An Ultrafinite Set Theory
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From: Frederick Williams on 8 Mar 2010 12:33 Transfer Principle wrote: > > [...] But of course, as I myself found > out, it's far easier to give a theory with both finite and > infinite models than it is to give one with arbitrarily large > finite models but no infinite models. Compactness forbids it. -- I can't go on, I'll go on.
From: RussellE on 8 Mar 2010 18:39 On Mar 7, 10:04 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Mar 3, 10:18 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > > > > > > Axiom of making sense: if x in y then Ur(x) and Set(y) or Set(x) and > > Class(y); and if x < y then Ur(x) and Ur(y). > > Axiom of well-ordering: < is a well-ordering of Ur; that is, < is a > > total order, and if Set(x) and there is a y in x, then there is a > > <-smallest y in x. > > Axiom of this or that: 0 is the <-least urelement; T is the <-greatest > > element. > > Comprehension for sets: if P(y) is a formula then all (universal > > closures of) formulas of the form > > (Ex)(Set(x) and (y)(y in x <--> Ur(x) & P)) > > with the usual proviso on free variables in P. > > Comprehension for classes: if P(y) if a formula then all (universal > > closures of) formulas of the form > > (Ex)(Class(x) and (y)(y in x <--> Set(x) & P)) > > with the usual proviso on free variables in P. > > There, a wonderful ultrafinite theory. (Those in the know may recognise > > this theory as a variant of third-order successor arithmetic with > > top. For arithmetic with top see e.g. Raatikainen's paper > > http://www.mv.helsinki.fi/home/praatika/finitetruth.pdf, > > Interesting theory. Of course, since RE is the poster who > objected to standard theory, it's up to RE to make the final > decision as to whether this theory is acceptable or not. > > According to the link given by Aatu in the post above, the > theory given by axioms A1-A10 has infinite models, including > the standard model of arithmetic. I'm not sure whether RE > will find that acceptable or whether he demands a theory all > of whose models are finite. Yes, I want a system where all models are finite. > But of course, as I myself found > out, it's far easier to give a theory with both finite and > infinite models than it is to give one with arbitrarily large > finite models but no infinite models. > > > Now what? What use are we to put this theory to? > > Presumably RE wants us to use this theory anywhere where we > normally use a standard theory, since he finds the standard > infinitary theory objectionable. > > Thanks for the theory and link, Aatu! Yes. Thanks Aatu! http://www.mv.helsinki.fi/home/praatika/finitetruth.pdf I do find infinite set objectionable. I have devised several proofs showing infinite sets lead to contradictions. Other people have to make up their own minds if these proofs are convincing. What are you left with, if, like me, you think infinite sets are contradictory? The only idea I can come with is natural numbers can not grow without limit and there exists a "largest" natural number. One thing I do like about Raatikainen's paper is the possibility of proving the consistency of a finite theory. I find it amazing so many people spend so much time and effort on systems like ZFC which could be proven inconsistent at any time. I would think we want to start with a theory that is provably consistent. I suspect a theory with only finite models will have to change some of our most basic assumptions about arithmetic. For example, such a theory can not allow us to add one to any natural number. I don't even want to go into what such a theory has to say about time. Russell - 2 many 2 count
From: RussellE on 8 Mar 2010 18:40 On Mar 8, 9:33 am, Frederick Williams <frederick.willia...(a)tesco.net> wrote: > Transfer Principle wrote: > > > [...] But of course, as I myself found > > out, it's far easier to give a theory with both finite and > > infinite models than it is to give one with arbitrarily large > > finite models but no infinite models. > > Compactness forbids it. Interesting. Would you please expnad on this? Russell - Integers are an illusion
From: Virgil on 8 Mar 2010 18:50 In article <92c8f83f-5156-493d-af02-407e091a71e8(a)s36g2000prf.googlegroups.com>, RussellE <reasterly(a)gmail.com> wrote: > I do find infinite set objectionable. > I have devised several proofs showing infinite sets > lead to contradictions. Other people have to make > up their own minds if these proofs are convincing. While you may have created arguments that infinite sets lead to contradictions, these are not proofs in any mathematical sense until they have been vetted and approved, which yours have not been. > > What are you left with, if, like me, you think > infinite sets are contradictory? The only > idea I can come with is natural numbers > can not grow without limit and there exists > a "largest" natural number. Insisting on a "largest possible natural" in a supposedly expanding universe seems a bit contrary. > > One thing I do like about Raatikainen's > paper is the possibility of proving the > consistency of a finite theory. > > I find it amazing so many people spend > so much time and effort on systems like > ZFC which could be proven inconsistent > at any time. I would think we want to start > with a theory that is provably consistent. What good would provable consistency be if your provably consistent system should impose restrictions making much of current mathematics impossible? So until someone actually proves those set theories with infinities now extant to be inconsistent, I prefer to keep them.
From: MoeBlee on 8 Mar 2010 20:21
On Mar 8, 5:39 pm, RussellE <reaste...(a)gmail.com> wrote: > I have devised several proofs showing infinite sets > lead to contradictions. Other people have to make > up their own minds if these proofs are convincing. Would you please stop insulting our intelligence with remarks like that. > > I find it amazing so many people spend > so much time and effort on systems like > ZFC which could be proven inconsistent > at any time. I would think we want to start > with a theory that is provably consistent. Proved consistent in WHAT manner? Proved consistent from what axioms or principles and by what logic? (Please refer to a little thing called 'the 2nd incompleteness theorem'.) And what will be the POWER of the theory you're trying to make? What amount of mathematics - arithmetic, calculus, etc. - can we derive in your theory? > I suspect a theory with only finite models > will have to change some of our most > basic assumptions about arithmetic. > For example, such a theory can not > allow us to add one to any natural number. So, do you blame people for prefering that we can add one to any natural number? It seems like a reasonable thing to do. You give me the natural number n and I add 1 to it. Just because you think infinity is the boogeyman, I have to give up that I can add 1 to any natural number? I don't see that there's much in it for me. > I don't even want to go into what such a > theory has to say about time. ZFC says NOTHING about time. MoeBlee |