This paper discusses a sequence of extensions ofNFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theoryNFof Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] thatNFdisproves Choice (and so proves Infinity). Even if one assumes the consistency ofNF, one is hampered by the lack of powerful methods for proofs of consistency and (...) independence such as are available for use withZFC; very clever work has been done with permutation methods, starting with [18] and [5], and exemplified more recently by [14], but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition ofNFUbelow for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent ofNF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting fromNF(see below for a statement of this axiom).We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in [13]), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects. (shrink)

Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with first-order logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to (...) computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and set-theoretic membership) by repeated applications of four operators that are analogues of the well-known operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; first-order logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. (shrink)

Hailed by the Bulletin of the American Mathematical Society as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields. Starting with an introduction to symbolic logic, the first eight chapters develop logic through the restricted predicate calculus. Topics include the statement calculus, the use of names, an axiomatic treatment of the statement (...) calculus, descriptions, and equality. Succeeding chapters explore abstract set theory—with examinations of class membership as well as relations and functions—cardinal and ordinal arithmetic, and the axiom of choice. An invaluable reference book for all mathematicians, this text is suitable for advanced undergraduates and graduate students. Numerous exercises make it particularly appropriate for classroom use. (shrink)

An ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory "New Foundations" (NF) is constructed. Marcel Crabbe has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for (...) any ordinal α. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such "tangled type theories" in general, exhibiting a "tangled type theory" (and also an extension of Zermelo set theory with Δ 0 comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that "tangled type theory with urelements" has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU. (shrink)