Cantor Confusion
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From: Franziska Neugebauer on 3 Jan 2007 12:44 mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: >> David R Tribble schrieb: >> >> Well, now I'm confused. Could you provide an example of a natural >> >> number that does not exist? >> > >> >> mueckenh wrote: >> > Take the first 10^100 digits of pi (if you can - but you cannot). >> > It is impossible to bring this number to existence in the whole >> > universe. >> >> Can you "bring into existence" the number 1? > > Here it is: > > . [X] a dot [ ] the number 1 F. N. -- xyz
From: William Hughes on 3 Jan 2007 14:22 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > Here is the EIT fort newcomers: > 0.1 > 0.11 > 0.111 > ... > > > 0.11111 is the unary representation of the natural nunmber 5 > > L_D is the diagonal (0.)111... > > > > I think it is of little value to continue this discussion. You always > > > implicitly assume the actual existence of all natural numbers. > > > > No. I use three properties: > > > > i: There is no last line > > Then also the number 1/9 = 0.111... does not exist as a decimal > representation? No, this merely says that there is not last 1 in 1/9 = 0.111.. . This says nothing about whether 0.111... exits > I agree! Irrelevant. I do not assume that 0.111... exists. I only assume that given that 0.111...1 exists you can show that 0.111...11 exists. > But this implies he nonexistence of the > diagonal. > > > > ii: L_D contains any element, that can be shown to exists > > in the diagonal. > > Yes. > > > iii: It is always possible to find another element that exists > > in the diagonal. > > That is the same with the lines. Why should the diagonal exist actually > but he system of lines should not exist actually? (1/9 = 0.111...) Indeed. It is the same with lines. It is always possible to find another line. However, iii says nothing about whether the diagonal "actally exists" (or equivalently whether the system of lines "acutally exists"). > > > > If you use these three properties (all of which come directly from > > you) then you can show a contradiction whithout making > > any assuption about whether all the elements of the diagonal exist. > > > > The fact that if you also assume > > > > iv: Not all the elements of the diagonal exist > > Not all natural numbers in unary representation 0.1, 0.11, 0.111, ... > exist. If all elements of the diagonal exist (which are the last digits > of the unary numbers) then these unary numbers must exist too, as far > as I understand existence. Any discussion of iv is completely irrelevant. iv is neither needed nor used. > > > > you can get a second contradiction. is true but irrelevant. One can > > reach a contradiction without making assumption iv, and making > > assumption > > iv does not make this first contradiction go away. > > The only contradiction I can see is that you assume the existence of > the last digits of all natural numbers (in unary representation, > forming your complete diagonal) while you do not accept (or pretend not > to accept) the existence of the numbers themselves. This is > unacceptable in my eyes. > The contradiction is: A: L_D must have the property that given any set of elements that exist in L_D one can always find another element of L_D [This follows immediately from i ii and iii] B: L_D has a largest element. [this follows from the fact the L_D is a line] It is not necessary to assume the existence of the last digits of all natural numbers (in unary representation). or to assume the existence of all natural numbers. The fact that you can use A along with other assumptions to lead to another contradiction is irrelevant, and does not make the contradiction between A and B go away. - William Hughes
From: Virgil on 3 Jan 2007 14:41 In article <1167843537.696213.50720(a)48g2000cwx.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > David R Tribble schrieb: > > >> Well, now I'm confused. Could you provide an example of a natural > > >> number that does not exist? > > > > > > > mueckenh wrote: > > > Take the first 10^100 digits of pi (if you can - but you cannot). It is > > > impossible to bring this number to existence in the whole universe. > > > > Can you "bring into existence" the number 1? > > Here it is: > > . > > Regards, WM Seems to have evaporated, WM. Try again.
From: Virgil on 3 Jan 2007 14:52 In article <1167844172.445217.317620(a)i12g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1167738133.535489.164800(a)n51g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1167493176.710056.286330(a)s34g2000cwa.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > Dik T. Winter schrieb: > > > > ... > > > > > > At what point is 1/3 represented? > > > > > > > > > > In the binary tree it is not represented, but it could be > > > > > represented > > > > > by using a ternary tree. > > > > > > > > Now you are arguing differently from before. When I stated that 1/3 > > > > was not in your tree, you argued that it was in your tree. Do you now > > > > think differently? > > > > > > No. I have been knowing for several years already that the decimal or > > > binary representation of 1/3 does not exist. > > > > You state so, contradicting the axiom of infinity. > > I do not contradit the axiom of infinity by free will. If you give up free will, you are less than human. > I am forced to > do so. The "Devil" made you do it? Hardly a satisfactory excuse. >Otherwise I had to assume that the union of all finite binary > trees is identical to the complete binary tree, as far as nodes and > egdes are concerned, but that the trees are not identical as far as > paths are concerned. What is wrong with assuming what is easily provable? As every edge and node is a member of some finite path of some finite tree, each must be in the union. But no infinite path is a member of any finite tree, so none can be members of a union of finite trees. That WM does not understand what a union is is a flaw in his onw understanding, not in the valid logic which he rejects. > > I cannot accept ghosts in mathematics, but, of course, I cannot prove > that they do not exist. So we are finshed with this topic. WE, on the other hand, can prove that what WM regards as ghosts do exist in several versions of set theory. As WM has no specific version of his own, he effectively has none at all.
From: Virgil on 3 Jan 2007 15:09
In article <1167845252.057193.113440(a)a3g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > Here is the EIT fort newcomers: > 0.1 > 0.11 > 0.111 > ... > > > 0.11111 is the unary representation of the natural nunmber 5 > > L_D is the diagonal (0.)111... The L_D, as WM originally described it, must have a last term, but the diagonal in a set theory such as ZFC or NBG cannot have a last term, so that there is no set well defined set theory in which WM's allegations can be valid. Unless WM can provide an axiom system for his alleged set theory. Well can you, punk? > > > > I think it is of little value to continue this discussion. You always > > > implicitly assume the actual existence of all natural numbers. > > > > No. I use three properties: > > > > i: There is no last line > > Then also the number 1/9 = 0.111... does not exist as a decimal > representation? False. The ellipsis "..." means "without end". > I agree! But this implies he nonexistence of the > diagonal. How so? It only means that is one enforces the WM assumption that there also is a last line. > > That is the same with the lines. Why should the diagonal exist actually > but he system of lines should not exist actually? (1/9 = 0.111...) WM misrepresents the actuality: In ZFC or NBG Every line 0,1, 0.11, ..., 0.111...1, ... exists, each having a last digit but there not being any last line, and also there is a diagonal 0.111... which, like the set of lines, does not end. > Not all natural numbers in unary representation 0.1, 0.11, 0.111, ... > exist. They do in ZFC or NBG. What axiom system for set theory does WM propose in which some of them do not exist? If all elements of the diagonal exist (which are the last digits > of the unary numbers) then these unary numbers must exist too, as far > as I understand existence. Both do, unless WM can produce an axiom system in which they don't. > The only contradiction I can see is that you assume the existence of > the last digits of all natural numbers (in unary representation, > forming your complete diagonal) while you do not accept (or pretend not > to accept) the existence of the numbers themselves. We accept both the existence of both the last unary digit of any natural number and the existence of that natural. Why do you falsely claim otherwise? > This is > unacceptable in my eyes. Your misrepresentations are unacceptable in our eyes, too. |