set theory and its philosophy a critical introduction mar 2004

[...]... of names 336 This page intentionally left blank Part I Sets This page intentionally left blank Introduction to Part I This book, as its title declares, is about sets; and sets, as we shall use the term here, are a sort of aggregate But, as a cursory glance at the literature makes clear, aggregation is far from being a univocal notion Just what sort of aggregate sets are is a somewhat technical matter,... however, is that what mathematicians say is no more reliable as a guide to the interpretation of their work than what artists say about their work, or musicians So we certainly should not automatically take mathematicians at their settheoretically reductionist word And there have in any case been notable recusants throughout the period, such as Mac Lane (1986) and Mayberry (1994) Nevertheless, we shall need... But once again it is at least debatable whether set theory can indeed have the role that is ascribed to it: it is far from clear that the axiom of choice is correctly regarded as a set- theoretic principle at all, and similar doubts may be raised about other purported applications of set- theoretic principles in mathematics These three roles for set theory — as a means of taming the infinite, as a supplier... set theory are littered with variants of this claim: one of them states baldly that set theory is the foundation of mathematics’ (Kunen 1980, p xi), and similar claims are to be found not just (as perhaps one might expect) in books written by set theorists but also in many mainstream mathematics books Indeed this role for set theory has become so familiar that hardly anybody who gets as far as reading... a theory that claims to be foundational If we embed mathematics in set theory and treat set theory implicationally, then mathematics — all mathematics — asserts only conditional truths about structures of a certain sort But our metalinguistic study of set- theoretic structures is plainly recognizable as a species of mathematics So we have no reason not to suppose that here too the correct interpretation... conditional At no point, then, will mathematics assert anything unconditionally, and any application of any part whatever of mathematics that depends on the unconditional existence of mathematical objects will be vitiated Thoroughgoing implicationism — the view that mathematics has no subject matter whatever and consists solely of the logical derivation of consequences from axioms — is thus a very harsh... influenced many mathematicians was the philosophy of mathematics which led Bourbaki to adopt a first-order formulation of his system This philosophy has at its core a conception of rigour that is essentially formalist in character In an unformalized mathematical text, he says, one is exposed to the danger of faulty reasoning arising from, for example, incorrect use of intuition or argument by analogy In practice,... corralling mathematics in such a way that nothing within its boundary is open to philosophical dispute This seems to be the content of the observation, which crops up repeatedly in the mathematical literature, that mathematicians are platonists on weekdays and formalists on Sundays: if a mathematical problem is represented as amounting to the question whether a particular sentence is a theorem of a certain... to a variant of Poincar´ ’s petitio since e we presuppose the notion of infinity in our attempt to characterize it But once again Poincar´ ’s objection can be met by distinguishing carefully between obe ject language and metalanguage We cannot in the language itself assert an infinite list of object language sentences, but we can in the metalanguage make a commitment to assert any member of such a list... have been in the least surprising, for we shall prove later in this book that if there are infinitely many objects, then (at least on the standard understanding of the second-order quantifier) there are uncountably many properties those objects may have A first-order scheme, on the other hand, can only have countably many instances (assuming, as we normally do, that the language of the theory is countable) .