Bulletin of Symbolic Logic

Research Article

Second-Order Logic and Foundations of Mathematics

Jouko Väänänen

Department of Mathematics, University of Helsinki, Helsinki, Finland, E-mail: jouko.vaananen@helsinki.fi


We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former.

(Received June 07 2001)

(Revised July 20 2001)