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From: Tony Orlow on 17 Oct 2005 10:40 stephen(a)nomail.com said: > Tony Orlow <aeo6(a)cornell.edu> wrote: > > stephen(a)nomail.com said: > >> albstorz(a)gmx.de wrote: > >> > >> > But there is a slight difference. Since there is no infinite natural in > >> > form of a set of Os and since after every set of #s there should be a > >> > O, the size of the set of the naturals as sets of #s could not extend > >> > the "biggest" number of the naturals in form of sets of Os. > >> > Since there is no biggest number and since there is no infinite number, > >> > the size of the set of numbers in form of sets of #s is undefined as > >> > the biggest natural number is undefined. > >> > >> Whoever said the size of a set has anything to do with the "biggest" > >> element? > > Stephen, did you even look at the diagrams he presented? Do you not see that > > the width of the square and the height are the same. Do you not see that the > > width is the count of naturals and the height is the value? The picture said > > so, that's who. > > What square? The sides of a square are line segments. > The four corners of the square are defined by the ends > of those line segments. If your lines extend indefinitely, > then there is no square. > > This is not a square: > +-----------..... > | > | > . > . > . > > A square has four corners. This only has one "corner". > Remember, infinite lines do not end. Not even "at infinity". (sigh) As Albrecht said, the square is defined by the diagonal at 45 degrees. For every natural value represented by 0's in the diagram there is an equal count represeted by #'s. This is the identity relationship between count and value that I've been talking about. Think of it as the limit of a square as the side goes to oo. Your objection is just another form of "No Largest Finite!! No Diagonal Corner!!! (jingle jangle)" Oh, nice wind chime!! > > <snip> > > > It says everything about the size of the set of naturals compared with the > > values in the set. Can the square be wider than it is tall? No? Well, then, > > what does that say to you? (probably nothing, of course) > > As there is no square in the first place, it is irrelevant > if a square can be wider than it is tall. There is no spoon. (sigh) > > Stephen > -- Smiles, Tony
From: David Kastrup on 17 Oct 2005 11:13 Tony Orlow <aeo6(a)cornell.edu> writes: > stephen(a)nomail.com said: >> Tony Orlow <aeo6(a)cornell.edu> wrote: >> > stephen(a)nomail.com said: >> >> albstorz(a)gmx.de wrote: >> >> >> >> > But there is a slight difference. Since there is no infinite >> >> > natural in form of a set of Os and since after every set of #s >> >> > there should be a O, the size of the set of the naturals as >> >> > sets of #s could not extend the "biggest" number of the >> >> > naturals in form of sets of Os. Since there is no biggest >> >> > number and since there is no infinite number, the size of the >> >> > set of numbers in form of sets of #s is undefined as the >> >> > biggest natural number is undefined. >> >> >> >> Whoever said the size of a set has anything to do with the >> >> "biggest" element? >> > Stephen, did you even look at the diagrams he presented? Do you >> > not see that the width of the square and the height are the >> > same. Do you not see that the width is the count of naturals and >> > the height is the value? The picture said so, that's who. >> >> What square? The sides of a square are line segments. The four >> corners of the square are defined by the ends of those line >> segments. If your lines extend indefinitely, then there is no >> square. >> >> This is not a square: >> +-----------..... >> | >> | >> . >> . >> . >> >> A square has four corners. This only has one "corner". >> Remember, infinite lines do not end. Not even "at infinity". > (sigh) As Albrecht said, the square is defined by the diagonal at 45 > degrees. There is no "diagonal" for something that has only one corner. > For every natural value represented by 0's in the diagram there is > an equal count represeted by #'s. It does not make sense to talk about "an equal count" for things that don't end. > This is the identity relationship between count and value that I've > been talking about. Think of it as the limit of a square as the side > goes to oo. Your objection is just another form of "No Largest > Finite!! No Diagonal Corner!!! (jingle jangle)" Oh, nice wind > chime!! Well, too bad that you insist on making the same mistake all over again. Small wonder you get your nose rubbed into it all over again. