An Ultrafinite Set Theory
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From: Aatu Koskensilta on 4 Mar 2010 00:51 Transfer Principle <lwalke3(a)lausd.net> writes: > I want to be able to write "<= is a total order" (or similar > statements involving equivalence relations, partial orders, possibly > wellorders depending on the circumstance) with the knowledge that what > I'm writing isn't formally an axiom, but is intended as _shorthand_ > for a longer formal expression that may be too cumbersome to write > (and corresponds to the _standard_ definition of the term I'm > using). But the standard theorists won't budge and insist that > everyone write in symbolic language (a generalization that might be > viewed as a "lie"). It is more appropriately viewed as bizarre fantasy. Who in their right mind insists that mathematical statements be written in a symbolic language in an ordinary context? What some people insist on is clarity; they want to know what various notions mean, what inferences involving the notions are justified, and so on. It's irrelevant whether these notions are "standard" or not. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 4 Mar 2010 01:18 Transfer Principle <lwalke3(a)lausd.net> writes: > RE's "urelements" lack elements, and so do urelements in standard > urelement theories like ZFCU or NFU. RE's natural numbers include > "1,2,3," and so do the standard natural numbers. Indeed, I informally > think of the set of RE natural numbers as being a proper subset of the > set of standard natural numbers, and if M is a subset of N, then the > elements of M are still called natural numbers. M being a finite set > doesn't change its elements status as natural numbers. Here's a boon for you, a veritable ultrafinite theory in your sense. (You might note that the axioms are not presented in symbolic language.) We have as primitives the binary relations < and in, three unary predicates Ur, Set and Class, and the individual symbols 0 and T. As axioms we have: 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 , H�jek and Pudl�k's _Metamathematics of First-Order Arithmetic_ , and sci.logic's own Andrew Boucher's logical studies into what exciting mathematics we may develop on the very cautious view that the successor function might not be total.) Now what? What use are we to put this theory to? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Gc on 4 Mar 2010 01:24 On 3 maalis, 18:37, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > As far as I know, first order PA is not adequate. (I'm open to being > > convinced otherwise, though.) > > Sink your teeth into the mumblings of Feferman on predicativism, ACA_0, > proof-theoretic reductions, what not, then! I just skimmed a paper by Feferman from his web page. He talks about his system W in which you an do analysis, which is proof theoritically reductible to PA, btw which is a concept I am not so familiar with. But W has a different language (It has set variables, of course, because you do analysis) than PA so I understand it is somekind of conservative extension of PA. Of course, PA is very probably enough for physics when you talk about naturals, but in physics you need a lot analysis too. How can you have sufficiently enough analysis only in PA?
From: Aatu Koskensilta on 4 Mar 2010 01:33 Gc <gcut667(a)hotmail.com> writes: > Of course, PA is very probably enough for physics when you talk about > naturals, but in physics you need a lot analysis too. How can you have > sufficiently enough analysis only in PA? We can't "do analysis" in PA in any very natural sense. What is at issue is rather that arithmetical results of use in physics -- e.g. experimental predictions -- derived by analytic means can be proved in PA, for example via a detour through the sort of conservative extensions you mentioned. You probably know more physics than I ever will; if you take an interest in these matters you need consult the literature on reverse mathematics and proof theoretic reductions and decide for yourself whether you find Feferman's view palatable. As noted, as regards the philosophy of science and philosophy of mathematics Feferman's claim has direct relevance to the Quinean idea that the justification for our mathematical theories lies in their use in our best scientific theories. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Gc on 4 Mar 2010 01:53
On 4 maalis, 08:33, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >As > noted, as regards the philosophy of science and philosophy of > mathematics Feferman's claim has direct relevance to the Quinean idea > that the justification for our mathematical theories lies in their use > in our best scientific theories. OK. PS. In my view the justification lies only in that most of these strong theories make sense to me. In harmonic analysis people investigate(d) a lot sets of divergence of Fourier series, even if you sum the series in some other way they knew that the series do converge a.e. I read the essay by the great Antoni Zygmund who said the reason being just "I like it". How to spread resources in mathematics is another thing. I hope that also people who investigate large cardinals can make a career. |