Set Theory

The chapter is a concise, practical presentation of the basics of set theory. The topics include set equivalence, countability, partially ordered, linearly ordered, and well-ordered sets, the axiom of choice, and Zorn’s lemma, as well as cardinal numbers and cardinal arithmetic. The first two sections are essential for a proper understanding of the rest of the book. In particular, a thorough understanding of countability and Zorn’s lemma are indispensable. Parts of the section on cardinal numbers may be included, but only an intuitive understanding of cardinal numbers is sufficient to follow such topics as the discussion on the existence of a vector space of arbitrary (infinite) dimension, and the existence of inseparable Hilbert spaces. Cardinal arithmetic can be omitted since its applications later in the book are limited. Ordinal numbers have been carefully avoided.

The Concise Oxford Dictionary of Mathematics

Over 4,000 entries This informative A to Z provides clear, jargon-free definitions of a wide variety of mathematical terms. Its articles cover both pure and applied mathematics and statistics, and include key theories, concepts, methods, programmes, people, and terminology. For this sixth edition, around 800 new terms have been defined, expanding on the dictionary’s coverage of algebra, differential geometry, algebraic geometry, representation theory, and statistics. Among this new material are articles such as cardinal arithmetic, first fundamental form, Lagrange’s theorem, Navier-Stokes equations, potential, and splitting field. The existing entries have also been revised and updated to account for developments in the field. Numerous supplementary features complement the text, including detailed appendices on basic algebra, areas and volumes, trigonometric formulae, and Roman numerals. Newly added to these sections is a historical timeline of significant mathematicians’ lives and the emergence of key theorems. There are also illustrations, graphs, and charts throughout the text, as well as useful web links to provide access to further reading.

Neo-Fregeanism and the Burali-Forti Paradox

This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the Burali-Forti Paradox) when combined with impredicative second-order logic. It continues by arguing that the fault lies in the use of impredicative logic rather than in the abstraction principle per se. As the focus is on a form of predicative logic which reflects a philosophical diagnosis of the source of the paradox, the chapter considers how far Hale and Wright’s neo-logicist programme in cardinal arithmetic can be carried out in that logic.

DENSE IDEALS AND CARDINAL ARITHMETIC

From large cardinals we show the consistency of normal, fine, κ-complete λ-dense ideals on ${{\cal P}_\kappa }\left( \lambda \right)$ for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.

Pcf and abelian groups

We deal with some pcf (possible cofinality theory) investigations mostly motivated by questions in abelian group theory. We concentrate on applications to test problems but we expect the combinatorics will have reasonably wide applications. The main test problem is the “trivial dual conjecture” which says that there is a quite free abelian group with trivial dual. The “quite free” stands for “-free” for a suitable cardinal , the first open case is . We almost always answer it positively, that is, prove the existence of -free abelian groups with trivial dual, i.e., with no non-trivial homomorphisms to the integers. Combinatorially, we prove that “almost always” there are which are quite free and have a relevant black box. The qualification “almost always” means except when we have strong restrictions on cardinal arithmetic, in fact restrictions which hold “everywhere”. The nicest combinatorial result is probably the so-called “Black Box Trichotomy Theorem” proved in ZFC. Also we may replace abelian groups by R-modules. Part of our motivation (in dealing with modules) is that in some sense the improvement over earlier results becomes clearer in this context.

TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.