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From: Virgil on 15 Oct 2005 01:57 In article <MPG.1db9c1cb9e6fd04198a4a7(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Can the square be wider than it is tall? TO seems to be a square much wider than he is tall.
From: albstorz on 15 Oct 2005 04:48 Tony Orlow wrote: > albstorz(a)gmx.de said: > > > > (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? > > I think your diagram is very nice, and your point pretty clear. That is a good > graphic illustration of the equality between element value and element count > for the natural numbers. It would seem very hard to argue that the array with > its diagonal is somehow longer than it is wide, using this unary notation.. I > believe that you have constructed a representation of a set which is > transfinite, but not infinite, unless the strings of 0's and #'s are allowed to > become infinite in both directions. Good job! Danke! > > > > > > Best regards > > > > Albrecht S. Storz, Germany > > > > > > -- > Smiles, > > Tony The idea results from the understanding, that every number is a set. A number is the unchanged aspect of a simultanity of endlessly many objects which only and absolutly only has common aspects in the number of their elements. That's the only, or one possible definition of a number. The similarity of my sketches with the usual representation of the Cantor diagonal argument is not an accidently happend effect. We could interpret the struktures in the sketches alternatively. In one sense, the lines or columns are natural numbers, in the form of the 1-adic system or in the other sense as sets which completeness follows the peano axiomes. Numbers are sets. A set without number isn't a set. Since there is no infinite numeral, there is no set with an infinite number of elements. But if someone don't like this interpretation, he may think, that infinity is something as "undefined". Infinity, endlessly, uncountability means the same aspect in different "dimensiones". Endlessly is the aspect of infinity in space and time, uncountability the aspcet of infinity of sets of discret things. If we accept the uncountability as a form of infinity, this leads to the paradoxon that the natural numbers are not countable. That's paradox since the natural numbers count themself. The most mathematics shurely say, that the word "uncountability" is just a word. It's accidentally another word for infinity. Since infinity is defined. Infinity is that, what could be biject to a part of itself. With infinity you can do very interesting things. You can find two of them: potential infinity and actual infinity. Perhaps three? Undefined? Infinity is just a "facon de parler". In this sense it's the strongest tool of math. I know, the reactions of the most other posters will be the usual one. You know them. A dream can be stronger than every argument. So I thank you for your kindness and your understanding. Glückauf. AS
From: albstorz on 15 Oct 2005 04:54 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? > > > 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. AS
From: albstorz on 15 Oct 2005 05:56 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. Consider the angle between the straight lines a and c. What is the rectangular distance between a and c in infinity? Infinity or undefined? If this real value is infinite, there must be a numeral, which is infinite. Or there is no infinity. To made this concept connected to my argument in the starting posting. Consider the rectangular triangle with the side c as a staircaise with steps 1 in length and hight. If side a has the length 10, the length of the x-components of the side c is equally 10. The lenght of the y-components of the side c is also 10 like the lenght of side b. 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. Regards Albrecht Storz
From: Peter Webb on 15 Oct 2005 08:16
As Stephen and several others have pointed out, the cardinality of a set is not defined by its largest element. You have to form an explicit bijection to prove equal cardinality; that's what cardinality is defined to mean. That the largest element of a set is a useless way to define something even resembling cardinality, consider the set of integers {..-3, -2, -1, 0, 1, 2 ...}. Now consider the those less than or equal to 1 {..-3, -2, -1, 0, 1}. B y your measure, this has "cardinality" of 1, so 1 = infinity. The countable set of Rationals x such that 0<x<1 has 0.9, 0.99, 0.999 etc but not 1.000, and so has a bounded value but no maximum element in the set. Whaddya going to do here? Keep on trying. |