On Sun, Apr 29, 2018 at 5:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Question: --------- So, does that mean that CH is neither really true nor really false, but some third option? Or what?
I'm curious what people think about this.
Well, obviously CH is independent of ZF; there are models in which it holds, and models in which it doesn't. By your use of the word "really", you bring up the question of whether we can model ZF with physics somehow. It's conceivable that the universe itself is infinite, even if the observable universe isn't; we can think of the observable universe as something like our Moore neighborhood. However, the universe is expanding and accelerating, so there are parts of the universe that are now causally separated. The Bekenstein bound says there's a limit on how much information one can pack into a volume before it collapses into a black hole, so it looks like reality's limited to at best countable models if you include the whole universe and only finite models if you include only the observable universe. Since there aren't finite models of ZF, I don't think ZF describes reality. (That said, nobody's quite sure what to make of the cardinality of the set of probability amplitudes yet; there aren't any models I know of where probability amplitude is quantized, but neither is it something you can directly measure.) -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com