An Ultrafinite Set Theory
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From: FredJeffries on 7 Mar 2010 14:34 On Feb 24, 5:52 pm, RussellE <reaste...(a)gmail.com> wrote: > I searched for "ultrafinite set theory" and all > I found was a remark by Zermelo: > "The 'ultrafinite antinomies of set theory', > which the scientific reactionaries and > anti-mathematicians eagerly and delightedly > call on in their campaign ..." > > I get the impression Zermelo didn't like > ultrafinitists. > You should not toss Zermelo off so lightly. If you would care to more carefully examine the context of the above quote, you may find a clue to a way of solving YOUR problem, i.e. finding a workable ultra- finitistic set theory. I took the liberty to search for the source of this quote and found out that Zermelo used the term "Ultrafinite" to modify "Set", or rather "non-set", and not "Set Theory" and seems to me to be refering to what is known as Proper Classes. The source is "Uber Grenzzahlen und Mengenbereiche: Neue Untersuchungen uber die Grundlagen der Mengenlehre", Fundamenta Mathematicae, 1930, 16: 29-47 which I won't even try owing to my high school German classes being so long ago. But I did find a translation of the relevant paragraph in Geoffrey Hellman, "The Many Worlds Interpretation of Set Theory", Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1988, Volume Two: Symposia and Invited Papers (1988), pp. 445-455 on page 447 <quote> The "ultrafinite antinomies of set theory", which the scientific reactionaries and anti-mathematicianse agerly and delightedly call on in their campaign against set theory, these specious "contradictions"a, rise solely from a confusion between the non- categorical axioms of set theory and the various particular models of them: What in one model appears as an "ultrafinite un- or super-set" is in the next higher domain a perfectly good "set" with a cardinal number and order type of its own, which serves as the foundation stone for the construction of the new domain. The boundless series of Cantor's ordinal numbers gives rise to an equally boundless series of essentially different models of set theory, in each of which the whole classical theory can be expressed. The polar opposite tendencies of the thinking mind, creative progress and all-embracing completeness, which lie at the root of Kant's "antinomies", find their symbolic expression and resolution in the concept of the well-ordered transfinite number-series, whose unrestricted progress comes to no real conclusion, but only to relative stopping-points, the "boundary numbers" that divide the lower from the higher models. And so the "antinomies" of set theory, properly understood, lead not to a restriction and mutilation, but rather to a further development (whose scope cannot yet be taken in) and enrichment of mathematical science. (Zermelo 1930, p. 47) </quote> See also José Ferreirós Domínguez, Labyrinth of thought: a history of set theory and its role in modern mathematics, p.376 http://books.google.com/books?id=DITy0nsYQQoC&pg=PA376&lpg=PA376&dq=ultrafinite&source=bl&ots=na0S3ACwwJ&sig=ZEcJdhmKiHQ0_y81UnvlWPDwKX0&hl=en&ei=j-GTS6GwL4mgsgPo9an9Aw&sa=X&oi=book_result&ct=result&resnum=10&ved=0CC0Q6AEwCQ#v=onepage&q=ultrafinite&f=false Take note particularly the explanation: The "ultrafinite non-sets" of one model become legitimate sets in the next model. and Zermelo's quoted above: What in one model appears as an "ultrafinite un- or super-set" is in the next higher domain a perfectly good "set" with a cardinal number and order type of its own, which serves as the foundation stone for the construction of the new domain. Others have pointed out the seeming shifting sand paradox of using numbers which are claimed not to exist or systems of which it is claimed that they are inconsistent to try to create a solidly based finitistic system. Let us rather take to heart the insights of Nelson, Yessenin-Volpin and , yes, even the notorious Cantorian Zermelo, and solidly build upon a solid foundation. Let us turn the Yessenin-Volpin method around: It has been conceded in this thread that there is a consistent system with one object. Use this one-object system to faithfully construct a two object system (a la Zermelo) and submit it for certification-of-quality to the inspectors and ask if it is a valid system. If you have done your work well, they will surely (although after a bit of time for the inspection added to the time spent constructing the extension) they must concede the verity of your construction. I they didn't they would be open to the criticism of unfaithfulness to the methods of Zermelo. Having a two-object system, proceed to construct a four object system and submit it for approval. Naturally, it