Skolem's paradox and constructivism (Q1095905)

The paper argues that problematic versions of the Löwenheim-Skolem Theorem (LST) are not forthcoming for intuitionist mathematics, and hence that ``Skolem's paradox'' is a problem for classical mathematics only. After a brief review of a recent discussion of the paradox [Proc. Aristotelian Soc., Supp. Vol. 68 (1985)], it notes that all extant proofs of the LST are intuitionistically invalid. It then proves a number of independence results for intuitionistic Zermelo-Fraenkel set theory (IZF), e.g. IZF \(+\) Fan Theorem \(+\) Bar Induction \(+\) Markov's Principle \(\nvdash\) `Every model has an elementarily equivalent countable model'. Some stronger results are announced but not proved, e.g.: IZF \(+\) Markov's Principle \(+\) Church's Thesis \(\vdash\) `Heyting Arithmetic is categorical'.