I've known others with views similar to Adam's. I'm not sure I understand the reason for treating (say) the set of real numbers differently from the integers, though. The reason I believe that [the Twin Prime Conjecture is either really true or really false] is that I can imagine a way of verifying the truth status of TPC by a census of all numbers of the form p+2 where p is prime — at least in principle. But likewise, I can imagine in principle a verification of the truth status of the Continuum Hypothesis. Suppose it's true. Then for every subset X of the reals: we need to check, for every function f : X —> Z *and* for every function g : X —> R, that *at least one of these* is a bijection. (Which itself can be described as a kind of transfinite induction.) Because I can imagine such a verification (whether of CH or not-CH), I find it impossible to imagine that the truth status of CH is indeterminate. And the concept of a "subset of the reals" seems quite well-defined to me. Flip a coin once independently for each real number, and you get some subset; every subset is obtainable in that way. Anyhow, that's my story and I'm sticking to it. —Dan Adam Goucher wrote: ----- My own philosophical position is Platonist/realist about first-order properties of the natural numbers, and formalist about everything else. So I believe Con(ZFC) has a definite truth-value (namely 'true'), but CH does not have a definite truth-value. This position means that, unlike a formalist, I reject the perfectly consistent theory "ZFC + ¬Con(ZFC)" as nonsensical on the basis that it proves false statements about the naturals (such as ¬Con(ZFC)). But I'll happily work in either "ZFC + CH" and "ZFC + ¬CH", as they are both (as far as we know) arithmetically sound. If you translate this position into an invented/discovered dichotomy, and then into German, then it simplifies to Kronecker's quotation "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk". -----