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)

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)

A strong coloring on a cardinal $\kappa $ is a function $f:[\kappa ]^2\to \kappa $ such that for every $A\subseteq \kappa $ of full size $\kappa $, every color $\unicode{x3b3} <\kappa $ is attained by $f\restriction [A]^2$. The symbol $$ \begin{align*} \kappa\nrightarrow[\kappa]^2_{\kappa} \end{align*} $$ asserts the existence of a strong coloring on $\kappa $.We introduce the symbol $$ \begin{align*} \kappa\nrightarrow_p[\kappa]^2_{\kappa} \end{align*} $$ which asserts the existence of a coloring $f:[\kappa ]^2\to \kappa $ which is strong over a partition $p:[\kappa ]^2\to (...) \theta $. A coloring f is strong over p if for every $A\in [\kappa ]^{\kappa }$ there is $i<\theta $ so that for every color $\unicode{x3b3} <\kappa $ is attained by $f\restriction )$.We prove that whenever $\kappa \nrightarrow [\kappa ]^2_{\kappa }$ holds, also $\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$ holds for an arbitrary finite partition p. Similarly, arbitrary finite p-s can be added to stronger symbols which hold in any model of ZFC. If $\kappa ^{\theta }=\kappa $, then $\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$ and stronger symbols, like $\operatorname {Pr}_1_p$ or $\operatorname {Pr}_0_p$, also hold for an arbitrary partition p to $\theta $ parts.The symbols $$ \begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0_p \end{gather*} $$ hold for an arbitrary countable partition p under the Continuum Hypothesis and are independent over ZFC $+ \neg $ CH. (shrink)

We revisit several results concerning club principles and nonsaturation of the nonstationary ideal, attempting to improve them in various ways. So we typically deal with a ideal J extending the nonstationary ideal on a regular uncountable cardinal \, our goal being to witness the nonsaturation of J by the existence of towers ).

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ⋄-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of ⋄. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of a strongly inaccessible cardinal. We (...) also present a new construction of a Souslin tree with an ascent path, thereby increasing the consistency strength of such a tree's nonexistence from a Mahlo cardinal to a weakly compact cardinal. Section 2 of this paper is targeted at newcomers with minimal background. It offers a comprehensive exposition of the subject of constructing Souslin trees and the challenges involved. (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)

It is relatively consistent with the existence of two supercompact cardinals that a special Aronszajn tree of height ${\aleph_{\omega_1+1}}$ is introduced by a cofinality-preserving forcing of size ${\aleph_3}$.

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure (...) of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for manyλ ⌝ PT(λ, ℵ1). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular λ which is a limit of supercompacts the bad stationary set concentrates on the “right” cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular λ {δ<+ : cfδ<λ} is good. It is proved in ZFC that ifλ=cfλ>ℵ1 then {α<+ : cfα<λ} is the union of λ sets on which there are squares. (shrink)

Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square _{\lambda}^{\ast}$ together with 2 λ = λ⁺ implies $\lozenge _{S}$ for every S ⊆ λ⁺ that reflects stationarily often. In this paper, for a set S ⊆ λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S; λ]. We say that the ideal is fat if it contains a stationary set. It is proved: 1. if I[S; λ] is fat, (...) then $\text{NS}_{\lambda ^{+}}$ ↾ S is non-saturated; 2. if I[S; λ] is fat and 2 λ = λ⁺, then $\lozenge _{S}$ holds; 3. $\square _{\lambda}^{\ast}$ implies that I[S; λ] is fat for every S ⊆ λ⁺ that reflects stationarily often; 4. it is relatively consistent with the existence of a supercompact cardinal that $\square _{\lambda}^{\ast}$ fails, while I[S; λ] is fat for every stationary S ⊆ λ⁺ that reflects stationarily often. The stronger principle $\lozenge _{\lambda ^{+}}^{\ast}$ is studied as well. (shrink)