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From: zuhair on 15 Oct 2005 15:11 ok, it seems what you want to say is that the last natural number is equivalent to the infinite number. Since I believe in a natural infinite number, therefore their is no contradiction on my side. In my views the smalles natural infinite number is the one that comes after all finite natural numbers horizontally placed. so at the end your rectangle will have w by w+1 dimentions , were w means the first natural infinite number which is equivalent to the number of finite natural numbers horizontally palced in ascending manner from zero. Best Zuhair
From: David R Tribble on 15 Oct 2005 18:06 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. 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.
From: stephen on 15 Oct 2005 19:11 albstorz(a)gmx.de wrote: > stephen(a)nomail.com wrote: >> 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? > You are unable to recognize the primitivest form of bijection? That is a meaningless sentence. Bijections make no mention of "biggest". >> >> > 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. >> >> No, the contradiction is assuming that cardinality has >> anything to do with the "biggest" element. Cardinality >> is not defined in terms of the largest element. It >> is defined in terms of bijections. Your post says >> nothing about bijections, so it says nothing about >> cardinality. >> >> Stephen > I did not use the word "bijection"? This must be really the flaw of my > proof. If you want to talk about cardinality, which is defined in terms of bijections, then you should be talking about bijections. You instead are talking about "biggest" elements which have nothing to do with cardinality. You claimed that the cardinality was undefined, because there is no biggest element. That is just simply and obviously wrong. Stephen
From: stephen on 15 Oct 2005 19:16 albstorz(a)gmx.de wrote: > albstorz(a)gmx.de wrote: >> (Hint: The sketches would not looks good in proportional fond) >> >> Let's start with a representation of the natural numbers in unitary >> (1-adic) system as follows: >> >> >> >> O O O O O O O O O ... >> O O O O O O O O ... >> O O O O O O O ... >> O O O O O O ... >> O O O O O ... >> O O O O ... >> O O O ... >> O O ... >> O ... >> . >> . >> . >> >> >> 1 2 3 4 5 6 7 8 9 ... >> >> >> Each vertical row shows a natural number. Horizontally, it is the >> sequence of the natural numbers. Since the Os, the elements, are local >> distinguished from each other, we can also look at the rows as sets. A >> set may contain the coordinates of the elements as their representation >> or may look like this: S3 = {O1, O2, O3} e.g. . >> >> >From the Peano axiomes follows that the set of all naturals is >> infinite. So, the set of the elements of the first horizontal row is >> infinite. Actually the set of the elements of every horizontal row is >> infinite. And the set of all the elements in this representation is >> also infinite. >> >> But there is no vertical row with infinite many elements since there is >> no infinite natural. >> >> Now let's fill the horizontal rows or lines with other symbols. We have >> to take into account that only lines should be filled with #s which >> containes Os. >> >> >> # O O O O O O O O O ... 1 >> # # O O O O O O O O ... 2 >> # # # O O O O O O O ... 3 >> # # # # O O O O O O ... 4 >> # # # # # O O O O O ... 5 >> # # # # # # O O O O ... 6 >> # # # # # # # O O O ... 7 >> # # # # # # # # O O ... 8 >> # # # # # # # # # O ... 9 >> . . . . . . . . . . . >> . . . . . . . . . . . >> . . . . . . . . . . . >> >> >> The vertical sequence of sets of #s fullfill the peano axiomes exactly >> as the horizontal sequence of the sets of Os does. >> 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. >> >> 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. >> >> Or let's say it in another form: The first vertical row of #s could not >> exceed the biggest vertical row of Os (and could not be smaller). So, >> the cardinality of this set is undefined like the biggest natural >> number. But the set of the elements of the first vertical row of #s has >> the same cardinality like the set of the natural numbers. >> --> Contradiction. >> >> Or did I construct a monster set which cardinality is subtransfinite? >> >> Comments? >> >> >> Best regards >> >> Albrecht S. Storz, Germany > Image a rectangular triangle with a = b. Than c = sqrt(2) * a. > Now expand the side a to infinity. What is the lenght of of the side b? > Since a = b, b must be equal infinity. > Some may argue, that there is no triangle with infinite sides. What does it mean for a line to be infinite in your infinite triangle? Presumably the line has one end point somewhere. Where is the other end point? At infinity? An infinite line does not end. It does not have another end point. You are German. Translate the following sentence into German: The line ends at infinity. <snip> > The lenghts of the sides of the sequence of this triangles follows the > peano axiomes. Consequently there are infinitely many steps if a is > infinite. So the sum of the x-components of c is infinite. But the sum > of the y-components of c isn't infinite. It is undefined. A really > logical concept. Infinite geometric figures really do not make much sense. You are just assuming conclusions about them without any sort of proof. Stephen
From: albstorz on 16 Oct 2005 14:41
David R Tribble wrote: > 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. > > 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. You are not able to understand that there is no difference between numerals and 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. Regards AS |