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Second-Order Logic and Foundations of Mathematics

Published online by Cambridge University Press:  15 January 2014

Jouko Väänänen
Jouko Väänänen*
Affiliation:
Department of Mathematics, University of Helsinki, Helsinki, Finland, E-mail: jouko.vaananen@helsinki.fi
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Abstract

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.

Information

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 4 , December 2001 , pp. 504 - 520
Copyright
Copyright © Association for Symbolic Logic 2001

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References

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