ETD

• arithmetic
• determinacy of reference
• model-theoretic argument
• set theory
• Skolem paradox

The work addresses the question ‘how do we fix the meaning of mathematical terms?’ by considering the cases of arithmetic and set theory. The point of departure is the basic consideration that we seem to be talking about numbers and sets as if they were sharp and definite concepts. By just looking at our best means to capture these notions, namely the mathematical theories of PA and ZFC, we encounter the disquieting phenomenon of nonstandard models. The same mathematical machinery that allows the construction of these models and the fact that set-theoretic semantics is itself set-theoretic historically gave rise to further skeptical doubts about the determinacy of reference in mathematics. The best-known way to secure determinacy for arithmetic is through second-order logic, but I argue that for our interests it does not fair better off than set theory. For set theory itself, on the other hand, there is no immediate solution to the problem. I survey the main proposals in the literature and conclude that none of them achieves what it sets out to do. Once radical skepticism is ruled out by epistemic considerations about the use of mathematical languages, I conclude that a moderate degree of indeterminacy is to be accepted in both arithmetic and set theory and that the request for absolute determinacy is to be abandoned as the last remnant of foundationalism.