•  11
    Infinite trace equivalence
    Annals of Pure and Applied Logic 151 (2-3): 170-198. 2008.
    We solve a longstanding problem by providing a denotational model for nondeterministic programs that identifies two programs iff they have the same range of possible behaviours. We discuss the difficulties with traditional approaches, where divergence is bottom or where a term denotes a function from a set of environments. We see that making forcing explicit, in the manner of game semantics, allows us to avoid these problems.We begin by modelling a first-order language with sequential I/O and un…Read more
  •  14
    The Price of Mathematical Scepticism
    Philosophia Mathematica 30 (3): 283-305. 2022.
    This paper argues that, insofar as we doubt the bivalence of the Continuum Hypothesis or the truth of the Axiom of Choice, we should also doubt the consistency of third-order arithmetic, both the classical and intuitionistic versions. Underlying this argument is the following philosophical view. Mathematical belief springs from certain intuitions, each of which can be either accepted or doubted in its entirety, but not half-accepted. Therefore, our beliefs about reality, bivalence, c…Read more
  •  28
    ZFC has sentences that quantify over all sets or all ordinals, without restriction. Some have argued that sentences of this kind lack a determinate meaning. We propose a set theory called TOPS, using Natural Deduction, that avoids this problem by speaking only about particular sets.