infinity ...
in [Math]
|
Prev: math
Next: The proof of mass vector.
From: albstorz on 14 Oct 2005 10:12 (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
From: stephen on 14 Oct 2005 10:23 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? > 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
From: Tony Orlow on 14 Oct 2005 14:21 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
From: Tony Orlow on 14 Oct 2005 14:24 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. > > > 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. 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) > > Stephen > > -- Smiles, Tony
From: stephen on 14 Oct 2005 14:53
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". <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. Stephen |