This is a survey paper giving a self-contained account of Shelah's theory of the pcf function pcf={cf:D is an ultrafilter on a}, where a is a set of regular cardinals such that a

Logic and Philosophy of Logic

Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic

Model Theory in Logic and Philosophy of Logic

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We study various aspects of the size, including the cardinality, of closed unbounded subsets of [λ]<κ, especially when λ = κ+n for n ε ω. The problem is resolved into the study of the size of certain stationary sets. Relative to the existence of an ω1-Erdös cardinal it is shown consistent that ωω3 < ωω13 and every closed unbounded subsetof [ω3]<ω2 has cardinality ωω13. A weakening of the ω1-Erdös property, ω1-remarkability, is defined and shown to be retained under a large (...) class of Easton-like forcings applied to ω1-Erdös cardinals. A class of reverse-Easton forcings preserving α-Erdösness is also described, with special attention to the establishment of □-principles. Article Outline. (shrink)

We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.

A forcing poset of size 221 which adds no new reals is described and shown to provide a Δ22 definable well-order of the reals . The encoding of this well-order is obtained by playing with products of Aronszajn trees: some products are special while other are Suslin trees. The paper also deals with the Magidor–Malitz logic: it is consistent that this logic is highly noncompact.

Assuming the existence of a supercompact cardinal and a weakly compact cardinal above it, we provide a generic extension where there are no Aronszajn trees of height ω 2 or ω 3 . On the other hand we show that some large cardinal assumptions are necessary for such a consistency result.

We show that all posets of uniform density ℵ 1 may have to add a Cohen real and develop some forcing machinery for obtaining this sort of result.

The cut and choose game is one of the infinitary games on a complete Boolean algebra B introduced by Jech. We prove that existence of a winning strategy for II in implies semiproperness of B. If the existence of a supercompact cardinal is consistent then so is “for every 1-distributive algebra B II has a winning strategy in ”.

The relationship between killing ideals and adding reals by forcings is analysed.

We continue [21] and study partition numbers of partial orderings which are related to /fin. In particular, we investigate Pf, be the suborder of /fin)ω containing only filtered elements, the Mathias partial order M, and , ω the lattice of partitions of ω, respectively. We show that Solomon's inequality holds for M and that it consistently fails for Pf. We show that the partition number of is C. We also show that consistently the distributivity number of ω is smaller than (...) the distributivity number of /fin. We also investigate partitions of a Polish space into closed sets. We show that such a partition either is countable or has size at least D, where D is the dominating number. We also show that the existence of a dominating family of size 1 does not imply that a Polish space can be partitioned into 1 many closed sets. (shrink)

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)

We define the property of Π2-compactness of a statement Φ of set theory, meaning roughly that the hard core of the impact of Φ on combinatorics of 1 can be isolated in a canonical model for the statement Φ. We show that the following statements are Π2-compact: “dominating NUMBER = 1,” “cofinality of the meager IDEAL = 1”, “cofinality of the null IDEAL = 1”, “bounding NUMBER = 1”, existence of various types of Souslin trees and variations on uniformity of (...) measure and CATEGORY = 1. Several important new metamathematical patterns among classical statements of set theory are pointed out. (shrink)

In this paper, we study the relationship between the cofinalityc(Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{b} $$ of an unbounded familyF of ω ω.

Schlindwein, C., Suslin's hypothesis does not imply stationary antichains, Annals of Pure and Applied Logic 64 153–167. Shelah has shown that Suslin's hypothesis does not imply every Aronszajn tree is special. We improve this result by constructing a model of Suslin's hypothesis in which some Aronszajn tree has no antichain whose levels constitute a stationary set. The main point is a new preservation theorem, the proof of which illustrates the usefulness of certain ideas in [8, Section 1].

We study the fine structure of the core model for one Woodin cardinal, building of the work of Mitchell and Steel on inner models of the form . We generalize to some combinatorial principles that were shown by Jensen to hold in L. We show that satisfies the statement: “□κ holds whenever κ the least measurable cardinal λ of order λ++”. We introduce a hierarchy of combinatorial principles □κ, λ for 1 λ κ such that □κ□κ, 1 □κ, λ □κ, (...) κ□κ*. We prove that if holds in V. As an application, we show that ZFC + PFA Con. We also obtain one Woodin cardinal as a lower bound on the consistency strength of stationary reflection at κ+ for a singular, countably closed limit cardinal κ such that # exists; likewise for the failure of □κ* at such a κ. (shrink)

Assume Q is a definable subset of a model of T. We define a notion of Q-isolated type, generalizing an earlier definition for countable Q. This notion is absolute. For superstable T, we give some sufficient conditions for the existence of Q-atomic models. We apply this to prove some results on weak categoricity over a predicate.

We pursue the study of families of functions on the natural numbers, with emphasis here on the bounded families. The situation being more complicated than the unbounded case, we attack the problem by classifying the families according to their bounding and dominating numbers, the traditional scheme for gaps. Many open questions remain.

If a˜cardinal κ1, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ0 and its new cofinality, ρ, is less than κ0, then, under some additional assumptions, each cardinal λ>κ1 less than cc/[κ1]<κ1) is collapsed to κ0 as well. If in addition N=M[f], where f : ρ→κ1 is an unbounded mapping, then N is a˜λ=κ0-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba.

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

We study the gameGfin related to weak distributivity of a given Boolean algebraB. Consider the following implication: ifBis weakly -distributive then player I does not have a winning strategy in the gameGfin. We show that this implication is true for properBbut it is not true in general.

We show the Borel Conjecture is consistent with the continuum large.

In this paper we probe the possibilities of creating a Kurepa tree in a generic extension of a ground model of CH plus no Kurepa trees by an ω1-preserving forcing notion of size at most ω1. In Section 1 we show that in the Lévy model obtained by collapsing all cardinals between ω1 and a strongly inaccessible cardinal by forcing with a countable support Lévy collapsing order, many ω1-preserving forcing notions of size at most ω1 including all ω-proper forcing notions (...) and some proper but not ω-proper forcing notions of size at most ω1 do not create Kurepa trees. In Section 2 we construct a model of CH plus no Kurepa trees, in which there is an ω-distributive Aronszajn tree such that forcing with that Aronszajn tree does create a Kurepa tree in the generic extension. At the end of the paper we ask three questions. (shrink)

Assuming Martin's Axiom, we compute the value of the cofinality of the symmetric group on the natural numbers. We also show that Martin's Axiom does not decide the value of the covering number of a related Mycielski ideal.

We show that it is consistent that the reaping number r is less than u , the size of the smallest base for an ultrafilter. To show that our forcing preserves certain ultrafilters, we prove a general partition theorem involving Ramsey ideals.

We show that o = k++ is necessary for ¬SCH. Together with previous results it provides the exact strenght of ¬SCH.

We isolate a condition on a class A of ordinals sufficient to Δ1-code it by a real in a class-generic extension of L. We then apply this condition to show that the class of ordinals of L-cofinality ω is Δ1 in a real of L-degree strictly below O#.

We isolate a condition on a class A of ordinals sufficient to Δ1-code it by a real in a class-generic extension of L. We then apply this condition to show that the class of ordinals of L-cofinality ω is Δ1 in a real of L-degree strictly below O#.

In this paper we consider whether L(R) has “enough information” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, non-reflecting stationary sets, generalizations of proper and semiproper forcing and Chang's conjecture.

It is shown to be consistent that countable, Fréchet,α 1-spaces are first countable. The result is obtained by using a countable support iteration of proper partial orders of lengthω 2.

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)