Cantor Confusion
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From: MoeBlee on 30 Oct 2006 00:18 mueckenh(a)rz.fh-augsburg.de wrote: > Every mathematician whose mind has not yet suffered the drill of set > theory you probably were exposed to knows that my argument is correct. How could they know it is correct regarding set theory if they don't know set theory? And you know the mind of every mathematician who has not studied set theory? Such a mind reader you are! > (And there are many mathematicians who never studied set theory but > every mathematician knows the geometric series.) > > However due to the strong pressure of the orthodox fraction it may be > disadvantageous for a mathematician to see his name printed here. > Further it is my general principle not to publish names of private > correspondents. Okay, we'll take your word for it then. Better yet, we'll just read your mind to witness your memories of correspondence you've have. Better yet, we'll just directly read the minds of every mathematician who has not studied set theory. Why not? It works for you! MoeBlee
From: MoeBlee on 30 Oct 2006 00:40 mueckenh(a)rz.fh-augsburg.de wrote: > As you think, until someone asks: what in heck is a wff ? These days, someone would ask that only if he hadn't read chapter one of a text in mathematical logic. Well formed formula is about as definite as it gets in mathematics. It's even basic in computing. Even if not called 'well formed formula', it's hard to imagine anyone who works with computer languages who is not familiar with the notion of a legitimate expression (legal formula', 'expression of proper syntax', or whatever it is called) in a programming language or other formal language. The notion of a well formed formula per a given formal language (such as the particular first order language for a given set theory) could not be more clear, since well formededness is recursively defined. A computer can check for well formededness in first order languages just as you can run a computer check for compliance with the syntax of a programming language. This should not be a problem for you to grasp. > Like: What in heck is a not defined well order? I think you're asking about well orderings; which is a different subject from well formed formulas. But I'm not in the mood today to talk about well orderings with someone who won't read the material that is prerequisite to the discussion. > > > I proved that there are not more real numbers than a countable set has > > > elements. I proved it in a manner which everybody with moderate > > > mathematical knowledge can understand. And, what is important, in a > > > manner completely independent of your special language. > > > > Then yours is a proof in an informal context of your own mathematical > > understanding (and what CLAIM to be a context shared by anyone with > > moderate mathematical knowledge). And, so, as I've said already about a > > half a dozen times by now, that is not a proof that set theory is > > inconsistent. To prove set theory is inconsistent, you must provide > > prove, IN set theory, a sentence P and a sentence ~P that are sentences > > in the language of set theory. > > > > > It is clear and proven *from outside* that their results > > > are wrong and, therefore, uninteresting for me. > > > > It matters little to me that set theory is uninteresting to you on > > account of your having convinced yourself that it conflicts with > > certain ideas of yours that are outside set theory. > > > > I view set theory (in any of a number of different formulations) > > Therefore I don't see any necessity to use a special language like ZFC > or NBG, but use mathematics which can be understood by any freshman. Non sequitor you just wrote to my sentence you broke in two to make it appear that there's some ground for your remark. That there are different formulations of set theory doesn't entail that if you claim something about set theory then we can't just a typical formulation and see how your claims hold up. You've proven nothing about set theory - neither about set theory in a very general sense nor about any particular formulation of set theory. > > as one > > among many possible formal axiomatizations to provide the usual > > theorems of real analysis. > > Obviously it does not provide the recursion f(n+1) = 1*E + f(n)/2 with > f(1) = 1*E. But that is the standard construction of the number of E > related to a segment of a path. And the limit for n --> oo is the > number of E related to an infinite path. It is ridiculous to require > transfinite induction to prove that. Who said anything about transfinite induction? Anyway, let me know when you have an argument that regards set theory. > Take the tree as a proof of consistency of ZFC, satisfying Skolem's > theorem: It is impossible from inside this