An uncountable countable set
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From: mueckenh on 18 Aug 2006 03:23 Dik T. Winter schrieb: > In article <virgil-D45A8C.12042316082006(a)news.usenetmonster.com> Virgil <virgil(a)comcast.net> writes: > > In article <1155725133.570350.44280(a)74g2000cwt.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > ... > > > The set of all constructible numbers including all real numbers in > > > Cantor-lists and all of their diagonal numbers is countable. > > > > This does not prove that the set of all real numbers is countable. > > Besides which, Cantor's "diagonal" proof is his second proof of the > > uncountability of the reals, and his first proof does not rely on any > > decimal or other representation. There are also several other proofs > > extant, no one of which has "Mueckenh" falsified. > > 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. I think > it was Zermelo who first modified the second proof to a proof about the > reals. I do not know whether you follow all, but at one time you asked > Mueckenheim whether Cantor considered dual repersentations in his diagonal > proof. 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. But he did not use irrational numbers: "Aus dem in § 2 Bewiesenen folgt nämlich ohne weiteres, daÃ? beispielsweise die Gesamtheit aller /reellen Zahlen/ eines beliebigen Intervalles sich nicht in der Reihenformï? ï?®ï?®ï?®ï? darstellen läÃ?t. Es läÃ?t sich aber von /jenem Satze/ ein viel einfacherer Beweis liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist." (The italics are mine) > > > > Nevertheless most people assert that the construction of a diagonal > > > number would show the uncountability of this countable set of > > > constructible numbers. > > > > The diagonal proof certainly shows that the set of constructible numbers > > cannot be listed in its entirety. > > That is wrong (as I see it). There exists a list of constructible numbers > (countability proves that). But that list is not constructible, so the > diagonal number is not constructible. No infinite list (of independent numbers) is constructible, because no one can construct infinitely many different real numbers. This argument shows that the whole Cantor diagonal proof is void. But if there are any lists of constructible numbers, then their diagonal numbers are constructible too. And this construction does *not* show that there are uncountably many constructible numbers. This argument shows again that the whole Cantor diagonal proof is void. Regards, WM.
From: mueckenh on 18 Aug 2006 03:27 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > An infinite sum of 1's is not infinite? > >> > >> n > >> lim sum 1 = lim n =def L > >> n -> oo i = 1 n -> oo > >> > >> There is no such L in N. > > > > Correct. > > The antecedent is true. > > > Therefore there are not infinitely many difference[s] of 1 > > between natural numbers. > > Your consequent is proven false (see below). Therefore your implication > is false, too. You are in error. You just proved it to be true. The set of natural numbers (i.e., finite numbers n, i.e., numbers with finitely many differences of 1 between 1 and n) does not yield infinitely many differences of 1. > > A difference of two numbers b and a is usually denoted as b - a. We > introduce the difference operator "-" action upon ordered pairs: > > -(a, b) def= b - a > > "How many differences there are" means the cardinality of the set > of all pairs {(a, b)}. > > Restricting a and b to omega and to "difference[s] of 1" one gets > > P def= {(a, b) | a, b e omega & -(a, b) = 1} > = {(a, a + 1) | a e omega } > > Since there is a bijection between P and omega, namely > > B: P x omega def= {((a, a + 1), a) | a e omega}, > > it follows that P ~ omega, meaning P is of same cardinality as omega. > > Thus there are "as many difference[s] of 1 between natural numbers as > there are natural numbers". Since the cardinality omega is infinite > there *are* "infinitely many difference[s] of 1 between natural > numbers". Correct, but your final conclusion is the typical mistake of set theorists. 1) The set exist. 2) The elements do not exist. You state it in the form: 1) There are infinitely many differences of 1. 2) The sum of these is not infinite. Why not? You should try to comprehend what "existence" means. Regards, WM
From: Virgil on 18 Aug 2006 03:30 In article <1155885500.337316.94880(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155725007.690845.21360(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > The representation of infinitely many natural numbers by > > > the stairs requires infinitely many stairs. Infinitely many stairs > > > require infinite height. > > > > Which in mathematics, a purely imagined world, is achievable, though not > > so in any physical world. > > But it is disliked very much in mathematics, because the set of only > *finite* numbers is pretended to be infinite. Does the set of points in a line have to "pretend" to be infinite? That infinite sets are disliked by "Mueckenh" does not in anyway indicate that they are disliked by mathematicians, nor that they cause any problems for mathematicians that have to been adequately dealt with. And "mathematics" itself, being inanimate, has no opinion on the issue at all.
From: Virgil on 18 Aug 2006 03:31 In article <1155885500.337316.94880(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155725007.690845.21360(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > The stair case is my proof that a union of infinitely many 1's gives an > > > infinite set. > > > > The union of infinitely many of any one thing is that thing. > > > > > The representation of infinitely many natural numbers by > > > the stairs requires infinitely many stairs. Infinitely many stairs > > > require infinite height. > > > > Which in mathematics, a purely imagined world, is achievable, though not > > so in any physical world. > > But it is disliked very much in mathematics, because the set of only > *finite* numbers is pretended to be infinite. Posting your wrongheaded ideas multiple times in no way improves them.
From: Virgil on 18 Aug 2006 03:38
In article <1155885581.167883.220630(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155725007.690845.21360(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > In article <1155640559.355146.166090(a)74g2000cwt.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > Dik T. Winter schrieb: > > > > ... > > > > > > Says who? You state that an infinite union of finite sets is > > > > > > finite. > > > > > > I ask you for a quote or a proof, and you refrain to give some. > > > > > > Are > > > > > > you > > > > > > not able to either prove that or give a quote? > > > > > > > > > > It is the definition of a natural number that it is a (positive) > > > > > finite > > > > > number. > > > > > > > > What is the relation with your statement that "an infinite union of > > > > finite sets is finit"? Where is the *proof* of that statement? Are > > > > you not able to either prove that or give a quote? > > > > > > The stair case is my proof that a union of infinitely many 1's gives an > > > infinite set. The representation of infinitely many natural numbers by > > > the stairs requires infinitely many stairs. Infinitely many stairs > > > require infinite height. > > > > > > You disagree. You state there is no stair of infinite height, so you > > > state that an infinite union of finite sets (1's) is not infinite. > > > > I state the opposite, that in the world of imagination that is > > mathematics, a staircase of infinite height is simple to achieve, but it > > does not require any stair to be infinitely high. The inability of > > anyone to find any internal inconsistency in ZF and NBG. while not > > absolute proof, is certainly compelling evidence in support of my > > position. > > Of course. If you pretend that one number of a set of numbers is > infinite without one of them being infinite I say no such thing. I say that no number need be infinite in order to have infinitely many numbers, just as I say that no step need be of infinite height in order to allow the set of steps to have infinite height. Similarly, the set of points in a finite line segment can be an infinite set without any point being by itself infinite. It is "Mueckenh" who is doing all the pretending here. he is pretending that others say what he wants them to have said instead of what they actually have said. > And you can be sure that ZFC and NBG will never be proven inconsistent. Cetainly not by anyone as inept as "Mueckenh". |