Cantor Confusion
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From: mueckenh on 30 Oct 2006 09:28 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > There is no computable list > > > > > of computable numbers. > > > > > > > > Why do you insist on this obvious fact? It does not support your > > > > position. There is a list of all finite constructions or definitions > > > > (encoded by numbers, Gödel). This is the definition of countablity. > > > > The diagonal number of this list shows the listed numbers are > > > > uncountable. Contradiction. > > > > > > > > > There is indeed a list of all possible finite constructions or > > > definitions. Call this list A. However, this list must contain things > > > that look > > > like definitions but are not because the method given to produce > > > a number does not halt. We cannot get a diagonal number from A, > > > because some of the members of A do not give numbers. > > > So there is no contradiction. > > > > A contains all finite words (construction fromulas, theorems). The > > diagonal (cannot be longer than the lines and hence) is a finite word > > too. That is enough to obtain a contradiction. > > > A also contains a number of things that look like contruction formulas > but are not (because they don't halt). Forget about Turing. It is not of interest here. All sequences (words) of A are finite / countable. > Thus there is no diagonal. The diagonal of these sequences (words) can be constructed and is finite / countable. > Thus there is no contradiction. You need a diagaonal before > you can get a contradiction, therefore you need set B. Why should we not construct the diagonal of these sequeces (words) of A? Regards, WM
From: mueckenh on 30 Oct 2006 09:37 William Hughes schrieb: > So there is no largest B, so there is no single B. A potentially > infinite > set is not finite. There is no largest B. But every B is finite. A potentially infinite set is always finite. > > > > > Try to find an example for the opposite assertion: Try to construct an > > actually infinite set with only finite numbers (which differ by a > > constant value at least). Then you will know it. > > > > The natural numbers. As you have pointed out this is a potentially > infinite > set so it is not a finite set. It is unbounded but always finite. Regards, WM
From: mueckenh on 30 Oct 2006 09:43 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > William Hughes schrieb: > > > > > > > If there is a limit to how many numbers we can define there > > > is also a limit to how large a number we can define. > > > > Why. Why does "there is a limited number of numbers" imply > > that "every number is limited"? > > > > Not by simply reversing the quantifier. > > The reason for believing that there are only a finite > number of integers is that there are only a finite number > of bits in the lifetime of the universe, and therefore only > a finite number of different integers will be defined. Correct. > > But note, to define an integer, the definition must be > communicated. We have only a finite number of bits > to use for this communication (this includes comunicating > the details of any compression scheme). So there are > only a limited number of integers that it is possible to define. > Among these is the largest integer that it is possible to > define. Sorry, I can't see that conclusion. You can always define a new and larger base., a new and more efficient way to use the bits. > > If we claim that the only integers that exist are the ones > that have been or will be defined, > then there is a largest possible integer. There is, or better there will be, a largest integer ever defined or communicated. But this integer will only be fixed after all life has been gone. Therefore it will never be fixed. So I don't think that it does exist. Regards, WM
From: mueckenh on 30 Oct 2006 09:48 Virgil schrieb: > > > Why. Why does "there are an infinite number of positions" imply > > > that "at least one position is infinite"? > > > > Try to find an example for the opposite assertion: Try to construct an > > actually infinite set with only finite numbers (which differ by a > > constant value at least). Then you will know it. > > I have in mind a list of infinitely many character strings with the nth > string having n characters. As all I have to do to "have" it is to > imagine it, I'm done. As long as your strings have a finite number of characters, you have a finite number of strings. Regards, WM
From: mueckenh on 30 Oct 2006 09:52
Virgil schrieb: > > Correct. And those positions have finite indexes too. > > Every set of natural numbers has a superset of natural numbers which is > > finite. Every! > > Not so. > > The set of even natural numbers has no finite superset. > > The set of prime natural numbers has no finite superset. Sorry, I meant "segment". > > > > > > It is simply purest nonsense, to believe that "all positions are > > > > finite" if "there are an infinite number of positions". > > > > > > There are certainly more than any finite number of positions. > > > And it is even purer nonsense to believe that in such a situation there > > > are only finitely many of them. > > > > Try to draw the graph of the function f(n) = n. The points lay on the > > diagonal of the first quadrant. As long as n is finite, f(n) is finite > > too and vice versa. > > So what. This shows that both are finite or both are not. Regards, WM |