> On 10 Dec 2018, at 07:53, Brent Meeker <[email protected]> wrote: > > > > On 12/9/2018 6:42 PM, Jason Resch wrote: >> >> >> On Sun, Dec 9, 2018 at 2:53 PM Brent Meeker <[email protected] >> <mailto:[email protected]>> wrote: >> They are fundamental only in the sense that one can use them as axioms. So >> their fundamentalism is circular. >> >> Brent >> >> On 12/9/2018 7:36 AM, Jason Resch wrote: >>> But I think truth plays an even more fundamental roll than this. e.g. >>> because the following statement is true "two has a successor" then there >>> exists a successor to 2 distinct from any previous number. Similarly, the >>> truth of "9 is not prime" implies the existence of a factor of 9 besides 1 >>> and 9. >> >> >> That position was defensible before Godel, but not after. He showed >> mathematical truth cannot be based on axioms. > > But he didn't show it could be based on something else.
Actually he did not but he provided the beginning of this. With the work of Skolem, Tarski, Mostowski, Robinson, it has been clear that Robinson Arithmetic is the least finitely axiomatizable Turing universal theory of numbers, and that without number or Turing equivalent, you cannot get them. Eliminate just one axiom, and you get an incomplete but complete-able theory. But RA is essentially undecidable, you cannot retract one axiom without losing Turing Universality, and you cannot get a finite or recursively enumerable addition of axioms to get the all of the arithmetical reality. Now Gödel was indeed quite open to the idea that Naturalism is wrong. Too bad he missed CT and also Mechanism, although he was not entirely close to it either. Arithmetic provides all you need to explain that With less than Arithmetic, you loss Turing Universality, and with more, you can only scratch the surface of the reality/truth of the universal machine. Bruno > > Brent > > -- > 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 [email protected] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
