ontolog-forum
[Top] [All Lists]

## Re: [ontolog-forum] Finitism vs. Transfinitism

 To: "[ontolog-forum]" Bill Andersen Thu, 6 Mar 2008 08:20:08 -0500 <8EDDB439-129D-421F-AF38-4F5B45ACB0C9@xxxxxxxxxxxxxxxxx>
 ```Hi Avril    (01) You used the phrase "potentially infinite" several times in your discussion below. I'm not sure I understand. Could you say more about what you mean by "potentially" so I can make sense of what a finitist could mean by "potentially infinite"?    (02) Cheers!    (03) Bill Andersen Ontology Works, Inc. 3600 O'Donnell Street, Suite 600 Baltimore, MD 21224 +1.410.675.1204 (w) +1.410.675.1201 (f) +1.443.858.6444 (m)    (04) On Mar 6, 2008, at 4:34 AM, Avril Styrman wrote:    (05) > Lainaus Christopher Menzel : > >>> "Having any set such as {1,2,3,...} that starts with number 1, and >>> has only successors of 1 as members, one after another, then, if the >>> set has a cardinality x, then x is also a member of the set" >>> >>> This axiom totally as objective as the interpretation of complete >>> induction in ZFC. >> >> Your "axiom" is incoherent twaddle and will remain so until you >> provide rigorous axioms or definitions for "number", "successor", >> "set", "member", and, especially, "cardinality" that are as rigorous >> as those found in ZFC. The only reason you are able to keep talking >> is that you refuse actually to cash your claims as mathematics. Your >> stock and trade is vagueness and ambiguity. The minute you try to >> turn it into real mathematics your "axiom" will vanish like a puff of >> smoke -- not that I expect you to try. > > I do not need to define things such as 'number' and 'successor', > because they are the most self-evident things in the world. > Instead, I can use them to define other things. For 'set' and > 'member', I can use the same axiom of extensionality that is in > ZFC -there is no transfinitism in extensionality. For cardinality > and rank, I can use constructive definitions, such as those in > Finitist set theory: www.cs.helsinki.fi/u/astyrman/FST.pdf > > >>> All the rest of the transf. hierarchy is built on omega-0. Having >>> omega-0 includes the very controversy of having a neverending as a >>> totality. >> >> There is no controversy among real mathematicians. The actual >> infinite is at the heart of nearly all contemporary mathematics, >> including in particular the real analysis that underlies physics. >> The >> existence of the transfinite is a simple consequence of the axioms of >> ZF set theory. > > The term "the natural numbers" can be used very well without > having to commit to anything transfinite, by maintaining that > the series is potentially infinite. It is in the heart of > mathematics of course. I'm sure that also you understand the > twist in having a never ending as a totality. > > * * * * * * * * * * * * * * * * * * * * > If you can show there are numbers bigger than the infinite, your head > whirls. [124] p.16. > > I have always said you can’t speak of all numbers, because > there’s no such > thing as ’all numbers’. But that’s only the expression of a > feeling. > Strictly, one should say, . . . ”In arithmetic we never are talking > about > all numbers, and if someone nevertheless does speak in that way, > then he so > to speak invents something - nonsensical - to supplement the > arithmetical > facts.” (Anything invented as a supplement to logic must of course > be > nonsense). [122] XII.129, [121] XII.448. > > . . . A searchlight sends out light into infinite space and so > illuminates > everything in its direction, but you can’t say it illuminates infini > ty. > [122] XII.142, [121] XII.490. > > The infinite number series is only the infinite possibility of > finite series > of numbers. It is senseless to speak of the whole infinite number > series, as > if it, too, were an extension. [122] XII.144, [121] XII.504. > > If I were to say ”If we were acquainted with an infinite extension, > then it > would be all right to talk of an actual infinite”, that would really > be like > saying, ”If there were a sense of abracadabra then it would be all r > ight to > talk about abracadabraic sense perception”. [122] XII.144, [121] XII > .511. > > But why is it easier to imagine life without end than an endless > series in > space? Somehow, it’s because we