A New Look at Quine on Set Theory

Abstract

This chapter first argues that while there are solid objections to be raised to Quine’s view, certain widespread arguments against result from overly crude and uncharitable interpretations of Quine. It then turns to the question of what kind of evidence it would take for a Quinean naturalist to change their mind about certain theses, such as the size of the set theoretic universe. It argues that Quineans might be moved to embrace further set-theoretic ontology in the light of the mathematical utility of large cardinals, and potentially even the ‘multiverse’ position on set theory.

Assessing the influence of the axiom of choices variations on mathematical foundations and subdisciplines

“…Researchers who believe in Occam's razor often choose to go with the second way. Specifically, the reverse mathematics program is one such example that seeks to determine the minimal axioms necessary to prove specific mathematical theorems [40]. Nevertheless, the problem with adopting weak variants of AC is that, while it might prevent certain unintuitive results or possible inconsistencies, certain important, useful and convenient theorems and results (like Tychonoff's theorem, Zorn's lemma, the statement "every vector space has a basis") could not be derived.…”