An uncountable countable set
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From: mueckenh on 23 Aug 2006 16:39 Dik T. Winter schrieb: > In article <J472w4.Lt1(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > > In article <1155885821.815144.187270(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > To be more correct. The first proof is not relying on the > > > > representation of reals, the second (diagonal) proof is not > > > > about reals at all. > ... > > > > The answer was: no. To clarify, in Cantor's diagonal proof > > > > there are no dual representations, so there was no need for Cantor > > > > to consider them. It is about infinite sequences of symbols. But > > > > Mueckenheim was insidious there, as he answered no, while not giving > > > > the clarification. > > > > > > Cantor considered his 2nd proof as being /valid/ for real numbers. > > With 2nd proof you mean the diagonal proof, I think? Yes. > > > > he did not use irrational numbers: "Aus dem in § 2 Bewiesenen folgt > > > nämlich ohne weiteres, daß beispielsweise die Gesamtheit aller > > > /reellen Zahlen/ eines beliebigen Intervalles sich nicht in der > > > Reihenform w_1, w_2, w_3, ..., w_v, ... darstellen läßt. > > > Es läßt sich aber von /jenem Satze/ ein viel einfacherer Beweis > > > liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist." > > > (The italics are mine) > > > Ok, found and reformatted. I will give an English translation: > From what has been proven in section 2 (*) follows that e.g. the > set of real numbers in an arbitrary interval can not be put in > a sequence w_1, w_2, w_3, ..., w_v, ... . > It is however possible to construct a much simpler proof for > that theorem (**) that is independent from the observation of irrational > numbers." > Now (*) means section 2 of the paper that shows the first proof. Yes. > What > does (**) mean? *Not* what you imply. What did I imply? > It talks about the theorem stated > in the first half of that paragraph, which I will give it first in German: > In dem Aufsatze, betitelt: "Über eine Eigenschaft des Inbegriffs aller > reellen algebraische Zahlen (Journ. Math. Bd. 77, S. 258) [hier S. 115], > findet sich wohl zum ersten Male ein Beweis für den Satz, daß es > unendliche Mannigfaltigkeiten gibt, die sich nicht gegenseitig auf die > Gesamtheit aller endlichen Zahlen 1, 2, 3, ...m v, ... beziehel lassen, > oder, wie ich much auszudrücken pflege, die nicht die Mächtigkeit der > Zahlenreihe 1, 2, 3, ..., v, ... haben. > translated: > In the article, titled: "Über eine Eigenschaft des Ombegriffs aller > reellen algebraischen Zahlen (Journ. Math. Bd. 77, S. 258) [hier > S. 115], can for the first time be found a proof of the theorem > that there are infinite sets that are not in bijection with the set of > natural numbers 1, 2, 3, ..., v, ..., or, as I say in general, do not > have the same cardinality as the natural numbers 1, 2, 3, ..., v, ... . > So he does *not* consider the diagonal proof adequate for the reals. > He considers it adequate and simpler than the first proof about the > theorem that there are infinite sets > with a cardinality not equal to > the natural numbers. Yes, of course. You only missed the clue. This first proof *is* the proof concerning the reals and nothing more. From *this alone* he made further conclusions (about numbers: so gibt es in jedem Intervall ....unendlich viele Zahlen, die nicht in Omega enthalten sind.) And all what it talks about in the final paragraph is lead back to *this proof concerning the reals (in §2)*: "In der Tat überzeugt man sich durch eine ähnliche Schlußweise wie in § 1, daß der Inbegriff ... sich in der Reihenform ... auffassen läßt, _woraus, mit Rücksicht auf diesen § 2_, die Richtigkeit des Satzes folgt." (Werke, p. 118) You should read his first paper. Until 1891 he had no other proof than that of this § 2 using real numbers. Regards, WM
From: MoeBlee on 23 Aug 2006 17:04 mueckenh(a)rz.fh-augsburg.de wrote: > In real mathematics (as opposed to matheology) > we have the valid foundation: What cannot be addressed, that does not > exist. Do you have axioms for this "real mathematics"? MoeBlee
