Midlands Logic Seminar

Week 7, 15 November

Mini workshop workshop on the significance of the soundness and completeness theorems

Study group: TBC

4:00-5:00 Walter Dean (Warwick)
"Arithmetical reflection, induction, and the provability of soundness"

A reflection principle is a statement or schema which seeks to express the soundness of a mathematical theory T within its own language. For instance, the so-called local reflection principle for Peano arithmetic (i.e. Prov(A) -> A) can be understood to assert that any sentence provable in PA is true in the standard model of arithmetic.

In this talk, I will first seek to highlight a tension between the original technical uses of reflection principles in proof theory (which primarily pertain to proofs of non-finite axiomatizability) and their more recent philosophical appropriation in debates about the role of the concept of truth in mathematical reasoning (wherein it is often claimed that acceptance of a theory T entails commitment to some form of reflection principle for T). I will next argue (on the basis of results of Kreisel, Lévy, Schmerl, and others) that the justification of reflection principles is closely related to the justification of induction (both mathematical and transfinite).

On this basis, I will suggest that the task of accounting for our (putative) knowledge of reflection principles may not be as straightforward as it might at first appear. I will additionally suggest that this motivates the consideration of various "non-canonical" definitions of arithmetical provability - e.g. based on cut-free or Herbrand provability - relative to which appropriate formulations of consistency are provable.

5:00-6:00 Richard Kaye (Birmingham)
"Circularity in Soundness and Completeness"

We raise an issue of circularity in the argument for the completeness of first order logic. An analysis of the problem sheds light on the development of mathematics, and suggests other possible directions for foundational research.