An uncountable countable set
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From: mueckenh on 18 Aug 2006 12:31 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 a"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. Regards, WM
From: mueckenh on 18 Aug 2006 12:32 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. Regards, WM
From: mueckenh on 18 Aug 2006 12:33 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? In particular what has it to do with the failure of Cantor's proofs? Regards, WM
From: mueckenh on 18 Aug 2006 12:35 Virgil schrieb: > 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. But the set of distances between 0 and the points cannot contain an infinite distance without at least one of the distances being infinite. And that is the point! Regards, WM
From: mueckenh on 18 Aug 2006 12:37
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 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. Learn: Infinitely many stairs require infinite height. But that it not admitted in mathematics. There have to be infinitely many stairs which give an infinitely long staircase but this staircase is allegedly of finite height. It seems that you are at the point to grasp that this is impossible and a contradiction of set teory (for stairs of unit length and height). Regards, WM |