> On 24 May 2020, at 01:47, Bruce Kellett <[email protected]> wrote: > > On Sun, May 24, 2020 at 9:37 AM Russell Standish <[email protected] > <mailto:[email protected]>> wrote: > On Sat, May 23, 2020 at 12:05:08PM -0700, 'Brent Meeker' via Everything List > 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, even > > ultrafinitism. But the idea that arithemtic exectues all programs certainly > > requires infinities. > > Only potential infinities, not actual infinities. For the UD (a finite > object) to execute any given program, one only needs to wait a finite > amount of time. > > > I thought the UD executing in arithmetic was timeless:
OK. > so all the infinity of possible programs have already been executed before > you even start thinking about it. OK. > So computationalism has actual infinities built in. Not really. But you need certainly MUCH MORE than RA, like PA. But PA is still finitiste (but not ultrafinitist). Computationalism, like physicalism, like mathematical logic, assumes as much math than we need. We know that for the phenomenology of numbers, there is no bound possible on the number and power of axioms needed. I “interview” PA and ZFC in RA, to get the laws of physics. Interviewing RA gives only the universal dovetlaing, which is not Löbian. So we agree, computationalism allows (and necessitates) an ultrafinitist ontology (or an ultraginit presentation of that ontology), but is not itself ultrafinitist. It needs at least one potential infinite, but the measure existence might need large cardinals in set theory, which is part of the phenomenology (quite beyond the ontology). With mechanism, we have a Skolem situation, with a small structure seen from outside (like a countable model of RA or PA), which, seen from inside is beyond all big transfinite set conceivable. It is counter-intuitive, but it is not so much different than imagining that a brain (finite small object) can coneviedd far away galaxies, and infinite possible universes. Bruno > > Bruce > However, I would think that ultrafinitism would change COMP's > predictions, and in a sense be incompatibe with it. Some programs will > not exist, because one would need to wait too long for them to be > executed by the UD. In fact, the choice of reference universal machine > would be significant in ultrafinitism, IIUC. > > -- > 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 view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/CAFxXSLSTHTP15AWvuW4uzvv2BiDuVbH-jS3b4Nak-1t1pEPXDg%40mail.gmail.com > > <https://groups.google.com/d/msgid/everything-list/CAFxXSLSTHTP15AWvuW4uzvv2BiDuVbH-jS3b4Nak-1t1pEPXDg%40mail.gmail.com?utm_medium=email&utm_source=footer>. -- 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/912C9147-8D3B-4F58-8AB6-4A94FA3DE28E%40ulb.ac.be.
