutual stationarity, Aronszajn trees, and superatomic Boolean Algebras. The author notes considerable overlap between this paper and the unpublished report submitted to the Banff Center for the Workshop on Singular Cardinals Combinatorics, May 1–5, 2004. (shrink)

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings,j, from the universe,V, into some transitive submodel,M. See Reinhardt–Solovay [7] for more details. Ifjis not the identity, andκis the first ordinal moved byj, thenκis a measurable cardinal. Conversely, Scott [8] showed that wheneverκis measurable, there is suchjandM. If we had assumed, in addition, that, thenκwould be theκth measurable cardinal; in general, the wider we assumeMto be, the largerκmust be.

We prove a number of consistency results complementary to the ZFC results from our paper [J. Cummings, M. Foreman, M. Magidor, Canonical structure in the universe of set theory: part one, Annals of Pure and Applied Logic 129 211–243]. We produce examples of non-tightly stationary mutually stationary sequences, sequences of cardinals on which every sequence of sets is mutually stationary, and mutually stationary sequences not concentrating on a fixed cofinality. We also give an alternative proof for the consistency of the (...) existence of stationarily many non-good points, show that diagonal Prikry forcing preserves certain stationary reflection properties, and study the relationship between some simultaneous reflection principles. Finally we show that the least cardinal where square fails can be the least inaccessible, and show that weak square is incompatible in a strong sense with generic supercompactness. (shrink)