An uncountable countable set
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From: Dik T. Winter on 17 Aug 2006 21:26 In article <1155817888.248664.91020(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > 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. What are you talking about? There is *no* infinite line. There are infinitely many finite lines. > f the real cannot be > completed, then it is unknown, whether it differs from every entry of > the list. The number is different from each finite line. And as there are only finite lines, it is different from all lines. I stated this before, but you do not agree, there are sets without a last element. This is what the axioms tell me. You may disagree with that, and reject the axiom of infinity, but if you want to show an inconsistency with the axiom of infinity you have to show how using that axiom you can prove A and ~A. Your statement above "It is not different for any infinite line" is a direct rejection of that axiom. So it is not valid when trying to detect an inconsistency. > > (There are indeed also many real numbers that are not > > computable..., but they are nevertheless real numbers, by the definition > > of "real number".) > > Like the devils which are devils by the definition of devil? Perhaps. What is that definition? > > > > > I think that 10 % must exist if the whole set = 100 % does exist? > > > > > No? > > > > > > > > No. > > > > > > Here you see the inconsistence of set theory. The set shall exist > > > actually = completely, but the first 10 % shall not exist. > > > > What is the inconsitency? Can you show what axiom of set theory or what > > proposition from set theory it violates? > > If you tell me what is to be understood by "set" and by "existence" in > set theory. Sorry. You made a statement. I asked you a question that you support your statement. Now you tell me that you can not support that statement unless I provide some definitions? To be precise, you state: Here you see the inconsistence of set theory. I ask: What is the inconsistency, can you prove it? So your initial statement is completely unfounded, because you stated it without additional information and want information to give a foundation. But you succeeded again in diverting the discussion from your initial assertion. The countable uncoutable set. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 18 Aug 2006 03:17 Dik T. Winter schrieb: > In article <1155725133.570350.44280(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > mike4ty4(a)yahoo.com schrieb: > ... > > The set of all constructible numbers including all real numbers in > > Cantor-lists and all of their diagonal numbers is countable. > > Nevertheless most people assert that the construction of a diagonal > > number would show the uncountability of this countable set of > > constructible numbers. > > But can the list of constructable numbers be constructed? No, of course it cannot, because only finite sets can be constructed, at least if there is not a law connecting infinitely many numbers. > That they > are countable comes from other considerations. It doesn't matter where that comes from. In fact all numbers which can be constructed form a countable set. And those which can be listed form a subset of this set. But the construction of one element of this set is taken as a proof that there are uncountably many constructible numbers. (Unconstructible numbers were not at all taken into account when Cantor's proof was published.) Regards, WM
From: mueckenh on 18 Aug 2006 03:18 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. Regards, WM
From: mueckenh on 18 Aug 2006 03:19 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, then you can be sure that the set theorists are plastered enough to celebrate you as a genius. And you can be sure that ZFC and NBG will never be proven inconsistent. Regards, WM
From: mueckenh on 18 Aug 2006 03:21
Virgil schrieb: > > > 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. But it is known that this set is countable. Don't panic, ZFC and NBG are without any contradiction and will remain so. Regards, WM |