Abstract
The Tarski paradox of the undefinability of truth is proved by a diagonalization argument similar to the argument of Russell’s paradox. In ZFC, Russell’s argument shows that the universal class (and large classes generally) do not exist. In other set theories, such as Jensen’s variant NFU of Quine’s “New Foundations”, large classes such as the universe may exist; the diagonalization arguments lead to somewhat different restrictions on the existence of sets in the presence of different axioms. In this paper, we explore the possibility that semantics expressed in NFU may have somewhat different restrictions imposed on them by the diagonalization argument of Tarski. A language L is definable in NFU, in which the stratified sentences of the language of NFU can be encoded (but, it should be noted, as a proper subclass of L). Truth for sentences in L is definable in NFU, and the reason that a suitably adapted Tarski argument fails to lead to paradox is not that truth for L is undefinable in NFU, but that quotation becomes a type-raising operation, causing the predicate needed for the “Tarski sentence” to be unstratified.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andrews, P. 1986 An introduction to mathematical logic and type theory: To truth through proof, Academic Press, New York, pp. 264–265.
Forster, T. 1992 Set theory with a universal set, Oxford Logic Guides, no. 20, Clarendon Press, London.
Grishin, V. N. 1972 The equivalence of Quine’s NF system to one of its fragments (in Russian), Nauchnotekhnicheskaya Informatsiya, series 2, vol. 1, pp. 22–24.
Hiller, A. P., and J. P. Zimbarg 1984 Self-reference with negative types, The Journal of Symbolic Logic, vol. 49, pp. 754–773.
Jensen, R. B. 1969 On the consistency of a slight(?) modification of Quine’s NF, Synthese, vol. 19, pp. 250–263.
Orey, S. 1964 New Foundations and the axiom of counting, Duke Mathematical Journal, vol. 31, pp. 655–660.
Quine, W. V. 1937 New foundations for mathematical logic, American Mathematical Monthly, vol. 44, pp. 70–80.
Rosser, J. B. 1953 Logic for mathematicians, McGraw-Hill, New York; and Chelsea, New York, 1978.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Holmes, M.R. (2001). Tarski’s Theorem and NFU . In: Anderson, C.A., Zelëny, M. (eds) Logic, Meaning and Computation. Synthese Library, vol 305. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0526-5_22
Download citation
DOI: https://doi.org/10.1007/978-94-010-0526-5_22
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-3891-1
Online ISBN: 978-94-010-0526-5
eBook Packages: Springer Book Archive