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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.

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

REFERENCES

[1] Barwise, K. Jon, Absolute logics and L∞ω , Annals of Mathematical Logic, vol. 4 (1972), pp. 309–340.CrossRefGoogle Scholar
[2] Henkin, Leon, Completeness in the theory of types, The Journal of Symbolic Logic, vol. 15 (1950), pp. 81–91.Google Scholar
[3] Hintikka, Jaakko, Reductions in the theory of types, Acta Philos. Fenn., vol. 8 (1955), pp. 57–115.Google Scholar
[4] Hintikka, Jaakko, The principles of mathematics revisited, Cambridge University Press, Cambridge, 1996, with an appendix by Gabriel Sandu.Google Scholar
[5] Kreisel, G., Informal rigour and completeness proofs, Proceedings of the international colloquium in the philosophy of science, London, 1965 (Lakatos, Imre, editor), vol. 1, North-Holland Publishing Co., Amsterdam, 1967, pp. 138–157.Google Scholar
[6] Lindström, Per, On extensions of elementary logic, Theoria, vol. 35 (1969), pp. 1–11.Google Scholar
[7] Lindström, Per, A note on weak second order logic with variables for elementarily definable relations, The proceedings of the Bertrand Russell memorial conference (Uldum, 1971), Bertrand Russell Memorial Logic Conference, Leeds, 1973, pp. 221–233.Google Scholar
[8] Montague, Richard, Reduction of higher-order logic, Theory of models (Proceedings of the 1963 international symposium Berkeley), North-Holland, Amsterdam, 1965, pp. 251–264.Google Scholar
[9] Montague, Richard, Set theory and higher-order logic, Formal systems and recursive functions (Proceedings of the eighth logic colloquium, Oxford, 1963), North-Holland, Amsterdam, 1965, pp. 131–148.CrossRefGoogle Scholar
[10] Mostowski, Andrzej, Concerning the problem of axiomatizability of the field of real numbers in the weak second order logic, Essays on the foundations of mathematics, Magnes Press, Hebrew University, Jerusalem, 1961, pp. 269–286.Google Scholar
[11] Shapiro, Stewart, Foundations without foundationalism, A case for second-order logic, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1991.Google Scholar
[12] Shelah, Saharon, The spectrum problem. I. ℵε -saturated models, the main gap, Israel Journal of Mathematics, vol. 43 (1982), no. 4, pp. 324–356.Google Scholar
[13] Shelah, Saharon, The spectrum problem. II. Totally transcendental and infinite depth, Israel Journal of Mathematics, vol. 43 (1982), no. 4, pp. 357–364.CrossRefGoogle Scholar
[14] Simpson, Stephen G., Subsystems of second order arithmetic, Springer-Verlag, Berlin, 1999.Google Scholar
[15] Tarski, Alfred, A decision method for elementary algebra and geometry, RAND Corporation, Santa Monica, California, 1948.Google Scholar
[16] Väänänen, Jouko, Set-theoretic definability of logics, Model-theoretic logics, Springer, New York, 1985, pp. 599–643.Google Scholar