An uncountable countable set
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From: Dik T. Winter on 23 Aug 2006 21:30 In article <1156365561.644074.5000(a)p79g2000cwp.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... Let me reformat the first paragraph completely in translation (I think you agree with the translation): > > In the article, titled: "?ber eine Eigenschaft des Imbegriffs 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, ... . > > 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." > > 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. You are also missing the clue. The first paragraph shows that he thought it was actually a proof of the theorem that there were sets with cardinality larger than the naturals (and it was, it did show such about the reals). That the first proof considers only reals and nothing more, does not matter at all. In this paragraph he is stating the theorem as is, and for *that* theorem he does give a simpler proof. The conclusion that it would also be a simpler proof about the reals is not warranted by the text. From the start of the second sentence we already immediately see that the reals were used as an example. In German: "beispielweise". > You should read his first paper. Did you think I have not read it? But the contents are not relevant here. From the paragraph quoted I can only conclude that in that context he considered it a proof of the theorem that there are sets with cardinality larger than that of the naturals. > Until 1891 he had no other proof than that of this ? 2 using real > numbers. So in 1891 he found a proof not using real numbers. That does not show that he thought his new proof could be applied without modification to the real numbers. And that is what you did imply. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 23 Aug 2006 21:48 In article <ecig8i$ta$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > Dik T. Winter wrote: .... > > 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. Sorry, you have got that wrong. We are allowed an infinite number of iterations indexed with finite numbers (whatever that may mean). > 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. This is approximately right. The number is larger than the number of finite naturals. Whether the reals can be linearly ordered in a discrete manner (eh? quite a few buzzwords here without definition) depends on the axiom of choice, and a few more things. > > (The second > > part shows that the cardinality of the set of subsets of a set is > > strictly larger than that of the original set. Actually this is the second part of again another article. I will look it up when I am close to the book again. > > 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? No. Nowhere in his papers did Cantor use decimals. > 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. You make here as much sense as in much earlier articles. > > 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. Pray, for once, provide a definition. A set is either finite or infinite. And if a set is finite, by the definitions there is a largest element. So, how do you define "unbounded but finite"? > > 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. Not at all. There is no question about "representation of quantities". > > 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. What does not surprise me at all is that apparently you do not understand the discussions at all. > > 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? Quite a lot. When was the last time you did read a book about set theory? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 23 Aug 2006 21:50 In article <ecih7f$33a$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Virgil wrote: > > In article <ecdsfu$om3$1(a)ruby.cit.cornell.edu>, > > Tony Orlow <aeo6(a)cornell.edu> wrote: > > > >> imaginatorium(a)despammed.com wrote: > > > >>> Therefore [ in a ring] there cannot be a nonzero p > >>> such that x . 0 = p, which is what we would need to have a > >>> multiplicative inverse of zero. > >> That is all very true of absolute 0 and oo. Where you instead substitute > >> a measurable infinity for oo, and its inverse for 0, then that > >> infinitesimal value is greater than 0, and the equation doesn't hold. > > > > That constraint holds in absolutely every ring. > > > > Until TO defines the entire arithmetical structure of his system and > > proves there is a model satisfying that definition, the zero object has > > no multiplicative inverse in any arithmetic containing non-zero elements. > > > >>> Of course, you can append an object to a ring and call it Bigun (or > >>> anything else) and investigate the resulting structure (see javascript > >>> and my lens calculators for a practical example), but this structure > >>> will not be a ring. > >> Big'un already exists in 2's complement as 1000... It's its own additive > >> inverse, and not 0. > > > > In what axiom system, TO? > > > > TO has not shown his alleged 'bigun' is even possible in ZF or NBG and > > he has no system of axioms of his own. > > > > So in what axiom system, TO? > > You ask for the chicken, I sit on the egg. TO often has laid an egg, but I suspect his sitting on one will produce nothing more productive that a flat egg.
From: Virgil on 23 Aug 2006 22:03 In article <44ece1fb(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > Tony Orlow wrote: > >> Set theory contradicts with: > >> > >> (1) E y e N, A x>y, x< 2*x < x^2 < 2^x (y=2) > >> > >> because: > >> > >> (2) A y e N, aleph_0>y > > > > I don't know what you intend '<' to stand for. For the domination > > relation? The less than relation on ordinals? > > The standard "less than" operator, commonly used for finite reals. But since one is only talking about naturals, which are separate from the reals in ZF and NBG, that makes the TO (1) statement nonsense. > > > > > I don't know what is meant by '(y=2)' in the larger formula. > > I mean that for x>2, 2*x < x^2 < 2^2. For x=2 they're equal. Then (1) is a very peculiar way of trying to convey that claim. > Is aleph_0>2? AS TO's comparison operator is according to TO defined only for reals, the question is nonsense. > > What is transfinitology? > > You know, Cantorian religion. That Hollywood stuff. ;) As separate from TO religion and all of that Broadway stuff? > > > What is the definition (and in what theory is > > this definition?) of '/' where w (omega) is in the numerator? > > Well, I was calling it Bigulosity. The definition of x/y is "how many > intervals of length y fit into an interval of length x?". As we have no definition of length valid for intervals whose endpoints are not finite reals, it means nothing. > This is > constructible using straightedge and compass. DO let us see your straightedge and compass construction of Aleph_0^2/Aleph_0, TO. I need a new wall hanging.
From: Virgil on 23 Aug 2006 22:13
In article <ecii1c$4cc$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > imaginatorium(a)despammed.com wrote: > > So should we understand that in Tonyspeak "absolute 0" just means what > > mathematicians call zero. What is the point of using different words? > > > > The point is that infinitesimal values are considered 0 in the standard > math, and where we're discussing infinitesimal, it's prudent to > distinguish between them and innfitesimal values. To misses the real point, which is that in what are called standard models of the naturals, integers, rationals and reals, infinite and infinitesimal elements simply do not exist at all. One has to create the so called non-standard models before one can even talk about them. > Then you put aside your taught assumptions for a moment, and consider a > different approach. I'll hold my breath in the meantime. Give is a complete set of assumptions which work together, TO, and we will have something to consider. As long as TO merely wants to patch his assumptions onto current systems it is not going to work. |