The Dennes Room

Event Detail

Fri Oct 26, 2018
4:10–5:15 PM
Logic Colloquium
Jouko Väänänen (University of Helsinki)
On an extension of a theorem of Zermelo

Zermelo (1930) proved the following categoricity result for set theory: Suppose M is a set and E, E’ are two binary relations on M. If both (M, E) and (M, E’) satisfy the second order Zermelo–Fraenkel axioms, then (M,E) and (M, E’) are isomorphic. Of course, the same is not true for first order ZFC. However, we show that if first order ZFC is formulated in the extended vocabulary {E,E’}, then Zermelo’s result holds even in the first order case. Similarly, Dedekind’s categoricity result (1888) for second order Peano arithmetic has an extension to a result about first order Peano.