Second-order logic and foundations of mathematics

Bulletin of Symbolic Logic 7 (4):504-520 (2001)
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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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Jouko A Vaananen
University of Helsinki

References found in this work

Completeness in the theory of types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
A Decision Method for Elementary Algebra and Geometry.Alfred Tarski - 1952 - Journal of Symbolic Logic 17 (3):207-207.
Absolute logics and L∞ω.K. Jon Barwise - 1972 - Annals of Mathematical Logic 4 (3):309-340.

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