Do We Have a Determinate Conception of
Finiteness and Natural Number?
I'm going to begin with a rather familiar worry, about whether we have a determinate understanding of second-order quantification. Then I will consider an analogous worry about whether we have a determinate understanding of the notion of finiteness. At first blush the case for the second worry seems quite analogous to the case for the first; but for various reasons doubts about the determinacy of the notion of finiteness may seem harder to take seriously than doubts about the determinacy of second-order quantification. I'll be considering a number of different reactions to this situation.
I'll start off the discussion not with second-order quantification, but with the notion of set; for the moment I'll tacitly take our logic to be first-order. How determinate is our understanding of the notion of set?
I think that familiar considerations against the determinacy of our notion of 'set' are fairly strong. I'll remind you of two of them. The first concerns very central set-theoretic questions such as the size of the continuum, that are unsettled not only by the standard axioms of set theory but by any axioms that anyone has been able to think of that are intuitively compelling. The continuum example seems to me an especially striking one because it isn't simply that there is an undecidable 'yes'/'no' question, of whether the____________________