Second order logic, set theory and foundations of mathematics

Jouko Väänänen

pp. 371-380

in: P. Dybjer, Sten Lindström, Erik Palmgren, Göran Sundholm (eds), Epistemology versus ontology, Berlin, Springer, 2012

Abstract

The question, whether second order logic is a better foundation for mathematics than set theory, is addressed. The main difference between second order logic and set theory is that set theory builds up a transfinite cumulative hierarchy while second order logic stays within one application of the power sets. It is argued that in many ways this difference is illusory. More importantly, it is argued that the often stated difference, that second order logic has categorical characterizations of relevant mathematical structures, while set theory has non-standard models, amounts to no difference at all. Second order logic and set theory permit quite similar categoricity results on one hand, and similar non-standard models on the other hand.