The Philosophical Impact of the Löwenheim-Skolem Theorem

Abstract

Not only that the theorem which Löwenheim proved 1915 was the first big result in what we now call Model Theory, but, primarily due to Skolem, who profoundly analyzed and understood the significance of its far-reaching consequences, the Löwenheim-Skolem Theorem made also a revolutionary impact on the history of the twentieth century mathematics, and philosophy of mathematics in particular. Among the consequences, the most disastrous were those that concerned Hilbert’s categoricity demand and Cantor’s concept of cardinality. In this article, it is argued that though it should be admitted that the first group of consequences, related to the possibility of non-standard models, clearly pointed to the expressive weakness of the (first-order) language in which, in the first three decades of the last century, the main mathematical theories were expected to be formalized, this lesson concerning language was only the first part of the story. The need for re-investigation of the concept of relational structure, and the concept of cardinality in particular, became acute only in view of results by Paul Cohen, Solomon Feferman and Azriel Lévy in the seventh decade of the century. It is shown how the relativity of cardinality should be understood and why instead of being attributed to sets as such it should be rather attributed to sets as basic sets of relational structures. It is also shown that, if properly understood, the relativity of cardinals may be relevant not only for the philosophy of mathematics but for metaphysics as well.