Thomas Jech

About the Axiom of Choice

Publisher Summary The axiom of choice is crucial not only in logic (set theory and model theory) but also in other modern disciplines as well such as point set topology, algebra, functional analysis, and measure theory. This chapter presents examples of fundamental theorems of abstract algebra and topology whose proofs use the axiom of choice. In some instances, the theorems are as strong as the axiom of choice—an example of a statement equivalent to the axiom of choice is the Tychonoff product theorem in point set topology. Some objections to the axiom of choice are based on the fact that the axiom has paradoxical consequences. The most famous example is Banach–Tarspkai paradox. The Banach–Tarspkai paradox states that using the axiom of choice, one can cut a ball into a finite number of pieces that can be so rearranged that one obtains two balls of the same size as the original ball. The chapter also sketches the proof of this paradox to show how the axiom of choice is used and that there is nothing paradoxical about this theorem.

More game-theoretic properties of boolean algebras

Abstract The following infinite game G was investigated in [5]: Let B be a Boolean algebra. Two players, White and Black, take turns to choose successively a sequence

Singular Cardinals and the pcf Theory

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Weak distributivity, a problem of Von Neumann and the mystery of measurability

�DedicatedtoDorothyMaharamStone This article investigates the weak distributivity of Booleano-algebras satisfying the countable chain condition. It addresses primarily the question when such algebras carry ao-additive measure. We use as a starting point the problem of John von Neumann stated in 1937 in the Scottish Book. He asked if the countable chain condition and weak distributivity are sufficient for the existence of such a measure. Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and the combinatorial problems. We shall survey the history of von Neumann’s Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam to whom we dedicate this article. §1. CompleteBooleanalgebrasandweakdistributivity. ABooleanalgebra

A game theoretic property of Boolean algebras1)

Publisher Summary This chapter introduces an infinite game played on a Boolean algebra, and investigates the properties of Boolean algebras defined in terms of existence of winning strategies in the game. The chapter demonstrates that the properties are closely related to distributivity laws for Boolean algebras. The chapter illustrates an example of a Boolean algebra for which the game presented in the chapter is not determined. The chapter defines the game GB for Boolean algebras, to compare the existence of strategies with standard Boolean algebraic properties.

Simple complete Boolean algebras

IfV=L, and κ is an uncountable regular non weakly compact cardinal, then there exists a simple complete Boolean algebra of cardinality κ.

Semi-Cohen Boolean algebras☆

Abstract 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.

OTTER experiments in a system of combinatory logic

This paper describes some experiments involving the automated theorem-proving program OTTER in the system TRC of illative combinatory logic. We show how OTTER can be steered to find a contradiction in an inconsistent variant of TRC, and present some experimentally discovered identities in TRC.

Full Reflection of Stationary Sets Below

It is consistent that, for every n ≥ 2, every stationary subset of ω n consisting of ordinals of cofinality ω κ , where κ = 0 or κ ≤ n − 3, reflects fully in the set of ordinals of cofinality ω n −1 . We also show that this result is best possible.

alephomega

Abstract We consider several infinite games involving a given κ-complete ideal over a regular uncountable cardinal κ. We give a new characterization of precipitous ideals and introduce the class of weakly precipitous and pseudo-precipitous ideals. We also define the notion of degree of functions and functionals and compare it with the Galvin-Hajnal norm.