On Sun, Apr 29, 2018 at 8:38 PM, Henry Baker <hbaker1@pipeline.com> wrote:
It's pretty clear: if there are consistent models with CH and consistent models w/o CH, then that makes CH *independent* -- to choose whichever way you please.
I don't think it's that simple. ZFC is an attempt to codify our intuitions about what a set is. There may be other things about sets that people agree are "true" in the intended model---that is, there could be new axioms that become generally accepted. While there are lots of non-isomorphic models of ZFC, it's pretty clear to me that some of them are other systems that happen to satisfy the axioms that are not the intended object that ZFC is an attempt to axiomatize. For example ZFC has countable models, but I think people generally agree that there are an uncountable number of sets---it's just an unfortunate property of first-order logic that any system with a model has a countable model. And you can't prove Consis(ZFC) in ZFC, so there are models where ZFC is true, but Consis(ZFC) is false, These are pretty clearly not the models we are interested in. I don't think anyone is interested at all in what you can prove from ZFC + not(Consis(ZFC)), while if you could prove CH from ZFC + Consis(ZFC) (unfortunately, you provably can't), then I think almost all mathematicians would accept CH as true. .
Now consider physics. I find CH completely antithetical to quantum theory -- e.g., "ultraviolet catastrophe" and all that.
What does the ultraviolet catastrophe have to do with CH?
Curiously, much of the math of QM seems to make liberal use of CH, so something's wrong with the math at the basis of QM. I believe that this was the essence of Fredkin's, Bennett's, Feynman's (?) etc., arguments.
I've never seen any QM math that made use of CH; can you point me to a reference to one of these uses? Andy