The Metamathematics of Putnam’s Model-Theoretic Arguments
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Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.
KeywordsCompleteness Theorem Transitive Model Metaphysical Realist Correspondence Relation Constructivisation Argument
I wish to thank Timothy Bays, Michael Potter and Peter Smith for patient suggestions, advice, and comments. I particularly want to thank Gerald Sacks, who taught me model theory, with whom I had many engaging discussions, and without whom I would probably have no proof of the Submodel Skolem Theorem.
- Bays, T. (2009). Skolem’s paradox. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. [http://plato.stanford.edu/entries/paradox-skolem/]—visited on 14.vi.2010.
- Benacerraf, P. (1985). Skolem and the Skeptic. Proceedings of the Aristotelian Society, 59, 85–115.Google Scholar
- Boolos, G. S., & Jeffrey, R. C. (1989). Computability and Logic, 3rd ed. Cambridge: Cambridge University Press.Google Scholar
- Franzen, T. (2004). Inexhaustibility: A non-exhaustive treatment. Wellesley, MA: Association for Symbolic Logic, Lecture Notes in Logic 16.Google Scholar
- Kunen, K. (1980). Set theory: An introduction to independence proofs. Volume 102, Studies in logic and the foundations of mathematics. Oxford: North-Holland.Google Scholar
- Mostowski, A. (1969). Constructible sets with applications. Amsterdam: North-Holland, Studies in Logic and the Foundations of Mathematics.Google Scholar
- Potter, M. (2004). Set theory and its philosophy. Oxford: Oxford University Press.Google Scholar
- Putnam, H. (1978). Meaning and the moral sciences. London: Routledge and Kegan Paul.Google Scholar
- Putnam, H. (1989). Model theory and the ‘factuality’ of semantics. In A. George (Ed.), Reflections on chomsky (pp. 213–232). Oxford: Basil Blackwell.Google Scholar
- Putnam, H. (1992). Replies. Philosophical Topics, 20, 347–408.Google Scholar
- Simpson, S. G. (1999). Subsystems of second-order arithmetic. Berlin: Springer-VerlagGoogle Scholar
- Skolem, T. (1922). Some remarks on axiomatised set theory. In: J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1931 (1967) (pp. 290–301). Cambridge, MA: Harvard University Press.Google Scholar
- Skolem, T. (1958). Une relativisation des notions mathématiques fondamentales. In: E. J. Fenstad (Ed.), Selected works in logic (1970) (pp. 633–638). Oslo: Universitetsforlaget.Google Scholar
- Velleman, D. J. (1998). ‘Review of Levin’s ‘Putnam on reference and constructible sets’ (1997)’. Mathematical Reviews, 98c:03015, 1364Google Scholar
- Wright, C. (1985). Skolem and the Skeptic. Proceedings of the Aristotelian Society, 59, 117–137.Google Scholar