An uncountable countable set
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From: Virgil on 16 Aug 2006 13:51 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.
From: Virgil on 16 Aug 2006 13:56 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.
From: Virgil on 16 Aug 2006 14:04 In article <1155725133.570350.44280(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > > A set can't be both uncountable and countable at the same time > > Are you sure? In either ZFC and NBG, quite sure. In ZF, absent C, the issue is slightly less clear. > 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. > 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.
From: Dik T. Winter on 16 Aug 2006 20:57 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. > > 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. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Franziska Neugebauer on 17 Aug 2006 06:10
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. 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". F. N. -- xyz |