LudovicoVan
Posts:
4,157
From:
London
Registered:
2/8/08


Re: The axioms for ultrafinitist number theory?
Posted:
Mar 23, 2014 3:20 AM


"Dan Christensen" <Dan_Christensen@sympatico.ca> wrote in message news:e92189e483ff40e88642ec82c9c267b5@googlegroups.com... > On Sunday, March 23, 2014 2:24:05 AM UTC4, Julio Di Egidio wrote: >> "Dan Christensen" <Dan_Christensen@sympatico.ca> wrote in message >> news:bf0643c248b6471cbb25d961178afb8f@googlegroups.com... >> >> > Axioms rewritten with binary predicate S for the successor relation and >> > set membership 'e' for epsilon: >> >> You have changed nothing of substance, > > True. They are cosmetic changes: S(x,y) instead of (x,y) in S. > >> and your induction schema is still crippled. > > I think you will find that it works.
Idiot. I have given you a counterargument that you have just ignored. Here it is again:
Take your proof that all elements but zero have a predecessor (*) then try the same proof with the added axiom that max has no predecessor: the proof goes through just as well! I.e. your induction axiom is crippled, and your system does not capture only finite models. And this was already explained upthread and with more details.
(*) <http://dcproof.com/ExistenceOfPredecessorsFinite.htm>
>> > It can be formally proven (374 lines in DC Proof format) that these >> > axioms >> > are satisfied by: >> >> > n = {0, max}, max=/=0 and S(x,y) <=> x=0 & y=max >> >> It can also be proved that omega_0+1 is a model for your axioms. > > Not sure what that means.
It means your axioms are not only satisfied by finite models. Or you just don't know what omega_0+1 is?
BTW, I'll take the chance, before I finally killfile you for good, to tell you something: Do you really think that being a patent liar and a obnoxious moron, and the insults to e.g. WM just because you feel strong that other morons stand on your side, are going to benefit your selfpromotion and that of your petty program? And, talking of that DC Proof, you may be a decent programmer assuming you did that yourself, but be reassured that you just suck at reasoning in general and at logic in particular: in fact, you do not even know what logic is, you think it's a calculus...
And now get lost, you have annoyed me one too many times and you'll never learn anyway.
Julio

