Obections to Cantor's Theory (Wikipedia article)
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From: Helene.Boucher on 22 Jul 2005 00:58 Helene.Boucher(a)wanadoo.fr wrote: > Virgil wrote: > > > > > (x)(y)(x = Sy => S(x + y)) > > > > Shouldn't the above read '(x)(y)(x + Sy => S(x + y))' ? > > Yes! > Well actually, no! It should read: (x)(y)(x + Sy = S(x + y)) Shame on you! Terrible proof reader!
From: Virgil on 22 Jul 2005 01:09 In article <1122007567.875427.14850(a)o13g2000cwo.googlegroups.com>, Helene.Boucher(a)wanadoo.fr wrote: > Virgil wrote: > > Good a proof reader! > > > In article <1122004787.370831.6550(a)o13g2000cwo.googlegroups.com>, > > Helene.Boucher(a)wanadoo.fr wrote: > > > > (x)(Sy = x*y + x) > > > > Shouldn't the above read '(x)(x*Sy = x*y + x)' ? > > Yes! > > > > > > (x)(!x = 0 => (there exists y)(Sy = x) > > > > Shouldn't the above read '(x)(x != 0 => (there exists y)(Sy = x)'? > > Unless I'm missing something, "! x = 0" is the same as "x != 0"... > > > > > > The last axiom can be proven by induction, so is not included in the > > > axioms of PA (at least when PA is defined straightaway, and not from > > > Q). For instance, here's a standard defintion of PA (Mendelson), with > > > S a total function: > > > (x)(y)(Sx = Sy) > > > (x)(!Sx = 0) > > > (x)(xy)(Sx = Sy => x = y) > > > > Shouldn't the above read '(x)(y)(Sx = Sy => x = y)' ? > > Yes! > > > > (x)(y)(x = Sy => S(x + y)) > > > > Shouldn't the above read '(x)(y)(x + Sy => S(x + y))' ? > > Yes! > > Thank you! I don't know what time it is where you are, but it was 5am > in the morning when I wrote that! I have made more grotesque typos with less excuse.
From: Helene.Boucher on 22 Jul 2005 01:59 Helene.Boucher(a)wanadoo.fr wrote: > > Shame on you! Terrible proof reader! On reading that back, it may give the wrong impression. Of course I am joking (the joke being the shame is on me for writing so badly). Any and all help is, was, and will be appreciated. Thanks for taking the trouble to point out my errors, and I am only surprised that anyone other than Jeffrey and I are still reading this part of the thread.
From: Ross A. Finlayson on 22 Jul 2005 02:10 Robert Kolker wrote: > Peter Webb wrote: > > > > > > How do you disagree with an axiom? > > By assuming a contrary axiom, as is done in non-Euclidean geometry. The > parallel postulate is denied in non-euclidean geometry. Axioms are > posits, or assumptions. They are NOT self evidence truths. > > Bob Kolker There are so many points possible to address, in a variety of ways, with regards to this increased amount of discussion (and confusion) about the infinite, sets infinite, theories of infinite sets, and various perceptions of perspectives of infinity, increase without bound, untrammeled induction, universality, etcetera. I am telling you so! Here I want to ask Bob: are there any self-evident truths? The answer may be yes, but in a reflexive kind of way. It would be interesting to diagram the progression of various arguments and stated opinions about basically "transfinite cardinals" as the obviously to some, not contradictory, the word, uh, ... people argue about it, uh, contentionary, conflictory, confrontational, ah, controversial flashpoint of discussion of infinity. One question that many ask themselves with regards to transfinite cardinals is: what good are they. What can they do. What can they show me. For many the answer is "not much". There is a group of methods called measure theory that does have some of its generally accepted formulations expressed in terms of transfinite cardinals, in one dimension, as basically the difference between cardinality of the continuum and everywhere discontinuous sets, and it is plain to say that what results there may be in terms of the continuum between various every discontinuous sets can be formulated without recourse to the transfinite cardinals. Another possible notion of the utillity of the transfinite cardinals is with regards to basically permutations in algorihtmic analysis, with regards to basically combinatorial explosion, again those things can be explained in different ways of generally the discrete calculi or logarithms. One nice thing about most applied mathematics is that it's possible to construct a physical experiment using definitions of laws of nature, or statistical experiments, to exhibit observed results that agree with prediction made based upon those applicable mathematical methods. In a nonstandard measure theory, Vitali's result may be seen to not hold, and non-measurable sets don't exist, and transfinite cardinals are ineffective and evne misleading as a nomenclature for various quantities in those models, using what is called a kind of infinitesimal, which is a similar, yet different, thing, as an infinity. Infinite sets are equivalent. Bob: the universe is infinite. Now, where that is so, that leads to various notions including, where basically the powerset, diagonal, and nested interval results are held as the epitome of pure mathematics about the infinite, direct contradictions of those things. I have addressed many or the most of the salient issues with regards to the pickling of transfinite cardinals and post-Cantorian theory of the infinite. Thus, I claim some knowledge of pure mathematics, math. So, the existence of a self-evident truth, yes or no, and why. My answer you have. I can keep typing like this for days. Hell, I have. This took fifteen minutes. "You must be able to strike like a hawk." So: what's new in this post? Ross -- "Careful study of the complete thread will show that my statements are largely correct."
From: Han de Bruijn on 22 Jul 2005 03:33
David Kastrup wrote: > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > >>stephen(a)nomail.com wrote: >> >>>Why are you bringing physics into this? Whether black >>>holes exist or not has nothing to do with set theory. >> >>Theoretically: no. In practice: yes. Because some consequences of >>set theory have invaded into physics by the fact that mathematics >>becomes somewhat _applied_ there, huh ! Geez ... > > Physics can influence where mathematics is heading, but not what it is > finding there. Its verdict on mathematics can't be "true"/"false", > but just "interesting"/"irrelevant". That's the other way around. Mathematics _influences_ where physics is heading and what it is finding there. Its verdict on physics is "true"/ "false". All physicists except one (: me) have no doubt that mathematics is quite reliable for this purpose. That's exactly what bothers me. Roger Penrose entered physics, together with his machinery called set theory, and he made predictions concerning the nature of black holes. Herewith the notion of 'infinity' is put into practice and it finds a physical interpretation, namely the singularity that resides behind the event horizon of a black hole. It is emphasized that the 'infinity' concept here is the one as understood by mainstream mathematics, since i.e. intiutionism hasn't any applications in physics and it is not interested in such. (As Brouwer has said: mathematics has nothing to do with reality. Intuitionism is extremely idealistic.) But suppose it had been otherwise. Suppose that intuitionism had become the mainstream in mathematics. Then we would have had quite a different concept of infinity, far more restrictive anyway. Now guess what would have happened to physics if Roger Penrose had been such an intuitionist or a constructivist ? Would we still have then those theories about the Cosmic Censor ? Personally, I don't think so. I find this disturbing. Our picture of the world should not depend upon the kind of mathematics accidentally used. Han de Bruijn |