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The Justification of Mathematical Induction

Published online by Cambridge University Press:  28 February 2022

Georg Boolos
Georg Boolos*
Affiliation:
M.I.T.
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Extract

There ie an old (c. 1967) argument due to Dana Scott tnat is not as well known to philosophers and logicians as it ought to be. I shall cone back to it later.

The principle of matheaatlcal induction asserts that every number belongs to any class tnat contains zero and also contains tne successor of any member.

Can the principle of mathematical induction be proved? That is to say, is there a way to show tnat every numoer oelongs to any class, that, etc?

Like any other statement, the principle of mathematical induction can be derived from itself, in zero lines. This quick and easy derivation is not a proof of mathematical induction: it does not show that induction is true.

Type
Part XI. New Directions in the Philosophy of Mathematics
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1984 , Issue 2: Volume Two: Symposia and Invited Papers 1984 , 1984 , pp. 469 - 475
Copyright
Copyright © 1985 by the Philosophy of Science Association

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Footnotes

1

This paper replaces my remarks on Prof. Maddy's paper; it was written while I was on a fellowship for Independent Study and Research from the National Endowment for the Humanities. I am grateful to Scott Weinstein for helpful comments.

References

Boolos, G.S (1971). “The Iterative Conception of Set.” Journal of Philosophy 68: 215-231.CrossRefGoogle Scholar
Scott, Dana (1974). “Axionatizing Set Theory.” In Axiomatic Set Theory. (Proceedings of Symposia In Pure Mathematics. Volume 13. Part II.) Edited by Jech, Thomas J. Providence: American Mathematical Society. Pages 207-214.CrossRefGoogle Scholar
Shoenfield, J.R (1977). “Axions of Set Theory.” In Handbook of Mathematical Logic. Edited by Barwise, Jo Amsterdam: North-Holland Publishing Company. Pages 321-344.CrossRefGoogle Scholar