Cantor Confusion
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From: David Marcus on 30 Oct 2006 20:53 Virgil wrote: > In article <1162216575.579280.178630(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > Either you have a proof that can be given in ZFC or you don't. Which is > > > it? This shouldn't be a difficult question to answer. > > > > But it is not an interesting question. > > We find it very interesting that WM will not, or more likely cannot, > answer it. That is rather interesting. Why wouldn't WM answer it if he could? And, if he couldn't, why wouldn't he admit it? -- David Marcus
From: David Marcus on 30 Oct 2006 21:10 Lester Zick wrote: > On Mon, 30 Oct 2006 16:43:03 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >Han de Bruijn wrote: > >> David Marcus wrote: > >> > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> >>And add this to the fact that noon and beyond cannot exist > >> >>in this problem. > >> > > >> > Are you still doing physics, water, and a finite number of molecules? > >> > Let us know when you switch to mathematics. > >> > >> I'm DOING mathematics. Mathematics is NOT independent of Physics. > > > >I guess your definition of "mathematics" is different from mine. > > I think the only relevant question is whether your definition of > "mathematics" is demonstrably true? If not I don't see that any > definition of mathematics is more virtuous one way or the other. That would depend on what your definition of "demonstrably true" is. My definition of "mathematics" agrees with what mathematicians do. Is it "demonstrably true" that "noon... cannot exist"? -- David Marcus
From: David Marcus on 30 Oct 2006 21:16 Sebastian Holzmann wrote: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Sebastian Holzmann schrieb: > >> No. A set x (which is here to denote an element of a model M of ZF-INF) > >> is called "finite" if x satisfies one of the following conditions: > >> > >> 0: x does not have an element > >> 1: x has exactly one element > >> 2: x has exactly two elements > >> and so on > >> > >> otherwise, x is called "infinite". Where do I need the axiom of infinity > >> to do this? > > > > It is impossible to prove any "otherwise". If x has not n or less > > elements, then it may have n+1 elements. You will never know that x has > > infinitely many elements because you can test only for finite numbers. > > It is the same as with the natural numbers. In fact you do no know that > > N has infinitely many elements without the axiom of infinity. Otherwise > > one would not need that axiom. > > 1. Any given set either satisfies one of the conditions, or it doesn't > any. If you have objections to that, you leave the paths of standard > logic, and your reasoning bears no relevance to standard mathematics. Once, WM admitted he doen't use standard logic ("It is outside of your model, independent of ZFC"). However, he never says what logic he is using (if any). > 2. The axiom of infinity is added to ensure that a set "like N" exists > in _every_ model of the theory. If it isn't added, there might be models > that do not have an "N" and some who do. Learn about logic, model theory > and set theory! -- David Marcus
From: Dik T. Winter on 30 Oct 2006 21:27 In article <md5ak294kscg4uk48a276jktc64lf430rq(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > On Sun, 29 Oct 2006 02:01:16 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > wrote: .... > >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". .... > 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. (That may exist or may not exist.) > 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. But that is not a mathematical definition, but a philosophical definition. Without domain of discourse, such a definition makes no sense, mathematically. > 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. This is wrong. You do not define the quality of infinity in specific objects, but as a general property within a set of axioms (part of your domain of discourse). As far as I know, "infinity" in mathematics has only been defined in the context of topology, where it is the single point used in the one-point compactification (but it is long ago that I did all this). And from there it has a derived definitions in projective geometry (the point at infinity, and also the line at infinity). The term is used, losely, in analysis, using limits, but is not really defined there as term. You will see oo used in analysis, but when it is used it does *not* mean "infinity". When that symbol is used the usage has a specific definition. > >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. No, there is something else. In time definitions in a particular field of mathematics can change over time. So when you are reading old books or articles you need to know whether the definitions actually have been changed since then or not. Especially Wolfgang Mueckenheim does not allow for such changes of definitions. But also in physics definitions do change in time. The definition of a metre (as measure) has changes at least two times, and I think three times, in the course of time. That definition changed, due to more sophistication; newer definitions could be more precise, and objects could be measured more correctly using the new definitions. In mathematics it is a bit different. At a certain time, 1 was considered a prime because it satisfied the naive definition (a prime is divisible only by 1 and itself). This became unsatisfactory because it complicated the formulation of major results (Unique Prime Factorisation), especially so when it was attempted to use it in other places than the natural numbers. So now, 1 is a unit and 2 is the smallest prime in the natural numbers. > >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. .... > 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. I think you are wrong. At best they tell that the system is not shown to be inconsistent. > The idea of a "standard" system really only > refers to whatever happens to be conventional and not what is true or > even best. That is false, at least in mathematics. There is no single standard system at all. There are, of course, conventional standard systems that are taught at universities because it is thought that they give the more direct information for progress. Only until now did they show better that they produced results than other systems. > 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. Of course. But when you want to argue agains a system (whatever system you wish), you should argue within the system. Unless you want to get phylosophical. So any argument on whether John Conways 'surreal numbers', or (based on Cantor) the ordinals and cardinals (using one of the many formulations of set theory), or Robinson non-standard reals, or Kuck's method is better is outside the realm of mathematics. You can not argue that Conways' is better than Cantor's because in Conways' method omega+1 = 1+omega, which is false in Cantor's. They are different views on things. > Nor is it possible to argue against those pieces or parts within the > system itself. It is possible to argue against parts of the system, as long as your argument stays within the system. > 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 axioma
From: David Marcus on 30 Oct 2006 21:55
Dik T. Winter wrote: > In article <md5ak294kscg4uk48a276jktc64lf430rq(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > 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. > > This is wrong. You do not define the quality of infinity in specific > objects, but as a general property within a set of axioms (part of your > domain of discourse). As far as I know, "infinity" in mathematics has > only been defined in the context of topology, where it is the single point > used in the one-point compactification (but it is long ago that I did all > this). And from there it has a derived definitions in projective geometry > (the point at infinity, and also the line at infinity). The term is used, > losely, in analysis, using limits, but is not really defined there as term. > You will see oo used in analysis, but when it is used it does *not* mean > "infinity". When that symbol is used the usage has a specific definition. I believe it is quite common in analysis to use the "extended real numbers". The extended real numbers is R with the addition of two elements +oo and -oo. E.g., see the book "Real Analysis" by H.L. Royden. > > >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. > > No, there is something else. In time definitions in a particular field > of mathematics can change over time. So when you are reading old books > or articles you need to know whether the definitions actually have been > changed since then or not. Especially Wolfgang Mueckenheim does not > allow for such changes of definitions. WM doesn't seem to even understand what the old books or articles meant when they were written. -- David Marcus |