simply take the endless life as nev > er > complete, whereas the infinite series in space ought, we feel, > already to > exist as a whole. [122] XII.145, [121] XII.515. > > Let’s imagine a man whose life goes back for an infinite time and wh > o says > to us: ‘I’m just writing down the last digit of p, and it’s a > 2. Every day > of his life he has written down a digit, without ever having begun; > he has > just finished. This seems utter nonsense, and a reductio ad absurdum > of the > concept of an infinite totality. [122] XII.145, [121] XII.516. > > . . . what is infinite about endlessness is only the endlessness > itself. > [122] XII.145, [121] XII.519. > * * * * * * * * * * * * * * * * * * * * > >> Although it cannot be proved mathematically, due to >> Gödel's theorem, over a century of rigorous testing and use suggests >> there is every reason to believe these axioms are consistent and only >> cranks like you argue there are "problems" without the least >> mathematical evidence or competence. If you think there are problems >> with ZF -- despite the fact that you cannot prove its inconsistency >> -- >> then the only rational mathematical response is to provide a >> theoretical alternative so there is actually something to discuss. >> But you obviously lack the ability to do this. > > The problems are evident, but you just deny them, similarly as a > priest in the year 1600 would deny alternative gods. This is very > natural for a human being: "My language is good, my country > is good, my theory is good". > > Why do you need an alternative for something that is useless? I'm not > aiming to give an alternative, but I'm only writing a thesis about > the problems of transfinity. The problems are not in the coherence > of the axioms, but the problems are in what the axioms say. Similarly, > Alice in the Wonderland is totally coherent, but it is only a story. > > Again, tell me one thing where transfinitism is really used, other > than > in turning contradiction into contraction, proving that Cantor's set > exists, and so forth. The proofs that require transfinitism are not > really required in practice. They are not required in space flights, > cosmology, physics, chemistry, computer science, you name it. > > >>> I have clearly argued that potential infinity is totally enough for >>> the needs of the man kind. >> >> You have done no such thing. Your talk of "potential infinity" is >> useless to science and a distraction to this forum until you actually >> provide a mathematical theory that realizes the idea and demonstrate >> its adequacy for natural science and the theory of computation. > > For the theory of computation, Turing machine is an implementation > of potential infinity. The memory tape of a TM is potentially > infinite. It is not actually infinite/transfinite. The potential > infinity is totally enough and very useful. Show me one actual > problem or application where it is not enough, and I'll turn into > a transfinitist. > > > Avril > > > > > > [124] Cora Diamond (editor): Wittgenstein’s Lectures on the Foundati > ons of > Mathematics, Cambridge, 1939. > > [122] Ludwig Wittgenstein: Philosophical Remarks. Edited by Rhus > Rhees and > translated into English by Raymond Hargreaves and Roger White. Basil > Blackwell, 1975. First German edition 1964. > > [121] LudwigWittgenstein: Filosofisia Huomautuksia (Philosophical > Remarks). > Translated by Heikki Nyman. Werner Söderström, 1983. > > _________________________________________________________________ > Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ > Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ > Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx > Shared Files: http://ontolog.cim3.net/file/ > Community Wiki: http://ontolog.cim3.net/wiki/ > To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx >    (06) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (07) ```
 Current Thread [ontolog-forum] Finitism vs. Transfinitism, Avril Styrman [ontolog-forum] Finitism vs. Transfinitism, Avril Styrman Re: [ontolog-forum] Finitism vs. Transfinitism, Christopher Menzel Re: [ontolog-forum] Finitism vs. Transfinitism, Avril Styrman Re: [ontolog-forum] Finitism vs. Transfinitism, Bill Andersen <= Re: [ontolog-forum] Finitism vs. Transfinitism, Avril Styrman