An uncountable countable set
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From: Virgil on 18 Aug 2006 15:04 In article <1155918918.402162.4860(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. No one in any geometry I am aware of ever claims that any distance is infinite, merely that the set of them is infinite. This is the case even when the distances themselves are bounded, the set of them can be and usually is infinite. > And that is the point! "Mueckenh"'s head has a point, but he is otherwise pointless. > > Regards, WM
From: Virgil on 18 Aug 2006 15:15 In article <1155919049.835564.6210(a)m73g2000cwd.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 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. Any finite number of successive stairs has finite height, but without any limit on height unless there is a corresponding limit on number. If we have infinitely many finite heights with no upper bound on them the the set of heights is unbounded, meaning not having any finite limit. This does not imply that the set contains any infinite value any more than having a set of naturals having no finite upper limit requires that a member of that set itself be infinite. Perhaps when "Mueckenh" can produced a list of all the specific statements about sets on which he bases his arguments, i.e., an explicit axiom system, he will begin to reason sanely.
From: David R Tribble on 18 Aug 2006 19:19 mueckenh wrote: > 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 ... Yes, infinitely many finite-sized steps produce a staircase of infinite height. That is admitted in mathematics. > ... but this staircase is allegedly of finite height. No, the staircase is infinite in height, but each step is finite in height. The distance between any two steps is finite, but the total height of all the steps together (the entire staircase) is infinite.
From: Tony Orlow on 18 Aug 2006 19:53 David R Tribble wrote: > mueckenh wrote: >> 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 ... > > Yes, infinitely many finite-sized steps produce a staircase of infinite > height. That is admitted in mathematics. All too often "admitted" that those infinities are equal. > >> ... but this staircase is allegedly of finite height. > > No, the staircase is infinite in height, but each step is finite in > height. The distance between any two steps is finite, but the total > height of all the steps together (the entire staircase) is infinite. > So, the greatest possible value that any natural number can reach, in height, is infinite? Certainly you claim that the width of this square, the count of all such values, is infinite. But are the values themselves, even potentially, infinite? Isn't The width the same as the height of the square, as drawn? I say the set, as defined, is the same size as its maximum element. :)
From: Tony Orlow on 18 Aug 2006 19:55
Tony Orlow wrote: > David R Tribble wrote: >> mueckenh wrote: >>> 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 ... >> >> Yes, infinitely many finite-sized steps produce a staircase of infinite >> height. That is admitted in mathematics. > > All too often "admitted" that those infinities are equal. > >> >>> ... but this staircase is allegedly of finite height. >> >> No, the staircase is infinite in height, but each step is finite in >> height. The distance between any two steps is finite, but the total >> height of all the steps together (the entire staircase) is infinite. >> > So, the greatest possible value that any natural number can reach, in > height, is infinite? Certainly you claim that the width of this square, > the count of all such values, is infinite. But are the values > themselves, even potentially, infinite? Isn't The width the same as the > height of the square, as drawn? I say the set, as defined, is the same > size as its maximum element. > > :) By the way, hello! Nice to see you all again, I think. Many peacefulnesses upon you. Smiles, Tony |