will take a bit longer to construct and have verified this system. You may ask how to proceed in constructing these extensions without running into the paradox of having to use numbers which don't yet exist in the system. Here is where to use Nelson's observation, with which anyone who has any knowledge of computer system should have noticed: Tally numbers (cardinals) are different sorts of numbers than positional numbers. (Ed Nelson, Predicative Arithmetic p.173 available online at http://www.math.princeton.edu/~nelson/books/pa.pdf) <quote> Originally, sequences of tally marks were used to count things. Then positional notation -- the most powerful achievement of mathematics-- was invented. Decimals (i.e., numbers written in positional notation) are simply canonical forms for variable-free terms of arithmetic. It has been universally assumed, on the basis of scant evidence, that decimals are the same kind of thing as sequences of tally marks, only expressed in a more practical and efficient notation. This assumption is based on the semantic view of mathematics, in which mathematical expressions, such as decimals and tally marks, are regarded as denoting abstract objects. But to one who takes a formalist view of mathematics, the subject matter of mathematics is the expressions themselves together with the rules for manipulating them-- nothing more. From this point of view, the invention of positional notation was the creation of a new kind of number. </quote> Instead of talking about tally numbers and decimals, I will follow Rudy Rucker and use "Counting Numbers" and "Information Numbers". In each system that you construct, consider the numbers therein as the counting numbers. From them create that system's information numbers using a positional notation scheme. Then when you create the next system we use the previous system's information numbers as the new system's counting numbers in a Zermeloization process. I leave the details (and the glory) to others. I will just quote Nelson again (Predicative Arithmetic p.75) as an answer to the "why bother/so what" questions: There is a story of a bank employee who was told to count a bundle of bills to verify that there was actually a thousand of them. The employee began to count them: 1, 2, 3, ..., 61, 62, 63-- and then stopped, being convinced that since it had checked perfectly all that way it must be correct.
From: FredJeffries on 7 Mar 2010 14:47 On Feb 24, 5:52 pm, RussellE <reaste...(a)gmail.com> wrote: > I searched for "ultrafinite set theory" and all > I found was a remark by Zermelo: > "The 'ultrafinite antinomies of set theory', > which the scientific reactionaries and > anti-mathematicians eagerly and delightedly > call on in their campaign ..." > > I get the impression Zermelo didn't like > ultrafinitists. > > There were some articles about Essenin-Volpin's > set theory as well as finite abelian groups. > I couldn't find an actual ultrafinite set thory. It is unfortunate that you used that search phrase as 'ultrafinite', at least in Zermelo's case, means something like beyond even Cantor's transfinite, which is the direct opposite of what you seem to be looking for. If instead, you use the term 'ultrafinitism', you would come up with the very helpful "FOM: Part I:Ultrafinitism,Naturalism,Vagueness" by Robert Tragesser http://www.cs.nyu.edu/pipermail/fom/1998-April/001825.html which lists "three well thought out ultrafinitisms, those of: J.Hjelmslev, P. Vopenka, and S. Lavine." Alas, I have never found part 2 of the article.
From: Transfer Principle on 7 Mar 2010 22:18 On Mar 4, 4:55 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > We're opponents, and so I expect nothing less. > Newberry and I disagree on whether his intuitions are reasonable. > I've never seen any real sense to his ideas, but I have no contempt > for him. (This is not true for certain others, Andrew Usher, for > example, who do behave in a truly contemptible manner.) Ah yes, Newberry. I've noticed that thread back in its early stages, but I didn't post there, since I didn't feel that avoiding vacuous truth was worth defending. But in that thread, Marshall Spight made a comment about Newberry's idea and explicitly stated that it applied equally to this thread as well. And so my next post in this thread will be a response directed to Spight in which I respond to that comment. Also, that thread is now suddenly turning into another discussion between Nam Nguyen and the anti-"cranks." I have yet to decide whether I wish to make any comments there now.
From: Transfer Principle on 8 Mar 2010 01:04 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. 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!
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. |