model of ZFC, i.e. the real > numbers, Now you're definitely just stringing together mathematical terminology in a nonsensical way. The real numbers are not a model of ZFC, and the mathematical world would just love to have you explain how you've proven the consistency of ZFC with your tree and what that has to do with Lowenheim-Skolem. > to proof its countability, because the function required to do > that is not available there. Outside this model the function is > available which relates fractions of edges to paths. > > Therefore, the real numbers show ZFC is consistent, because it has a > countable model. The real numbers do no such thing. You have no idea what you're talking about. It's offensive, actually, the way you throw jargon of set theory and mathematical logic around though you have no idea what it means. MoeBlee
From: mueckenh on 30 Oct 2006 01:44 Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > > > Sebastian Holzmann schrieb: > > > >> Virgil <virgil(a)comcast.net> wrote: > >> > In article <1161883732.413718.244570(a)k70g2000cwa.googlegroups.com>, > >> > mueckenh(a)rz.fh-augsburg.de wrote: > >> >> What you propose, namely the infinity of ZF without the axiom INF would > >> >> not be an advance. But meanwhile you may have recognized that your > >> >> assertion (ZF even without INF is not finite) is false. > >> > It is, however, quite true that ZF without INF need not be finite. > >> It is, more than that, quite true that ZF without INF _is_ infinite > > Are you really sure? > > I am. > So you also must be sure that ZF is yellow. > >> (the axiom schema of separation alone provides infinitely many axioms). > > Are you really sure? > > I am. But most of them are very dirty? > > >> The point is: ZF without INF does not prohibit the existence of infinite > >> sets, nor does it force them to exist. > > > > It prohibits to speak of infinite sets and to recognize such sets. So > > one cannot be sure, but you are? > > Given any model of ZF-INF, one cannot be sure if there are infinite > sets. In particular the sets consisting of infinitely many rabbits would be of great interest. Regards, WM
From: Virgil on 30 Oct 2006 02:57 In article <1162190663.531903.51440(a)e64g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Sebastian Holzmann schrieb: > > > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > > > > > Sebastian Holzmann schrieb: > > > > > >> Virgil <virgil(a)comcast.net> wrote: > > >> > In article <1161883732.413718.244570(a)k70g2000cwa.googlegroups.com>, > > >> > mueckenh(a)rz.fh-augsburg.de wrote: > > >> >> What you propose, namely the infinity of ZF without the axiom INF > > >> >> would > > >> >> not be an advance. But meanwhile you may have recognized that your > > >> >> assertion (ZF even without INF is not finite) is false. > > >> > It is, however, quite true that ZF without INF need not be finite. > > >> It is, more than that, quite true that ZF without INF _is_ infinite > > > Are you really sure? > > > > I am. > > > So you also must be sure that ZF is yellow. And WM is equally sure that what natural numbers which he allows might exist are all purple with pink polka dots. > > >> (the axiom schema of separation alone provides infinitely many axioms). > > > Are you really sure? > > > > I am. > > But most of them are very dirty? > > > > >> The point is: ZF without INF does not prohibit the existence of infinite > > >> sets, nor does it force them to exist. > > > > > > It prohibits to speak of infinite sets and to recognize such sets. So > > > one cannot be sure, but you are? > > > > Given any model of ZF-INF, one cannot be sure if there are infinite > > sets. > > In particular the sets consisting of infinitely many rabbits would be > of great interest. And like most of WM's notions, such sets would be hare today and gone tomorrow.
From: Han de Bruijn on 30 Oct 2006 03:39
David Marcus wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>mueckenh(a)rz.fh-augsburg.de wrote: >> >>>stephen(a)nomail.com schrieb: >>> >>>>Can you describe a continuous version of the problem where each >>>>"unit" of water has a well defined exit time? A key part of >>>>the original problem is that the time at which each ball is >>>>removed is defined and reached. This is crucial to the problem. It is >>>>not just a matter of rates. If you added balls 1-10, then 2-20, >>>>3-30, ... but you removed balls 2,4,6,8, ... then the vase is >>>>not empty at noon, even though the rates of insertions and removals >>>>are the same as in the original problem. So you cannot just >>>>say the rate is 10 in and 1 out and base an answer on that. >>> >>>The answer for any time t *before noon* is independent of the chosen >>>enumeration of the balls. Doesn't that fact make you think a bit >>>deeper? >> >>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. Han de Bruijn |