ABSTRACT
S K O L E M ’S P A R A DOX A ND C O N S T R U C T I V I S M
Skolem’s paradox has not been shown to arise for constructivism. Indeed, considerations that we shall advance below indicate that the main result cited to produce the paradox could be obtained only by methods not in mainstream intuitionistic practice. It strikes us as an important fact for the philosophy of mathematics. The intuitionistic conception of the mathematical universe appears, as far as we know, to be free from Skolemite stress. If one could discover reasons for believing this to be no accident, it would be an im portan t new con sideration, in addition to the meaning-theoretic ones advanced by Dummett (1978, ‘Concluding philosophical rem arks’), tha t ought to be assessed when trying to reach a view on the question whether intuitionism is the correct philosophy o f mathematics. We give below the detailed reasons why we believe tha t intuitionistic
mathematics is indeed free o f the Skolem paradox. They culm inate in a strong independence result: even a very powerful version o f intuitionis tic set theory does not yield any o f the usual forms o f a countable downward Lowenheim -Skolem theorem. The p roo f draws on the general equivalence described in McCarty (1984) between intuitionistic mathematics and classical recursive mathematics. But first we set the stage by explaining the history of the (classical) paradox, and the philosophical reflections on the foundations o f set theory that it has provoked. The recent symposium between Paul Benacerraf and Crispin Wright provides a focus for these considerations. Then we inspect the known proofs o f the Low enheim -Skolem theorem , and reveal them all to be constructively unacceptable. Finally we set out the independence results. They yield, we believe,
the deep reasons for the localised constructive failures. Besides showing
that weak versions of the Lowenheim — Skolem Theorem cannot be proved in extensions of the intuitionistic set theory IZF, we prove that the Theorem entails principles which many constructivists would reject (e.g., Kripke’s Schema) and is falsified outright by principles (Church’s Thesis and M arkov’s Scheme) which a number o f constructiv ists would accept. Let us also emphasize at the outset that the m eta mathematics which we adopt in giving our proofs is itself constructive.