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: David R Tribble on 17 Oct 2005 11:34 Albrecht S. Storz wrote: >> [...] >> Since there is no biggest number and since there is no infinite number, >> the size of the set of numbers in form of sets of #s is undefined as >> the biggest natural number is undefined. >> >> But the sequence of the sets of # fullfill the peano axiomes. So this >> set must be infinite. >> >> The cardinality of a set is not able to be infinite and "not defined" >> at the same time. >> This is the contradiction. > David R Tribble wrote: >> I don't see the contradiction. The size of the set is "not defined" >> to be the same as any natural number, and the set size is obviously >> infinite. This is no contradiction, since no natural number is >> infinite. >> >> The thing that is "not defined" is the largest natural, which obviously >> does not exist. But the set size is infinite, and is nicely defined >> by an infinite cardinal. >> >> You seem to be mixing the two concepts of "natural" and "cardinal" >> numbers to create a supposed contradiction, but that does not work. > Albrecht S. Storz wrote: > You are not able to understand that there is no difference between > numerals and sets. I have no problem seeing the correspondence between natural numbers and von Neumann sets. But neither of these are the same as cardinalities, which are not numbers, but measures (sizes) of sets. > My sketches shows this exactly. > Cantor proofs his wrong conclusion with the same mix of potential > infinity and actual infinity. But there is no bijection between this > two concepts. The antidiagonal is an unicorn. > There is no stringend concept about infinity. And there is no aleph_1, > aleph_2, ... or any other infinity. For that to be true, there must be a bijection between an infinite set (any infinite set) and its powerset. Bitte, show us a bijection between N and P(N).
From: Tony Orlow on 17 Oct 2005 16:54 David R Tribble said: > Albrecht S. Storz wrote: > >> [...] > >> Since there is no biggest number and since there is no infinite number, > >> the size of the set of numbers in form of sets of #s is undefined as > >> the biggest natural number is undefined. > >> > >> But the sequence of the sets of # fullfill the peano axiomes. So this > >> set must be infinite. > >> > >> The cardinality of a set is not able to be infinite and "not defined" > >> at the same time. > >> This is the contradiction. > > > > David R Tribble wrote: > >> I don't see the contradiction. The size of the set is "not defined" > >> to be the same as any natural number, and the set size is obviously > >> infinite. This is no contradiction, since no natural number is > >> infinite. > >> > >> The thing that is "not defined" is the largest natural, which obviously > >> does not exist. But the set size is infinite, and is nicely defined > >> by an infinite cardinal. > >> > >> You seem to be mixing the two concepts of "natural" and "cardinal" > >> numbers to create a supposed contradiction, but that does not work. > > > > Albrecht S. Storz wrote: > > You are not able to understand that there is no difference between > > numerals and sets. > > I have no problem seeing the correspondence between natural numbers > and von Neumann sets. But neither of these are the same as > cardinalities, which are not numbers, but measures (sizes) of sets. > > > > My sketches shows this exactly. > > Cantor proofs his wrong conclusion with the same mix of potential > > infinity and actual infinity. But there is no bijection between this > > two concepts. The antidiagonal is an unicorn. > > There is no stringend concept about infinity. And there is no aleph_1, > > aleph_2, ... or any other infinity. > > For that to be true, there must be a bijection between an infinite > set (any infinite set) and its powerset. Bitte, show us a bijection > between N and P(N). > > I already showed you the bijection between binary *N and P(*N). What didn't you like about it? It is valid. -- Smiles, Tony
From: David R Tribble on 17 Oct 2005 17:39
Albrecht S. Storz wrote: >> Cantor proofs his wrong conclusion with the same mix of potential >> infinity and actual infinity. But there is no bijection between this >> two concepts. The antidiagonal is an unicorn. >> There is no stringend concept about infinity. And there is no aleph_1, >> aleph_2, ... or any other infinity. > David R Tribble said: >> For that to be true, there must be a bijection between an infinite >> set (any infinite set) and its powerset. Bitte, show us a bijection >> between N and P(N). > Tony Orlow wrote: > I already showed you the bijection between binary *N and P(*N). > What didn't you like about it? It is valid. No, you showed a mapping between *N and R, which is equivalent to a mapping between *N and P(N). That's easy. But you have not provided a mapping between any set and its powerset, infinite or otherwise. |