> On 27 Oct 2019, at 02:33, Russell Standish <[email protected]> wrote: > > On Sat, Oct 26, 2019 at 05:25:39PM -0700, 'Brent Meeker' via Everything List > wrote: >> >> >> On 10/26/2019 5:05 PM, Russell Standish wrote: >>> On Sat, Oct 26, 2019 at 12:50:10PM -0700, 'Brent Meeker' via Everything >>> List wrote: >>>> I'm glad you recognize there's a difference between "potential" and "real"; >>>> a distinction this list is founded to obfuscate. >>>> >>>> Brent >>> I don't know about that. The everything is "all finite things" is a >>> perfectly rational starting point for an ensemble theory. >>> >>> Even though Max's original "all mathematical objects" theory is >>> ambivalent on the point, I've always interpreted that as "all finite >>> axiomatic systems", even in my first paper on the subject: "Why Occams >>> Razor". >> >> Peano arithmetic is not a finite axiomatic system. It has infinitely many >> axioms of the form (s...(s(s(s)))...). >> So I'm not sure what your refer to. I doubt that "all finite things" is >> well defined. Is the set of all finite things finite? >> > > It was a way of characterising a type of theory, such as "all finite > axiomatic systems", or "all turing machines". > > Peano arithmetic is a finite axiomatic system.
Robinson arithmetic is finitely axiomatisable, like NBG set theory. But Peano arithmetic is probably not finitely axiomatisable. Of course, it is recursively enumerable, and can be programmed on computer. I guess that is what you meant by saying it can be a finite system. > However, the integers > is not - as shown by Goedel's incompleteness theorem. The arithmetical truth (about the natural numbers) is not *completely*axiomatisable. Integers, natural numbers, programs, digital machine, are all on the same par. There is no complete theory about what they can and cannot do. Bruno > > > Cheers > > -- > > ---------------------------------------------------------------------------- > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders > Visiting Senior Research Fellow [email protected] > Economics, Kingston University http://www.hpcoders.com.au > ---------------------------------------------------------------------------- > > -- > 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 view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/20191027003338.GI12720%40zen. -- 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 view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/46A48AA1-30A4-4B09-A458-304CEFE345B4%40ulb.ac.be.
