The importance of Georg Cantor’s religious convictions is often neglected in discussions of his mathematics and metaphysics. Herein I argue, pace Jan ́e (1995), that due to the importance of Christianity to Cantor, he would have never thought of absolutely infinite collections/inconsistent multiplicities,as being merely potential, or as being purely mathematical entities. I begin by considering and rejecting two arguments due to Ignacio Jan ́e based on letters to Hilbert and the generating principles for ordinals, respectively, showing that my reading (...) of Cantor is consistent with that evidence. I then argue that evidence from Cantor’s later writings shows that he was still very religious later in his career, and thus would not have given up on the reality of the absolute, as that would imply an imperfection on the part of God. (shrink)

Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of (...) arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism. (shrink)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)