Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Lester Zick on 29 Oct 2006 17:00 On Sun, 29 Oct 2006 02:01:16 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <gld7k21vutgpcr78bt42a6490mvqht1vb1(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > On Sat, 28 Oct 2006 00:36:01 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: > > > > However one can certainly show the square root of 2 without > > > > transfinity through rac construction even though its decimal expansion > > > > is infinite. > > > > > >You need not tell that to me. You should tell that to Wolfgang > > >Mueckenheim who insists that sqrt(2) does not exist because it is > > >impossible to know all the decimals in its decimal expansion. > > > > Well you know I talked to WM about this a year or two ago and of > > course I disagree with him in general terms. But an interesting aspect > > of the problem popped up just a few days ago when another person > > suggested that mathematical definitions have to designate a "domain of > > discourse" in what appear as particular terms such as card(x) = . . . > >In all discussions you need a "domain of discourse". This, however, does >not mean that you need to flag every term with the "domain of discourse". >Otherwise you need to flag with each term whether the "domain of discourse" >is the English language, the French language, or perhaps French translated >to English. When I read message in this newsgroup, I generally associate >"positive numbers" with "numbers larger than 0". Because I understand the >main domain is Anglo-Saxon mathematics. This is in contradiction to what >I did learn at university (0 is both positive and negative). But if you >are reading mathematics books in English translated from French you need >to be aware of this, because the main domain of discourse has change from >Anglo-Saxon mathematical terms to translated French mathematical terms. >(For some in this group Bourbaki is a filthy word, but the group did a >lot to put together quite a bit of mathematics.) Of course you need a domain of discourse. In mathematics that is established either by true definitions or axioms. But what I was specifically referring to is a process of particular definition rather than general definition. Particular definition is simple enumeration of some quality to be defined as properties of particular objects. A general definition just defines the subject without reference to objects defined in such terms. If I say "infinity is . . ." it's a non specific general definition applicable to whatever domain applies. Whereas if I try to define the quaity of infinity in specific objects as in "infinity(x) is . . ." I'm stuck with whatever x may turnout to be and without specifying what x is there is no way to determine whether the definition can be correct or not. >So, when I use the word positive in this newsgroup I will in general not >flag it as Anglo-Saxon, because the main domain of discourse here is >Anglo-Saxon mathematics and mathematical terms. > >The same holds for the Godbach conjecture. When you read current accounts, >they are clear, but when you read the original, it is nonsense. Until you >realise that the original was written when 1 was considered to be prime. >According to the tables of D. H. Lehmer, there are five primes smaller >than 10. But don't forget the process of comprehension itself improves vastly with time and experience of expression. When I look back over my own posts the level and sophistication of expression and application has improved enormously. >You need some assumptions about the domain of discourse in which the >terminology sits. In general that is clear, but some people bemuddle >that and try to proof some inconsistency of (say) ZFC using something >that is completely false in ZFC, and can not be proven within that >theory. Also Wolfgang Mueckenheim can talk about sqrt(2) not really >being a number, but if he does show he should know that he is not >talking within some standard theory, but only in a theory of his own. >That theory may be valuable, but it is not known whether that theory >is valuable until it has been developed sufficiently. Yet on the other hand those with a great deal of experience in any system often claim the system itself is true when it is in fact only problematic at best. The idea of a "standard" system really only refers to whatever happens to be conventional and not what is true or even best. It is often argued that one cannot argue against a system unless one understands the system and one cannot understand the system without having studied the system, the implication being that one cannot argue against a system as long as there is anyone who understands and has studied the system longer than critics of it. This is not necessarily true if one argues against isolated pieces or parts of the system and not the system as a whole. Nor is it possible to argue against those pieces or parts within the system itself. It isn't reasonable to demand one argue against a paradigm within the paradigm unless the paradigm itself can be shown to be true in mechanically exhaustive terms. And paradigms which rest on axiomatic foundations cannot be shown to be true in exhaustive terms which depend on assumptions of axiomatic truth. > > Now without debating the issue of mathematical definition in general I > > will say that if mathematikers insist on definitions drawn exclusively > > in particular terms then WM may have some basis for his conclusions in > > that the infinity of mathematical expressions cannot be accommodated > > in a physically limited universe. > >He is in a sense right, of course. But that is even true when you do >not draw exclusively in particular terms. The number of mathematical >expressions and inferences always will be finite. What this does not >mean is that all expressions and inferences are only about the finite. Of course not unless mathematical definition is restricted to particulars. >The whole point about using definitions in particular terms is that >everybody should know what you are talking about. When you use terms >without such (implicit or explicit) definitions you are likely to be >misunderstood. But that's where the idea of exhaustive truth for subjects comes in. > > In other > > words if mathematics insists its definitions can only be valid if cast > > in particul
From: William Hughes on 29 Oct 2006 17:53 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > And once again WM deliberately confuses, "all positions are > > > > finite", with "there are a finite number of positions". > > > > > > There is no confusing! Every finite position belongs to a finite > > > segment of positions (indexes). > > > > Correct. > > > > But > > every position belongs fo a finite set of positions, > > does not mean > > there are only a finite set of positions. > > > > The first means: > > > > for every position N, there exists a finite set > > of positions A(N) such that N belongs to A(N) > > > > The second means: > > > > there exists a finite set of positions B, such that > > for every position N, N belongs to B > > No it means only potential infinity: Every set B of positions is > finite, although there is no largest set B but every set B has a larger > superset B'. So there is no largest B, so there is no single B. A potentially infinite set is not finite. > Actual infinity would yield: There is a completed, finished infinite > set. That is nonsense, whther you call it finite set B or infinite set > |N. Finished infinity is a contradiction in itself, even worse than > quantifier changing. A bald statement is not an argument. In any case, you have admitted that the set of positions is potentially infinite. This is not a finite set. If you want to call it an unbounded set, rather than an infinite set, knock yourself out. > > > You cannot simply exchange the quantifiers. > > It has nothing to do with that. It is the only argument you have presented. > > > 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. > The natural numbers. As you have pointed out this is a potentially infinite set so it is not a finite set. If you break out in hives when you say the I word, call it an unbounded set. - William Hughes
From: William Hughes on 29 Oct 2006 18:07 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. 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. 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. - William Hughes
From: Virgil on 29 Oct 2006 19:04 In article <1162156688.091123.321640(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > The first means: > > > > for every position N, there exists a finite set > > of positions A(N) such that N belongs to A(N) > > > > The second means: > > > > there exists a finite set of positions B, such that > > for every position N, N belongs to B > > No it means only potential infinity: Every set B of positions is > finite, although there is no largest set B but every set B has a larger > superset B'. > Actual infinity would yield: There is a completed, finished infinite > set. That is nonsense, whther you call it finite set B or infinite set > |N. Finished infinity is a contradiction in itself, even worse than > quantifier changing. Only in WM's imagination. If one accepts ZF or NBG, and there is no reason but prejudice not to, then there ARE completed infinities. I have no objection to WM accepting whatever axioms he wants, but I strongly object to his attempts to deny me the same prerogative. > > > You cannot simply exchange the quantifiers. > > It has nothing to do with that. It has everything to do with WM trying to impose his beliefs on others. > > > 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. > > Regards, WM
From: Virgil on 29 Oct 2006 19:05
In article <1162156779.441703.8250(a)i42g2000cwa.googlegroups.com>, 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"? Is Wm too dumb to work it out for himself? |