> On 23 May 2020, at 21:05, 'Brent Meeker' via Everything List > <[email protected]> wrote: > > > > On 5/23/2020 4:42 AM, Bruno Marchal wrote: >> >> Well, those are theorem provable in very weak theories. It is more a >> question of grasping the proof than subscribing to a philosophical idea. >> That arithmetic executes all programs is a theorem similar to Euclid’s >> theorem that there is no biggest prima numbers. It is more a fact, than an >> idea which could be debated. I insist on this as I realise this is less >> known by the general scientists than 20 years ago. We knew this implicitly >> since Gödel 1931, and explicitly since Church, Turing and Kleene 1936. > > Recently you have said that your theory is consistent with finitism,
It has always been a finitism. Judson Webb wrote a book explaining exactly this. > even ultrafinitism. Yes. This I have realised more recently. But that was obvious, given that we assume RA for the ontology, and it has no axiom of infinity, nor the induction axioms. So it is consistent with the idea of a biggest natural numbers. Of course, to prove this to be consistent, you need the notion of model. But “in real life” we can never prove that we are consistent. We need to separate the assumptions from the meta level goal like showing that the assumptions make sense. That is the BABA of mathematical logic, but I have realised few people understand this, and confuse easy theories and the meta theories? Of course, Gödel did confuse the two for proving incompleteness, but that confusion is only partial, and done with all the needed precautions. > But the idea that arithemtic exectues all programs certainly requires > infinities. At the meta level, yes. But with mechanism, the “meta-level” is brought by the observer, whose existence is guarantied by the ground level (or ontology). The axioms of RA are just CL + 1) 0 ≠ s(x) 2) x ≠ y -> s(x) ≠ s(y) 3) x ≠ 0 -> Ey(x = s(y)) 4) x+0 = x 5) x+s(y) = s(x+y) 6) x*0=0 7) x*s(y)=(x*y)+x There is no axiom of infinity, nor induction axioms. All ultrafinitst people accept them, and you can add the axiom 8) Ex(Ay(x bigger-or-equal y) without being led to a contradiction. 1)-7) is the mathematical ontology. Of course, to prove this to be Turing universal, you will need at least the induction axiom. But RA is Turing universal even if you don’t prove it … Note that RA can prove the existence of machine believing in RA + induction, or in ZFC, even of ZFC + large cardinal axiom, etc. We must not confuse what RA can prove, and what the creature (whose existence can be proved by RA) can prove. RA can simulate ZF, does not make it possible toi identify RA and ZF, like the fact that I can imitate Einstein’s brain does not make me equivalent with Einstein. On the contrary, if I emulate Einstein’s brain, the best I can get is a conversation with Einstein. That does not entail that I will agree with him, or understand what he says. 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]. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/26965238-f906-a128-28d6-616336849466%40verizon.net. -- 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/977A2A60-D1C7-4723-91B1-6CCB0CADB6A1%40ulb.ac.be.
