The Review of Symbolic Logic

Research Article

ULTIMATE TRUTH VIS-À-VIS STABLE TRUTH

P. D. WELCHa1 c1

a1 School of Mathematics, University of Bristol

Abstract

We show that the set of ultimately true sentences in Hartry Field's Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger's revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second-order number theory is needed to establish the semantic values of sentences in Field's relative consistency proof of his theory over the ground model of the standard natural numbers: \Delta _3^1-CA0 (second-order number theory with a \Delta _3^1-comprehension axiom scheme) is insufficient. We briefly consider his claim to have produced a ‘revenge-immune’ solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a ‘determinately true’ operator can be introduced in other settings.

(Received April 18 2008)

Correspondence:

c1 SCHOOL OF MATHEMATICS, UNIVERSITY OF BRISTOL, BRISTOL BS8 1TW, UK. E-mail: p.welch@bristol.ac.uk