From: Tony Orlow on 23 Aug 2006 17:09 Dik T. Winter wrote: > Some clarification. Thanks, Dik! :) > > In article <J4CnBH.DqI(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > > In article <J472w4.Lt1(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > > > In article <1155885821.815144.187270(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > ... > > > > > To be more correct. The first proof is not relying on the > > > > > representation of reals, the second (diagonal) proof is not > > > > > about reals at all. > ... > [ About Cantor's proofs:] > I was not even completely correct in that statement. After completely > reading his first proof and his second proof (the diagonal proof), I am > quite sure that both his proofs were targeted at the following theorem: > There are sets with a cardinality larger than that of the natural > numbers. Yes, that would seem to be about the size of it. > The first part of his first proof shows that a complete ordered field > has cardinality larger than the natural numbers. In his proof he did > not rely an any properties of the reals other than that they form a > complete ordered field (he uses reals to exemplify). Specifically, he demonstrates that there will always exist an unaccessible real in any finite interval, given that we are only allowed a finite (natural, that is) number of iterations. Therefore, indeed, the number of reals in a finite interval is greater than any finite natural. That proof is essentially valid, but it's not a proof that the reals cannot be linearly ordered in a discrete manner. > (The second > part shows that the cardinality of the set of subsets of a set is > strictly larger than that of the original set. This proof comes > close to the proof provided by Hessenberg, but is, in my opinion, > a bit less strict.) It is related to the powerset through |P(S)|=2^|S|, but he did it in decimal, no? In any case, the powerset relation boils down to the symbolc equation N=S^L, where S is the number of logical states allowed (there can be more than two in various systems) and where L ultimately corresponds to the size of the root set. So, the two are related. > > What he states in the quote provided by Mueckenheim is that that proof > can be easily used to show that the cardinality of any set of reals in > an interval is larger than the cardinality of the natural numbers. I don't disagree that the first is infinite while the second is unbounded bt finite, and therefore smaller. > > His second proof is about sets of sequences, and his diagonal proof > shows that the cardinality of the set of infinite sequences with > two possible elements is strictly larger than the cardinality of > the natural numbers. Zermelo has indicated how that can be converted > to a proof that the reals have cardinality larger than that of the > natural numbers. But Cantor had *no* intention to prove that at that > point at all, and did nowhere write that it could be applied for that > purpose. Yes, the second is really a proof about power set and/or symbolic representations of quantites. > > Cantor's one and only purpose was a proof of the theorem: > There are sets with cardinality larger than that of the natural numbers. Considering that the set of finite naturals is ultimately finite, as Wolfgang has been trying to express, that's not a surprising result. What's surprising is Dr. M's reluctance to admit the actually infinite in the real interval. > > Has the book by Zermelo on Cantor's work ever been translated into English? > If not, the non-German speakers are certainly missing something. As with > the untranslated book by O. Perron on continued fractions. (BTW, with > Zermelo's book also the non-French speaking are also missing something; > parts of it are in French.) I dunno. Did I miss anything? :) Tony
From: Tony Orlow on 23 Aug 2006 17:13 Virgil wrote: > In article <ecdofb$i4g$1(a)ruby.cit.cornell.edu>, > Tony Orlow <aeo6(a)cornell.edu> wrote: > >> Virgil wrote: > >>> It is only the partial sums, a_k = sum(n=1->k: 9*10^-n), k in N, which >>> never reach 1. >>> The limit actually is EXACTLY 1, by reason of how limits are defined. >> The limit is the least upper bound of the sequence of values, Virgil. >> The limit of 0.9999... is 1. :) > > How does that differ from what I said? Hi Virgil! Your hair looks nice tonight. ;) It doesn't differ at all. It's so exactly he same, I think it's kismet. And, it's so nice to know you've finally come around to realizing there's a difference between 0.999... and 1, that 1 is the limit of that sequence, but that it never actually reaches it. I'm starting to have hope for you again. OK, maybe that's just because I'm in a delusional mood. Don't spoil it. This was nice. Thanks. ;) Tony
From: Tony Orlow on 23 Aug 2006 17:25
