On Tuesday, May 1, 2018 at 3:37:15 PM UTC-5, Brent wrote: > > An interesting proof by Hamkins and a lot of discussion of its > significance on John Baez's blog. It agrees with my intuition that the > mathematical idea of "finite" is not so obvious. > > Brent > > This gets into the rarefied atmosphere of degrees of unprovability. I have a book by Lerman on the subject, which I can read maybe 25 pages into before I am largely confused and lost. I would really need to be far better grounded in this. The idea is that one may ask if things are diagonal up to ω ordinarlity, which is standard Gödel/Turing machine stuff. Then we might however have Halting or provability out to ω + n, or 2ω to nω and then how about ω^n and then n^ω and now make is bigger with ω^ω and so forth. Then this in principle may continue onwards beyond the alephs into least accessible cardinals and so forth. One has this vast and maybe endless tower of greater transfinite models.
Finite systems that are well defined are cyclic groups and related structures. A mathematical system that has some artificial bound on it is not going to satisfy any universal requirements. The most one can have is finite but unbounded. So long as one does not have some series or progression that grows endlessly this can work. LC > > -------- Forwarded Message -------- > > On Tue, May 1, 2018 at 1:13 PM, James wrote: > > On Tue, May 1, 2018 at 7:19 AM, Cris wrote: > >> > >> ... For any set of axioms, there is a Turing machine which 1) never > halts and 2) that set of axioms cannot prove that it never halts. ... > > > >> But don’t you agree that the Halting Problem has a definite truth > value? In other words, that a given Turing machine (with a given input) > either runs forever or doesn’t, regardless of our ability to prove it? ... > > > > To answer the question posed, shouldn't we ask if, given any > > *particular* TM, there exists *some* consistent system/set of axioms > > that can prove whether it halts or not? I was under the impression > > that the answer here was "yes", regardless of any individual > > consistent system being unable to tackle the general problem. > > The problem is when you have nonstandard natural numbers. It's > perfectly valid, for instance, to have a Turing machine halt after ω + > 3 steps. You can say, "oh, but we use the unique standard model > defined by the second-order theory", but then the second order theory > has to live in some universe, and there are universes in which what's > uncomputable in your universe can be computable in mine: > > http://jdh.hamkins.org/every-function-can-be-computable/ > https://johncarlosbaez.wordpress.com/2016/04/02/computing-the-uncomputable/ > > So as soon as you move away from "only physically implementable math > is real", then you have do deal with all these other models. > -- > Mike > _______________________________________________ > math-fun mailing list > math...@mailman.xmission.com <javascript:> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
