Tarski’s Theorem and NFU

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.