Cantor Confusion
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From: Virgil on 10 Dec 2006 14:57 In article <1165763298.462685.83360(a)73g2000cwn.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > step...(a)nomail.com schreef: > > > Do you or do you not wish to abolish any mathematics > > that involves infinity? If you are perfectly content > > to let others freely explore whatever they wish, then > > why are you so aggressive? > > _What_ infinity. That's the question. Mainstream mathematics has mixed > up infinity so much that it's not a sensible notion anymore. > > Han de Bruijn It is not a single notion, certainly. The 'infinity' of the 'size' of the set of integers is quite distinct from the 'infinities' required for the two point compactification of the reals. Each notion by itself, and in its appropriate context, is quite sensible, but conflating them all into a single notin is not.
From: David Marcus on 10 Dec 2006 14:58 Tony Orlow wrote: > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > David Marcus schreef: > > > >> If you've been reading the threads, then you should know that WM has so > >> far failed to present any mathematics at all. All he does is present > >> incorrect arguments that he insists follow from the standard axioms and > >> then proclaim: "Behold, standard mathematics is inconsistent." Big deal. > > > > Nonsense. The arguments presented by WM are quite reasonable. > > I agree. He perceives the same problem with the von Neumann ordinals as > I do, he just has a different answer. His objection is valid, however. If you say so, then please state the objection. > >> Fine. Give some evidence that it "narrows your thinking". I.e., present > >> some mathematics that can't be done using ZFC as a foundation. > > > > �'m pretty sure you can twist and bend any thought in such a way that > > it fits into your set theoretical paradigm. And as soon it doesn't fit, > > you'd simply say that it is not mathematics. Wolfgang Mueckenheim has > > come up with several of such examples. You'd not be better off with me. > > See e.g. the "Probability XOR Calculus" thread, which was initiated by > > this author: > > > > http://groups.google.nl/group/sci.math/msg/b686cb8d04d44962 > > > > Han de Bruijn > > Yes, that was a very good thread, pointing out the need for > infinitesimal probabilities within infinite sets of alternatives, as > well as a number of other anomalies that came up. If you need something, then you have to either create it yourself or convince others that it would be useful enough for them to spend time trying to create it for you. -- David Marcus
From: David Marcus on 10 Dec 2006 15:05 Tony Orlow wrote: > In any case, I think his objections are specific enough to warrant > some attention. "Calculus XOR Probability" was about the loss of > additive probability measure, when you have an infinite set of equally > likely possibilities, Can you give me a procedure for generating a natural number at random? I can give you a simple procedure using a coin for generating a real number in the interval [0,1] at random. (By "at random", I mean each number is equally likely.) -- David Marcus
From: David Marcus on 10 Dec 2006 15:12 imaginatorium(a)despammed.com wrote: > David Marcus wrote: > > Tony Orlow wrote: > > > > Now, sequences may be said to derive from ordered sets, but sets are > > > said to be determined solely by membership, with order unimportant. So, > > > the notion of a sequence derives really from an inductive definition > > > such as Peano's, and not from the one primitive in set theory, > > > membership, alone. The notion of order is not captured by "is an element > > > of". Do you disagree? > > > > Of course I don't agree. You seem to be saying that infinite sequences > > can't be handled in ZFC. Since ZFC has no trouble modeling the natural > > numbers and defining functions, it clearly has no trouble acting as a > > foundation for all of calculus and analysis. > > But ZFC does have considerable difficulty dealing with infinite > sequences, when the Orlovian axioms are added in. I _think_ that Tony's > "infinite induction" thing means for a start that if a sequence of > elements has a particular property ("staircase length is 2") then the > limit of the sequence must have the same property. There doesn't seem > to be a definition of the Orlovian limit, except that in any particular > case Tony will construct an ad hoc story to make something having the > property being talked about. Thus the Tlimit of the staircase sequence > is a [search the archive for TO's words] "sort of infinitesimal > staircase-thingy of length 2". > > I don't think ZFC will handle Orlovian "positive infinite quantities" > too well, either. Tony gets to infinite values by simply advancing > along the real line for, um, an infinite distance, through the tunnel > of love (where it's too dark to see properly). Despite the fact that > any finite quantity (integer) can be represented as a "finite length" > two-ended string of digits, and the fact that each new integer is > formed by adding one, the naive expectation that this would enable a > proof by induction that these "infinite quantities" were also simply > finite quantities, it doesn't work like this, because, um, there's a > principle of somethingorother that excludes this. I wonder if in fact > just as electromagnetic radiation is mediated by photons, induction is > mediated by inductons, and the tunnel of love just happens to block the > passage of inductons. Perhaps. Clearly, the problems are more conceptual rather than any particular lack of ZFC. -- David Marcus
From: Virgil on 10 Dec 2006 15:15
In article <1165775199.395250.78780(a)j72g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > <statements which make it clear that certain > > things which were though to be settled are not settled> > > > > Terminology: If we say that X exists > > then we can use X in a proof. > > That already depends on what you understand by "a set exists". I > suspect that you understand that all its elements exist. Why should they not? > > > > > > On Dec 4 I wrote: > > > > You now agree that a potentially infinite set can have > > a cardinal number and that this cardinal is not > > a natural number. > > > > As your latest post points out, this is not (or > > no longer) true. > > I never agreed. oo is not a number, so it is not a cardinal number, it > is at most a "cardinal number". "oo" does not represents the number in question. The number of naturals may be Card(N) or aleph_0, and in some contexts may be omega as well, but "oo" is undefined as a number or a cardinality or an ordinality. > > > > > Stop me when I make a statement you disagree with > > We can then discuss this statement before proceding. .. . . > > -a cardinal number is an equivalence class on > > sets with respect to the equivalence relation > > bijection > > Yes, that is correct. But it does not satisfy the order-relation > "greater than" with natural cardinals. How not? What part of Cantor's definition of the ordering of cardinalities does WM claim fails? > > > > > -the equivalence relation bijection can be extended > > to include potentially infinite sets > > > > -given a potentially infinite set A, the set C > > of ordered pairs (a,a) exists, where C has > > the property > > > > if > > a is an element of A > > then > > (a,a) is an element of C > > > > Call C the identity function. C is a bijection > > on A. > > That again depends on what you understand by "to exist". The identity > function f(n) = n does not provide the existence of all natural > numbers. Does it provide a bijection on all the naturals which do exist? We see it by the fact that otherwise finite (every number > including the limit) = infinite (the limit of initial segments of > natural numbers). How does WM conjure up "every number including the limit" when there is no limit? > > > > > -A belongs to an equivalence > > class with respect to the equivalence relation > > bijection > > > > -A has a cardinal number > > > > -the cardinal number of A is not a natural number > > neither it is larger than every (or even any) natural number. By what definition of 'largeness' is the cardinal of A 'not larger than' any natural number? > > > > > -given two sets of natural numbers E and F where E is a > > potentially > > infinite set, and F has a largest element. there does > > not exist a bijection between E and F > > > > -the diagonal is the potentially infinite set of natural > > numbers. > > > > -every line L has a largest number > > > > -there is no bijection between the diagonal and a line L > > There is no complete diagonal. Then there is no complete 1, nor any other set in ZFC or NBG. WM can't have 1 without the other. |