An uncountable countable set
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From: Virgil on 18 Aug 2006 14:40 In article <1155900344.389115.79050(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155817888.248664.91020(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > > > > > Nobody ever has said that the diagonal is ever completed. There > > > > > > is a > > > > > > definition of that number, and it is easy to show, with that > > > > > > definition, > > > > > > that that number (when completed) would be different from all > > > > > > numbers > > > > > > on > > > > > > the list. > > > > > > > > > > But if it is impossible to complete it, then the result is void. > > > > > > > > There are many real numbers that can not be completed. But by the > > > > definition of "real number", each sequence of decimal digits defines > > > > a real number, and it is easily shown, by the above, that that real > > > > number is not in the list because it is different from each number > > > > in the list. > > > > > > > > > Oh no. It is not different for any infinite line but only for some > > > finite lines. > > > > > > Which numbers in the list is it NOT different from? > > > > We can show it is different from the first. > > > > We can show that if it is different from any one of them, it is also > > different from the next one. > > > > By the inductive principle, which is valid in our axiom systems, it is > > thus different from each and every single one of them. > > By the induction principle every line has a finite distance from the > first one, hence there is no infinite set of lines. Non sequitur. All you have managed to prove is that each distanced is finite, not that the set of distances is finite. > That is potential > infinity. The induction principle does not cover the whole set of > natural numbers. It does in ZF, ZFC and NBG. If "Mueckenh" is working in some other axiom system, "Mueckenh" should let us know what system that is. > > But if you admit that the induction principle is valid for the whole > set of natural numbers, then you can prove that lim[n-->oo] |{2, 4, 6, > ..., 2n}| / 2n > 1 > i.e. lim[n-->oo] 1/2 > 1. Why should I even try to prove true what I believe to be false? If by |{2,4,6,...,2n}| "Mueckenh" means Card({2,4,6,...,2n}), then induction will prove "Mueckenh"'s claim false for every n \in N, presuming N starts with 1, if we use von Neumanns N which includes 0, divisin by 0 is not defined.
From: Virgil on 18 Aug 2006 14:44 In article <1155900603.431779.61950(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > 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? > > > It has to pretend to exist (while it is only a line). Then it must have a consciousness of self, as objects without such consciousness cannot pretend to be what they are not. > > > > > 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. > > Fortunately for you. Otherwise it would curse many of those today's > "mathematicians" and kick them out of the temple. It is a "temple" from which "Mueckenh" excludes himself.
From: Virgil on 18 Aug 2006 14:51 In article <1155918702.907256.270320(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155886069.568472.204170(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > 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. > > > > It does in ZF or NBG. What axiom system is "Mueckenh" assuming? > > It does in ZFC and NBG? Numbers which yield only finitely many > differences of 1 between 1 and n yield infinitely many differences of > 1. "Mueckenh" ignores the important question to concentrate on trivialities. There are infinitely many n \in N, and thus infinitely many differences, Succ(n) - n, which equal 1. What axiom system is "Mueckenh" assuming? I repeat! What axiom system is "Mueckenh" assuming? Until we know that we might as well assume that everything "Mueckenh" claims is one of his axioms. In which case, "Mueckenh" has long since provided sufficient evidence that his axiom system is inconsistent.
From: Virgil on 18 Aug 2006 14:56 In article <1155918741.064502.307650(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155885821.815144.187270(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > 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. > > > > In what axiom system does "Mueckenh" claim that this is the case? > > In reality, of course. What are the axioms of that alleged system? Reality is not a mathematical system at all as far as I can tell. Mathematics and "reality" are effectively disjoint. Mathematics exists only in imaginary worlds, not in any world of "reality". So that "Mueckenh" is effectively saying that he does not have any axiom system at all, but is making things up as he goes along.
From: Virgil on 18 Aug 2006 14:59
In article <1155918790.826323.227510(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1155885684.223459.232190(a)74g2000cwt.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > But it is known that this set is countable. > > > > Is the set of all subsets of a countably infinite set itself countable? > > What has that to do with the question whether the set of constructible > numbers is countable or not? It tests what countability means to someone like "Mueckenh" who is flying without a net. > In particular what has it to do with the failure of Cantor's proofs? Without a concrete axiom system in which to analyse Cantor's proofs, "Mueckenh" has not the tools to either prove or disprove Cantor's work. |