Cantor Confusion
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From: Arturo Magidin on 1 Oct 2006 14:33 In article <LWOTg.5030$3E2.3848(a)tornado.rdc-kc.rr.com>, Poker Joker <Poker(a)wi.rr.com> wrote: > >"Arturo Magidin" <magidin(a)math.berkeley.edu> wrote in message >news:efmf1m$c88$1(a)agate.berkeley.edu... > >>>So if considering a single specific list >>>shows a flaw, then looking at ANY (ALL of them) list doesn't >>>help. > >> .. since no flaw has been exhibited by looking at any specific >> list (and "specific" in this case must mean explicit and specific, not >> a putative list with putative properties whose existence cannot be >> established a priori; otherwise, we might just say "take a list for >> which the argument does not work", which is of course nonsense), >> discussions about this are a waste of time. > >First you obfuscate the discussion by saying that specific cases >don't matter. No. FIRST, I presented a proof that holds for an arbitrary list. Then YOU obfuscated the matter by pretending your own shortcomings in understanding basic mathematical terminology somehow invalidates that proof. > Now you seem to imply they do, Still your own shortcomings being projected. If you have a proof that holds for an arbitrary list, in which the ONLY property of the list being used is the fact that it is a list, then consideration of particular specific cases is immaterial. You are free to check particular specific cases if it helps in YOUR understanding (or lack thereof) of the proof. If you can somehow exhibit a specific (explicit) instance in which the argument does not hold, then you would have shown that what was presented was an invalid argument. But in order to do so, one must exhibit a specific COUNTEREXAMPLE. In the case of a "proof" that attempted to show that for every real number x there exists a real number y such that x*y = 1, you would exhibit x=0, run through the proof (perhaps) and point out exactly which step is invalid with that specific number. That's fine. On the other hand, if we had a proof that for every real number x there exists a real number y such that x+y = 0, then a "particular case" would not be "let x be some real number, which might or might exist, which has no additive inverse; then your proof is wrong because your proof would imply it has, contradicting the fact that x does not have an additive inverse." At that point, your "particular case" is nothing but hot air and irrelevancy. > but in this discussion >they still don't because of the obfustated argument that somehow >they don't. > >After all, you never showed how step #2 isn't self-referential After all, you never showed you understood what step 2 really was; your claims that it was "self-referential" were hollow and as such they need not be addressed. The burden of proof is on you, since I have discharged mine by offering a valid proof. >in >the case that the process input is all real numbers. You would have to establish that such an input is possible, given the proof that was shown. You haven't. All you are doing is spinning your wheels. >Now I understand how someone can reject a fields medal. No doubt, your understanding of that is about as accurate as your understanding of Cantor's proof or of mathematical arguments in general. I wouldn't try to borrow against its value. -- ====================================================================== "It's not denial. I'm just very selective about what I accept as reality." --- Calvin ("Calvin and Hobbes" by Bill Watterson) ====================================================================== Arturo Magidin magidin-at-member-ams-org
From: William Hughes on 1 Oct 2006 14:43 Han.deBruijn(a)DTO.TUDelft.NL wrote: > William Hughes schreef: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > If logic and set theory clash, abandon set theory. > > > > Indeed, but logic and set theory do not clash. > > Set theory and intuition about infinite sets > > clash. > > Then abandon _both_ (formal / mathematical) logic _and_ set theory. > We are left with intuition. Fine. Oh by the way we are going to use my intuition. If you don't like it, too bad. Only I can tell what my intuition is. - William Hughes
