Abstract: Dedekind’s "Was sind und was sollen die Zahlen?" is well known as presenting second order axioms for natural numbers, and proving their categoricity. This is also the work where the notion of “Dedekind finite set” is introduced. Perhaps less well known is that Dedekind also gives a full proof of the recursion theorem to justify the iteration of functions over the naturals. (Dedekind’s iteration of functions is nearly, but not quite the same as the familiar scheme of primitive recursion. The differences are quite interesting.) Perhaps even more interesting is that Dedekind also gives a “well-ordering theorem” for Dedekind finite sets. Of course (as we now know) this result requires some form of the axiom of choice and it is instructive to see where AC is implicitly used in Dedekind’s proof. I will lead this discussion, attempting also to put this work into historical context as far as possible. This should be of interest to many people involved in logic, history of maths, philosophy and computer science, and all are welcome.
Title : Incompleteness via paradox (and completeness)
Abstract: This talk will explore a method for uniformly transforming the paradoxes of naive set theory and semantics into formal incompleteness results originally due to Georg Kreisel and Hao Wang. I will first trace the origins of this method in relation to Gödel’s proof of the completeness theorem for first-order logic and its subsequent arithmetization by Hilbert and Bernays in their Grundlagen der Mathematik. I will then describe how the method can be applied to construct arithmetical statements formally independent of systems of set theory and second-order arithmetic via formalizations of Russell’s paradox and the Liar (and time permitting also the Skolem and Richard paradoxes). Finally, I will consider the significance of these results relative to both the Hilbert program and subsequent work in predicative mathematics.
Abstract: We articulate the view that mutually interpretable statements have identical meanings. We then argue against this view by showing that it would make unintelligible an important mathematical practice, namely that of purity of methods.
Dr Walter Dean
Department of Philosophy
University of Warwick