On Ultrafinitism
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From: Rupert on 31 Oct 2006 02:19 MoeBlee wrote: > Rupert wrote: > > Edward Nelson, on the other hand, has done > > some interesting research about certain weak axiomatic theories in > > arithmetic, which may embody his stance. See his book "Predicative > > Arithmetic". > Check out Chapter 31, where he reports that he is trying to prove in predicative arithmetic that exponentiation is not total. This would imply that Elementary Function Arithmetic is inconsistent. Crazy stuff, eh?
From: MoeBlee on 31 Oct 2006 12:43 Rupert wrote: > He announced that he had a > consistency proof for ZF with any finite number of inaccessible > cardinals I don't understand that. If he's an ultrafinitist, then how could he use inaccessible cardinals? Or, is he claiming to prove the inconsistency of ZF by proving that ZF + "inaccessible cardinals exist" (which is relatively consistent to ZF?) proves the consistency of ZF? > http://math.ucsd.edu/~sbuss/ResearchWeb/nelson/paper.pdf Thanks, that one is free and it looks like good reading. > I don't think they would go about it that way. They could use a very > weak arithmetic in which only functions of very low computational > complexity could be proved total. Then only fairly small numbers could > be feasibly defined. They could even add an axiom saying that e.g. the > exponential function is not total, which would give rise to a theory > with nonstandard models. Nelson developed an alternative foundation for > probability theory using such an arithmetic. Hmm, I think I get the gist of that. Eventually, I'll look into it. MoeBlee
From: Rupert on 31 Oct 2006 19:49 MoeBlee wrote: > Rupert wrote: > > He announced that he had a > > consistency proof for ZF with any finite number of inaccessible > > cardinals > > I don't understand that. If he's an ultrafinitist, then how could he > use inaccessible cardinals? Well, it's a pretty startling claim he's made and frankly I'm a bit skeptical. If ZF+{"n inaccessible cardinals exist|n>=0} (let's call this T) is in fact consistent, then of course this fact cannot be proved in any subtheory of the theory in question. (Or at least, so it can be proved in EFA, and I'll assume in what follows that we accept at least EFA). So, if the theory he's working in is consistent, it must not be a subtheory of T. It is just conceivable that there might exist some methods of proof which should properly be called "ultrafinist" which cannot be formalized in T, but this is rather hard to imagine. > Or, is he claiming to prove the > inconsistency of ZF by proving that ZF + "inaccessible cardinals exist" > (which is relatively consistent to ZF?) proves the consistency of ZF? > If ZF+Con(ZF) is consistent, then it cannot be proved in ZF that Con(ZF) implies Con(ZF+"an inaccessible cardinal exists"). (This fact can be proved in EFA). In that sense ZF+"an inaccessible cardinal exists" has stronger consistency strength than ZF. In fact ZF+"an inaccessible cardinal exists" does prove the consistency of ZF (and also of ZF+Con(ZF), and this combined with Goedel's second incompleteness theorem can be used to prove the result I just stated). One simply takes V_kappa where kappa is the first inaccessible cardinal and proves that it is a model of ZF. And then, since Con(ZF) is true and therefore true in V_kappa, it is also a model of ZF+Con(ZF). This is no problem. (Unless of course someone did manage to prove in ZF that Con(ZF) implies Con(ZF+IC). That would imply that ZF+Con(ZF) is inconsistent. But that's not very likely to happen). Yessenin-Volpin is not claiming to prove that ZF is inconsistent. He's claiming to prove that ZF is consistent (and that an extension of it is consistent), but he hasn't specified in what theory. That's what would be interesting to know, since the theory in question couldn't be a subtheory of ZF, unless ZF is inconsistent. I don't know about Yessenin-Volpin, to be honest. I read a paper of which he was the co-author which had some serious mistakes in it. And the claim to be able to prove the consistency of ZF sounds a bit cranky as well. Also, it's hard to get clear from reading his work exactly what part of mathematics his stance would accept. At least with Nelson that's made clear. Nelson's work indisputably has some serious mathematical content which is of interest to mathematicians regardless of their foundational stance. It's really quite startling how serious his skepticism is: he is actually making serious efforts to prove the inconsistency of EFA. You might be interested in looking at the final chapter of "Predicative Arithmetic", called "A modified Hilbert program", where he discusses how a nonstandard extension of one of his theories can be used to develop nonstandard analysis. > > http://math.ucsd.edu/~sbuss/ResearchWeb/nelson/paper.pdf > > Thanks, that one is free and it looks like good reading. > > > I don't think they would go about it that way. They could use a very > > weak arithmetic in which only functions of very low computational > > complexity could be proved total. Then only fairly small numbers could > > be feasibly defined. They could even add an axiom saying that e.g. the > > exponential function is not total, which would give rise to a theory > > with nonstandard models. Nelson developed an alternative foundation for > > probability theory using such an arithmetic. > > Hmm, I think I get the gist of that. Eventually, I'll look into it. > > MoeBlee
From: Eckard Blumschein on 1 Nov 2006 04:46 On 10/31/2006 6:45 AM, galathaea wrote: > Eckard Blumschein wrote: >> On 10/29/2006 3:04 AM, galathaea wrote: >> >> > ultrafinitsm takes a conceptual step beyond finitism >> > by stressing that >> > not only is mathematics a finite process >> > but there exist hard limits >> > >> > there is no potential infinity >> >> Mueckenheim was blamed an ultrafinitist. However, he denies the actual >> infinity while the potential infinity seems to be obvious to anybody. > > plutarch was caught writing > "as the indifferent > is the mean between good and evil > so there is some mean > between finite and infinite" You are quite right. Finite and infinite exclude each other. While natural counting is clearly an infinite process, it makes nonetheless sense to imagine this process perfectly finished. This inherent contradiction to the notion of counting provides the key for the relationship between two mutually complementing worlds: number (countable) and continuum (uncountable, just fictitious numbers). >> Mueckenheim obviously shares this view. I do not understand why he >> cannot accept mathematics like dealing with the two abstract ideas >> number and continuum. >> >> One alternative after the other failed to unmask Cantors paradise as >> what I consider the Dedekind-Cantor Utopia. Kronecker even called the >> natural numbers given by the Lord. Brouwer even intended to improve set >> theory. Weyl suggested an atomist continuum. >> >> Even Cantor and Hilbert started at some sound finitist views. Now >> ultrafinitism is rumored to be the most silly counterpoint to formalism. >> >> Tell me please whether or not there is a drawer you may put me in? >> I consider the world of (countable) numbers quite different from the >> complementing world of (uncountable) continuum. In principle Cantor was >> conjecturing almost the same when he believed that there is nothing >> between aleph_0 and aleph_1. > > at some point > we all must return to democrites > and the dilemma of the cone I do not support wrong decisions by the mean rather than mainstream.
From: MoeBlee on 2 Nov 2006 17:07
Rupert wrote: > At least with Nelson > that's made clear. Nelson's work indisputably has some serious > mathematical content which is of interest to mathematicians regardless > of their foundational stance. It's really quite startling how serious > his skepticism is: he is actually making serious efforts to prove the > inconsistency of EFA. EFA? Is that PA without induction schema but with two axioms for exponentiation? Thanks also for the rest of your remarks, which will require though some time for me to get caught up on more set theory for me to fully understand what you wrote. MoeBlee |