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". Best wishes, Adam P. Goucher
Sent: Monday, April 30, 2018 at 3:18 PM From: "John Golden" <goldenj@gvsu.edu> To: "math-fun@mailman.xmission.com" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Continuum Hypothesis
I’ve been loving this discussion. With students (in a capstone course) we talk about Cantor and Gödel in sequence, and the CH comes off as an example of incompleteness. I.e. it could be true or false but unverifiable. So you have to change the axioms if you want it to be one or the other and provable. Am I propagating a misconception? On the other hand, most mathematicians with whom I’ve discussed this have a personal belief that it’s true or not. It often segues into a discussion on invented or discovered. I remember a great weeklong lecture from Alain Connes on why math is discovered, in which I think he identified as a CH believer. Hardcore invented types seem to stick to the CH being unverifiable, nothing more to say. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun