An uncountable countable set
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From: mueckenh on 25 Aug 2006 06:21 Virgil schrieb: > In article <1156189552.184903.323170(a)m79g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > But often some expression like 0.111... is ivolved where not all digit > > positions can be indexed by natural numbers and n is understood to > > approach something I do not know. > > Your ignorance is not an acceptable excuse to those of us who have no > trouble indexing every digit of "0.111..." by a natural number. The problem is not indexing but "indexing without covering". That is easy to prove impossible for any finite natural number in unary representation. And, alas, there are only finite natural numbers. > > Division was possible and was practised in fact long before rings and > > fields were known. > > Not outside of what later became known as rings. > > > Division of rationals by rationals or reals by reals is quite different. > So we have no reason to suppose that divisions involving other than > naturals need behave like division of naturals, or even be possible > without specific definitions of what it is and how it works.. > The old Greek already developed the method of geometric division, using the so-called Gnomon. This method makes no difference between division of naturals, rationals, reals or whatever deserves the name "number". Every well educated mathematician knows it. (Geometry is a certain language of mathematics.) > > > > One must have a very restricted mind to believe that without fields and > > rings division was impossible. > > The set of positive naturals is not even a ring, but it allows at least > two forms of division, and this has been recognized by most of us from > well before "Mueckenh". Your frequence of self contradicitions increases. You just said "Not outside of what later became known as rings." Regards, WM
From: mueckenh on 25 Aug 2006 06:22 Virgil schrieb: > > But as we just investigate consistency, you cannot presuppose it. With > > your attitude it is impossible to find any inconsistency even in an > > inconsistent theory. Deplorably you are too simple to recognize that. > > If "Mueckenh" can deduce from any axiom system both a statement within > the system and its negation, "Mueckenh" will have found his > inconsistency. Which part of my proof concerning the binary tree is not in accordance with the ZFC axioms in your opinion? Regards, WM
From: mueckenh on 25 Aug 2006 06:27 Dik T. Winter schrieb: > > It does to any practicable approximation (not by my logic but in > > reality). > > Mathematics is not about reality... It is. You have not yet recognized it. > > > > And why is the notation lim n --> oo not clear. You have defined it just > > > a few days ago: > > > > > > The same is meant as in sequences and series like: > > > > > > lim [n --> oo] (1/n) = 0 > > > > > > n becoming arbitrarily large, running through all natural numbers, but > > > > > > always remaining a finite number < aleph_0. > > > > That is correct as long as n is a natural number. > > You said (see above) that the notation was not clear, while it was clearly > a natural number. The notion n --> oo is not clear, often, because it is intermingled: 1/n = 0 in the limit and n remaining a natural which leads to 1/n > 0 in the limit. > How do you *define* remainings? And I see now that you switched from your > original, which read: > > lim [n --> oo] |{2, 4, 6, ..., 2n}| / |{1, 2, 3, ..., n}| = 1 (because > > omega is a fixed, well defined and well deteremined quantum, according > > to Cantor) > What are the remainings in this case? There are none. > > > > > Division was possible and was practised in fact long before rings and > > fields were known. > > Yes? On cardinal numbers? Of course, on all numbers which were known at that time, on all numbers which deserved the name number. The old Greek practiced it in geometric form. > > For *finite* cardinals. Let's see: > |{1, 2, 3, ...}| / |{2, 4, 6, ...}| > I can do (among many others): > (1) I start with a bijection between {1, 2, 3, ...} and {2, 4, 6, ...}; > so the quotient is 1 without remainder. > (2) I start with a bijection between {1, 3, 5, ...} and {2, 4, 6, ...}; > so the quotient is 1 with a remaining that also yields a bijection, > so the total quotient is 2 without remainder. > (3) I start with a bijection between {2, 3, 4, ...} and {2, 4, 6, ...}; > so the quotient is 1, with remainder 1. > Depending on how I chose my bijections I can get every finite quotient > with every finite (non-negative) remainder. So, which one should I chose? We need not get into these details. It is enough to know that the result is > 1 or not. But we can do without any cardinal division: A n e |N: |{2, 4, 6, ..., 2n}| / 2n < 1 but lim[n-->oo] |{2, 4, 6, ..., 2n}| / 2n > 10 (in order to avoid infinity) > And, what is: > |{2, 3, 5, 7, 11, 13, ...}| / |{1, 4, 6, 8, 9, 10, 12, ...}| ? Not necessary to define for my proof. Regards, WM
From: mueckenh on 25 Aug 2006 06:31 Dik T. Winter schrieb: > Some clarification. > > [ 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. > 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). "Wenn eine nach irgendeinem Gesetze gegebene unendliche Reihe voneinander verschiedener reeller Zahlgrößen .... vorliegt, so läßt sich in jedem vorgegeben Intervalle ...eine Zahl ... (und folglich unendlich viele solcher Zahlen) bestimmen, welche in der Reihe (4) nicht vorkommt; dies soll nun bewiesen werden. Can you read German? Reelle Zahl, Zahlgröße, Zahl, Zahl, Zahl. 10 times "Zahl" appears in this paper. Cantor uses neither "field" nor "ordered" nor "complete". He uses only the reals and reals and reals. This is not only indicated by the headline "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" but also by the note that this was a proof of the existence of Liouville's transfinite nunmbers. What text did you read? > (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.) > > 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. > > 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. > > Cantor's one and only purpose was a proof of the theorem: > There are sets with cardinality larger than that of the natural numbers. He wrote that it was a simpler proof for his first theorem, i.e. the theorem that the reals are uncountable. Regards, WM
From: Tony Orlow on 25 Aug 2006 08:42
Virgil wrote: > 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. If the naturals are not a subset of the reals in ZFC and NBG, then those theories are even more screwed up than they already seemed. >>> 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. What's peculiar about it? > >> Is aleph_0>2? > > AS TO's comparison operator is according to TO defined only for reals, > the question is nonsense. > Then aleph_0 is not a count of any sort, or it would exist on the hyperreal line. It's not a number of any real sort. > > >>> What is transfinitology? >> You know, Cantorian religion. That Hollywood stuff. ;) > > As separate from TO religion and all of that Broadway stuff? Broadway was my stomping ground as a kid. In New York, we tell it straight up. You don't see New York's actors into all that Hollywood nonsense, and you don't see scientists accepting transfinitology either. >>> 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. > That's because you have no infinite values which behave sufficiently like numbers to support such a definition. > >> 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. The selection of any unit is done by simply choosing a point separate from the origin. When it comes to division using infinite values, one translates the geometric definition into symbolic form and applies induction formulaically. Tony |