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Second-Order Logic and Foundations of Mathematics


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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[1] K. Jon Barwise , Absolute logics and L∞ω, Annals of Mathematical Logic, vol. 4 (1972), pp. 309340.

[4] Jaakko Hintikka , The principles of mathematics revisited, Cambridge University Press, Cambridge, 1996, with an appendix by Gabriel Sandu.

[6] Per Lindström , On extensions of elementary logic, Theoria, vol. 35 (1969), pp. 111.

[9] Richard Montague , Set theory and higher-order logic, Formal systems and recursive functions (Proceedings of the eighth logic colloquium, Oxford, 1963), North-Holland, Amsterdam, 1965, pp. 131148.

[12] Saharon Shelah , The spectrum problem. I. ℵε -saturated models, the main gap, Israel Journal of Mathematics, vol. 43 (1982), no. 4, pp. 324356.

[13] Saharon Shelah , The spectrum problem. II. Totally transcendental and infinite depth, Israel Journal of Mathematics, vol. 43 (1982), no. 4, pp. 357364.

[14] Stephen G. Simpson , Subsystems of second order arithmetic, Springer-Verlag, Berlin, 1999.

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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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