infinity ...
in [Math]
|
Prev: math
Next: The proof of mass vector.
From: Tony Orlow on 17 Oct 2005 09:28 albstorz(a)gmx.de said: > 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. I agree with this statement in the sense that all numbers represent some measure of a set of units, and the only common feature that all sets share is size. or number of these units. I wouldn't sat a quantity IS set, but rather a salient FEATURE of a set. Particular sets have other features and properties, depending on the properties of the elements. > 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. The antidiagonal serves to prove that there are more digital strings than whatever maximum length string you choose. The list of digital numbers is longer than it is wide. So, given an infinite number of digits, you have a larger infinite number of strings. That's all. > There is no stringend concept about infinity. And there is no aleph_1, > aleph_2, ... or any other infinity. Here we disagree. How many points, how many real numbers, are on the number line betweeen 0 and 1? > > Regards > > AS > > -- Smiles, Tony
From: Tony Orlow on 17 Oct 2005 09:33 stephen(a)nomail.com said: > 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. That is not necessarily the case. Consider all reals in [0,1], and infinite ordered set with distinct endpoints. It is valid to speak of what happens at infinity. > > 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. Actually, infinite geometric figures make wonderful sense. It is clear from his diagram that whatever length/value you allow the unary string elements to assume, that is also equal to the number of strings in the set. It's a very clear visual argument, the likes of which should be more prevalent in mathematics. It illsutrates exactly what I've been trying to get through. > > Stephen > -- Smiles, Tony
From: Tony Orlow on 17 Oct 2005 09:41 stephen(a)nomail.com said: > 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". One would think you'd be used to speaking with people whose English is not so great. He is making a bijection between the lengths of the strings of 0's and #'s, the one representing the values of the elements, and the other repreenting the count of the elements. The values of both are equal. That's the point. > > >> > >> > 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. because he is drawing a direct identity bijection between the value and count, he is perfectly justified in saying that if one is not defined, then neither is the other. > > Stephen > -- Smiles, Tony
From: William Hughes on 17 Oct 2005 10:29 albstorz(a)gmx.de wrote: <snip> > > If we accept the uncountability as a form of infinity, this leads to > the paradoxon that the natural numbers are not countable. No, the natural numbers are countable precisely because they do count themselves. The fact that there is no natural number that repsresents this "count" is not a paradox because the "count" is defined in terms of bijections. [You may not like the use of the terms "count" and "countable" because you think they should imply something different. So be it. However, you cannot say "you are using a term which I think should mean somthing different, so you must mean not what you mean but what I mean"] - William Hughes
From: Tony Orlow on 17 Oct 2005 10:34
albstorz(a)gmx.de said: > 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 =3D {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 co= > unt > > 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 allo= > wed 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. I agree. Even if you are talking about real measurements rather than discrete counting operations, those measurements are only pssobile given discrete units of measurement, and amount to the equivalent of a size of a set of units. > > The similarity of my sketches with the usual representation of the > Cantor diagonal argument is not an accidently happend effect. True. Cantor's diagonal argument requires the use of a base higher than 1. Using base 1 kind of destroys the argument. > > 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. What's that now? There is no infinite numeral? What makes you say that? Is 1/3 the same in decimal as 0.3333...? Isn't that an infinite numeral? Is there any reason I cannot have an infinite number of digits to the elft of the digital point, like ...3333.0? Okay, this can get ambiguous, so can't I declare a digit at some infinite position in common with other numbers and compare them? Think of the points in (0,1] as going from 0.000...001 through 1.000...000. Now multiply by N=1:000...000.0. We get 1.000...000 through 1:000...000.0, for a complete bijection between the infinite set of reals in (0,1] and the set of natural numbers, finite and infinite. Again, isn't the set of points in (0,1] infinite, or is there some finite number of points in that set? > > 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. The dimension of quantity can also have infinite values. Why is it different from space or time in that respect? > > 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? In my mind, the countable infinities are potentially, but not actually, infinite, since they are all restricted to finite numbers of iterations, and thus elements. Uncountable infinities are actually infinite, but seem countable in a variety of ways. The whole countability criterion is misguided, IMHO. > > 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. Gl=FCckauf. > > > AS > > -- Smiles, Tony |