Cantor Confusion
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From: mueckenh on 28 Oct 2006 16:50 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > You are so much caught inside your theory that you are unable to look > > at it from outside. > > No, I'm not. I think about philosophical problems with set theory > frequently. And I keep in mind my agenda to learn about other > foundational proposals as I try to get at least some of the outline or > even just the flavor (if that's all I'm capable of) of such alternative > proposals while I study to put myself in a position to rigorously > understand them. If you think objectively then you cannot avoid the conclusion that fractions of edges can be summed and you cannot avoid the conclusion that lim {n-->oo} (1 - 1/2^n)/(1 - 1/2) = 2 is the number of edges per path. > > > In the formal theory, there is the only predicate > > infinite used for 'is actually infinite' and there is no use of 'is > > potentially infinite'. But, alas, if there appear contradictions, then > > the interpretation of the predicate infinite is quickly adjusted. > > There's no such adjustment in the formal theory. Please stop spouting > misinformation. Of course it is. Look at the discussion here: After some have grasped that my binary tree is really dangerous, they talk about transfinite induction as necessary to get from (1 - 1/2^n)/(1 - 1/2) to the limit 2! Ridiculous! Regards, WM
From: David Marcus on 28 Oct 2006 16:54 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > 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. Since you decline to name any mathematicians who understand what you are saying and agree with it, please name a freshman who understands it and agrees with it. -- David Marcus
From: mueckenh on 28 Oct 2006 16:55 MoeBlee schrieb: > > According to set theorists, naming it is having it. > > Please stop spouting blatant misinformation. Since Russell wrote that > letter to Frege, there is no such thing in set theory as proving the > existence of a set having a certain property just by referring to "the > set that has property P". What is the set N? Where are all the natural numbers? They are nowhere and they will not come along anywhere. You write N or, equivalently, the axiom INF. That is all. You name these things like infinite sets and then you think that they existed and that you had it. But you don't. > > > At least the > > natural numbers and other infinite sets are present by naming N etc. > > How else should an infinite set come to existence in any primordial > > model without an axiom of infinity securing its existence? > > Oh, please, just read a damn book on the subject already! "Primordial > model". Oy vey. I am not a native English speaker. Perhaps this is not an original English word. I used it for "prior to the ordinary theory". If you can't understand it, then ask. If you understand it, then do just understand it. > > I know that. But some "experts" here are of the opinion that even ZF > > without INF is an infinite theory. > > You don't mean an 'infinite theory' (a theory is a set of sentences > closed under entailment). A theory is a systems of ideas. The axioms belong to it, not necessarily sentences which may or may not haven been derived. > You mean a theory that has a theorem that > there exists an infinite set. And I have no idea who the "experts" are > that think there is a theorem that there exists an infinite set without > using the axiom of infinity to prove the theorem. One expert wrote: I wouldn't necessarily call ZFC - AoI "finite set theory". And I replied: It is a theory without the actual infinite, a theory without omega. One may execute any operation. The finite domain will never be left. Which of the remaining axioms should yield infinity? Please adhere to what I wrote, if you are so bad in interpreting. Regards, WM
From: mueckenh on 28 Oct 2006 16:57 Virgil schrieb: > > They do not end anywhere. Correct. They split and split and split. But > > at every split another pair of edges is created. And that does not end > > too. Like the growth of the input of the vase. Why do you only look at > > one side, not at the other? > > The only relevant question is "According to the rules set up in the > problem, is each ball inserted before noon also removed before noon?" > > An affirmative answer confirms that the vase is empty at noon. > A negative answer directly violates the conditions of the problem. > > How does "WMueckenheim"answer? One condition of the problem is that the contents of the vase grows. It has nothing to do with the "infinitely many integer jump discontinuities which cluster around noon". If those would prohibit the continuous growth of the contents then they would prohibit also the calculation of the removed set. Regards, WM
From: mueckenh on 28 Oct 2006 17:02
Virgil schrieb: > > > > Correct. And therefore no such thing can exist unless it exists in the > > > > mind. But we know that there is no well order of the reals. in any > > > > mind, because it is proven non-definable. > > > > > > It is perfectly definable, and perfecty defined, it is merely incapable > > > of being instanciated. > > > > Then let me know one of the perfect definitions, please. > > A set is well ordered when every nonempty subset has a first member. Which is the first member of the subset of positive reals? If you can't answer: How would you order a set? If you can't define an order, what is your alternative? Writing a countable list? Scarcely sufficient for R. An uncountable catalogue? Difficult to print. What is your answer? Regards, WM |