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Second-Order Logic and Foundations of Mathematics

Jouko Vaananen
The Bulletin of Symbolic Logic
Vol. 7, No. 4 (Dec., 2001), pp. 504-520
DOI: 10.2307/2687796
Stable URL: http://www.jstor.org/stable/2687796
Page Count: 17
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Second-Order Logic and Foundations of Mathematics
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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.

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