Skolem's Criticisms of Set Theory

Abstract

In 1915 Leopold Ldwenheim formulated and gave a flawed proof of a result which for my purposes I express as follows: If a first order (FO) formula is satisfiable it is satisfiable in a denumerable domain.' L6wenheim had but vague inklings of the philosophic perplexities that were to arise in dealing with the implications of this theorem and its extensions and generarizations. Bringing out those perplexities was left to Thoraf Skolem. In 1920 Skolem gave a improved proof of the above result and proved the generalization of it for the case of denumerably many FO formulae. This generalization, that if denumerably many FO formulae are jointly satisfiable they are jointly satisfiable in a denumerable domain, is the Lowenheim-Skolem theorem (L-SK theorem). In 1922 Skolem proved a slightly different version of the L-SK theorem. Formulated in a misleading way, it is: Any consistent set of FO formulae has a denumerable model. Later in the paper, I say something about the differences between these results and their proofs. In his 1922 paper Some Remarks on Axiomatized Set Theory, Skolem raised certain difficulties arising from the L-SK theorem. The problems and perplexities surrounding this theorem come from its application to set theory. Set theory can be formulated as a FO theory. Hence, it has either finitely many FO formulae or a denumerably infinite number of FO formulae as its axioms. In such a theory there are theorems that assert the existence of nondenumerable sets. However, by the L-SK theorem such a set theory has a denumerable model, i.e., one in which only denumerably many things appear. This apparently paradoxical state of affairs became known as Skolem's paradox. Skolem himself was quite clear that there was no paradox.