From: Virgil on 1 Oct 2006 14:50 In article <1159698057.723006.10500(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > That was later conflated to a proof about the reals. > > > > > > > > It was later shown that it could be modified to form a proof that the > > > > set of all reals is uncountable. > > > > > > This was *not* "later shown", but at the very time of publishing in > > > 1890/91 Cantor considered this very proof as the proof of the > > > uncountability of he reals. > > > > > > Cantor, in the first paragraph: " Es l??t sich aber von jenem Satze > > > [uncountability of the reals] ein viel einfacherer Beweis liefern, der > > > unabh?ngig von der Betrachtung der Irrationalzahlen ist." > > > My translation: "Here is a much simpler proof of the theorem > > > [uncountability of the reals] which is independent of the reference to > > > irrational numbers" > > > > As it is not clear that this sentence refers to any such theorem, I take > > leave to doubt "Mueckenh"'s claim. > > Here is the full text: > > In dem Aufsatze, betitelt: ?ber eine Eigenschaft des Inbegriffs aller > reellen algebraischen Zahlen (Journ. Math. Bd. 77, S. 258), findet sich > wohl zum ersten Male ein Beweis f?r den Satz, da? es unendliche > Mannigfaltigkeiten gibt, die sich nicht gegenseitig eindeutig auf die > Gesamtheit aller endlichen ganzen Zahlen 1, 2, 3, ..., n?, ... > beziehen lassen, oder, wie ich mich auszudr?cken pflege, die nicht die > M?chtigkeit der Zahlenreihe 1, 2, 3, ..., n?, ... haben. Aus dem in > ? 2 Bewiesenen folgt n?mlich ohne weiteres, da? beispielsweise die > Gesamtheit aller reellen Zahlen eines beliebigen Intervalles > (alpha...beta)sich nicht in der Reihenform > > w1, w2, ... wn?, ... > > darstellen l??t. > Es l??t sich aber von jenem Satze ein viel einfacherer Beweis > liefern, der unabh?ngig von der Betrachtung der Irrationalzahlen ist. > > Regards, WM In the essay, calls: Over a characteristic of the epitome of all real algebraic numbers (Journ. Math. Bd. 77, S. 258), a proof probably is for the sentence for the first time that there is infinite variousnesses, which cannot be referred mutually clearly to the whole of all finite whole numbers of 1, 2, 3..., nue..., or, as I tend to be expressed, those not the power of the zahlenreihe 1, 2, 3..., nue... to have. From in 2 proving it follows easily that for example the whole of all real numbers of any interval (alpha... beta) sich not in the row form w1, w2... wnue... to represent leaves. However a much simpler proof can be supplied by that sentence, which is independent of the view of the irrational numbers.
From: Han.deBruijn on 1 Oct 2006 14:52 William Hughes wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > William Hughes schreef: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > If logic and set theory clash, abandon set theory. > > > > > > Indeed, but logic and set theory do not clash. > > > Set theory and intuition about infinite sets > > > clash. > > > > Then abandon _both_ (formal / mathematical) logic _and_ set theory. > > We are left with intuition. Fine. Oh by the way > we are going to use my intuition. If you don't like > it, too bad. Only I can tell what my intuition is. Better read better the add-on between parentheses. We are left with common speech logic and no set theory. Han de Bruijn
From: Virgil on 1 Oct 2006 14:55
In article <p5PTg.5102$3E2.2885(a)tornado.rdc-kc.rr.com>, "Poker Joker" <Poker(a)wi.rr.com> wrote: > "Peter Webb" <webbfamily-diespamdie(a)optusnet.com.au> wrote in message > news:451e79ad$0$28952$afc38c87(a)news.optusnet.com.au... > > > > "Poker Joker" <Poker(a)wi.rr.com> wrote in message > > news:3HmTg.25601$QT.205(a)tornado.rdc-kc.rr.com... > > >> I never tried to refute the uncountability of the reals. Too bad > >> you've never been able to understand that. > > > No, but you introduced a specious point that seemed to support this > > argument though. > > No I didn't. Showing a flaw in a proof is *FAR* from proving the > opposite. > > > You said that for any real x exists y such that x/y=0. > > > > This statement is false, at least within standard arithmetic, as you point > > out. > > Not if you neglect when x = 0. Then its true in general. > > > Cantor is different in that the statement is true, and easily proved. > > If you neglect x = 0. Then my statement is true and easily proved. Neglecting x = 0 does not eliminate x = 0. To eliminate x = 0, one must specifically include a requirement that excludes x = 0. |