Virgil wrote: < snippety doo dah > >>> Which is sufficient justification for refusing to allow TO's extension >>> of induction beyond standard induction. >> Or, alternatively, sufficient justification for rejecting >> transfinitology and the notion of a specific boundary between finite and >> infinite. > > TO is the one who demands a "specific boundary" between finite and > infinite, what we rerquire and what we have is specific definitions of > what constitutes a finite set and what constitutes an infinite set. No, what my system requires is a workable variable that's consistent whether it's finite or infinite. >>>>> The only acceptable forms of induction in ZF or NBG are finite >>>>> induction and transfinite induction, neither of which has TO shown he >>>>> knows how to apply. >>>> But I am explicitly suggesting the alternative, that if f(x)>g(x) for >>>> all x>y, then it is true for all infinite x. This leads to a clearer set >>>> of conclusions. >>> TO's alternative also leads to inconsistencies with the status quo ante, >>> which is sufficient reason for rejecting it. >> Huh? You mean, if I reject your axiom system, and concoct another, the >> fact that yours came first means that contradictions between the two are >> the fault of mine? > > What I mean is that you cannot impose your axioms/assumptions on top of > our ZF, ZFC of NBG and then object to the contradictions which arise as > being solely due to our axioms. I don't. I freely adimt regularly that the contradictions arising from my assumptions are between those and the standard assumptions. Others claim all contradictions arise from within my assumptions, which isn't the case. > > Our axioms sets do not allow your form of induction. Assuming your > version in addition to ours gives rise to contradictions, so we reject > your assumptions as contradictory. You're absolutely correct. I do not want you to consider infinite-case induction in ADDITION to ZFC or NBG, but as an ALTERNATIVE. They are clearly mutually incompatible. > > If ever TO completes his own system of axioms, including if he wishes > his version of induction, we shall see whether it has obvious > inconsistencies. Ours don't. > Okay, well, when the baby's back in daycare (1 week) and the boys back in school (2 weeks) then I'll have more time, but this is good prep work. Thanks. :) > >> Hmmmm..... Nah. If my ideas all fit together and >> provide at least as robust a system as the status quo, then seniority >> really doesn't count. > > Having a system does count, and while we have several, TO has none. I have one, but not polished and published yet. Still, it's enough for you to stop objecting out of pure habit. > >>> As TO has shown no talent for recognizing truth, and even seems to have >>> a talent for avoiding it, his judgements on where mathematical "truth" >>> lies are, at best, untrustworthy. >>> >> Define "truth". > > The tautologies of formal logic are truths. Those are the only ones > mathematicians qua mathematicians can be sure of. > Anything else is uncertain. Then the axioms of ZFC and NGB are not "true". So, stop pretending they are. There are other valid perspectives on the matter. > >>>> No, transfinitology doesn't satisfy my spiritual needs. >>> For spiritual needs, one needs something a little less literal minded >>> than mathematics. >> What makes you think mathematics is literal? It's certainly not >> concrete, but very abstract. What the numbers refer to is anyone's guess >> when they're doing the algebra. That's the beauty. It only becomes >> literal upon application. > > Absolutely backwards! Mathematics is at its most literal minded when > most abstract. To the extent that it becomes applied it loses its > literal mindedness. Define "literal". I read it as "actual". >>>>> What does one do for sets which are not ordered, or ordered sets which >>>>> have no max?, both of which are abundant. >>> >>>> Examples? Symbolic sets (languages) need not be ordered to be measured, >>>> necessarily, and sets which have no distinct range have no distinct >>>> size, unless the elements themselves have no relative measure. >>> The multidimensional point sets of multidimnsonal spaces, such as real >>> vector spaces, are not ordered and are not orderable in any way >>> consistent with their geometric properties. >> . It's not linear > > Thus, being non-linear, it does not fit on any number line. > > By which TO admits his error. Uh, no. With the snip of context, that doesn't even make enough sense to respond to. TO |