Hi Russel, Thank you for these remarks! I would see that closure under diagonalization is important, but i wonder if there is a bit of neglect to the uniqueness of this set. There are some indications that there may exist a continuum of sets with this property if we assume some version of the anti-catorian hypothesis (for example see http://www.plover.com/misc/CSF/sdarticle.pdf ) or, (I think) equivalently) some weakening of the Tennenbaum Theorem.
On Mon, May 13, 2013 at 6:57 PM, Russell Standish <li...@hpcoders.com.au>wrote: > On Mon, May 13, 2013 at 03:24:09PM -0700, meekerdb wrote: > > On 5/13/2013 2:49 PM, Stephen Paul King wrote: > > >Does the UD compute *all* functions or only those that are > > >recursively enumerable? > > > > It computes all of them. > > > > Brent > > > > Sorry - it does not compute all functions, just all partially > recursive ones. As Stephen says, there are only countably many > recursive functions, but a continuum of functions from N->N. > > As for Stephen's question of why we might want to single out that set > - it so happens that that set is closed under diagonalisation - which > is Goedel's "miracle". > > Its an aesthetic thing - just like Einstein's theory of general > relativity is the simplest, and most elegant, formulation of geometric > spacetime theories of gravitation. > > It doesn't mean its right, of course, but elegant theories have a > habit of being more likely right than inelegant ones. > > PS - I am unsure whether the set of partially recursive functions is > the only such set closed under diagonalisation - do you know Bruno? > > Cheers > > -- > > > ---------------------------------------------------------------------------- > Prof Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders > Visiting Professor of Mathematics hpco...@hpcoders.com.au > University of New South Wales http://www.hpcoders.com.au > > ---------------------------------------------------------------------------- > > -- > You received this message because you are subscribed to a topic in the > Google Groups "Everything List" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/everything-list/K7E-Vfwj4QU/unsubscribe?hl=en > . > To unsubscribe from this group and all its topics, 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 http://groups.google.com/group/everything-list?hl=en. > For more options, visit https://groups.google.com/groups/opt_out. > > > -- 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 http://groups.google.com/group/everything-list?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
