On Mon, Apr 30, 2018 at 3:28 PM, Cris Moore <moore@santafe.edu> wrote:
On Apr 30, 2018, at 3:21 PM, Cris Moore <moore@santafe.edu> wrote:
However, we should be careful, since even questions about the halting of Turing machines — which are certainly first-order claims about the integers — can be independent of ZFC: https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf> <https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf>>
Sorry, continuing: note that if a Turing machine halts in finite time, there is a finite proof of that fact — namely, run it and see what happens. But if it runs forever, proving that it does so might be difficult.
But surely there is a truth of the matter here! The Turing machine will either run forever, or it won’t. I find it hard to be a pluralist here.
Well, physically we can't build a Turing machine, so it depends on your idealization. Once you're idealizing, "really" stops having much meaning. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com