Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 25 Jul 2005 15:27 In article <MPG.1d4eb8a520ca63b1989f6a(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > The problem comes whenever someone wants to use the word "infinite" > in any other context. There ARE no infinite numbers, only infinite > sets, in your lexicon. There are no infinite natural numbers, but there are plenty of infinite 'cardinal numbers', infinitely many of them, at least in ZFC. > Talk about needing to learn about language. What is so hard about > infinite numbers, or trying to define the word itself, independent of > bijection genuflection? We already have cardinalities, which are sufficient to our needs. Let's see TO's definition of an "infinite" number.
From: Tony Orlow on 25 Jul 2005 15:31 Virgil said: > In article <MPG.1d483af164930cd4989f46(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > Virgil said: > > > In article <MPG.1d4733ca67be65e3989f31(a)newsstand.cit.cornell.edu>, > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > > > David Kastrup said: > > > > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > > > > > > > > David Kastrup said: > > > > > >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > > >> > > > > > >> > Now, I am not familiar, I think, with the proof concerning > > > > > >> > subsets of the natural numbers. Certainly a power set is a larger > > > > > >> > set than the set it's derived from, but that is no proof that it > > > > > >> > cannot be enumerated. > > > > > >> > > > > > >> Uh, not? > > > > > > > > > > > Yes, not. "Larger" is not a synonym for "uncountable" except in > > > > > > Cantorland, and that is a leap and an assumption. > > > > > > > > > > "Larger" is a synonymon for "can't be surjected onto from" in set > > > > > theory. And "uncountable" is a synonymon for "larger than the set of > > > > > naturals". It is not a leap or an assumption, but simply a > > > > > definition. > > > > > > > > > > >> > Is this the same as the proof concerning the "uncountability" of > > > > > >> > the reals? > > > > > >> > > > > > >> It's pretty similar. > > > > > > Figures. > > > > > >> > > > > > >> Assume a set X can be put into complete bijection with its powerset > > > > > >> P(X) such that we have a mapping x->f(x) where x is an element from > > > > > >> X > > > > > >> and f(x) is an element from P(X). Now consider > > > > > >> Q = {x in X|x not in f(x)}. Clearly, for all x in X we have > > > > > >> Q unequal to f(x), since x is a member of exactly one of f(x) and Q. > > > > > >> So Q is missing from the bijection. > > > > > >> > > > > > >> > > > > > > Again with the "Clearly". You might want to refrain from using the > > > > > > word, and just try to be clear, without hand-waving. > > > > > > > > > > > > There is no requirement that subset number x include x as a member, > > > > > > > > > > Quite so. But there is a requirement that subset number x _either_ > > > > > include x as a member _or_ not include x as a member. Only one of > > > > > those two statements can be true. And then Q _either_ not includes x > > > > > as a member _or_ does include it, respectively. > > > > > > > > > > You are free to choose your mapping as you want to. But once you have > > > > > chosen your mapping, each subset number x _either_ includes x as a > > > > > member _or_ it doesn't. Whether it does, can be chosen independently > > > > > for every x. But once you are through, for every particular x, x will > > > > > be in f(x), or it won't. And depending on that, x won't be in Q, or > > > > > it will. > > > > Actually, as I think about it, given this natural mapping of the naturals > > > > to > > > > the subsets of naturals, subset number x will ONLY include x as a member > > > > for > > > > subsets 0, 1 and 2. Beyond that, subset x will NEVER contain x. So, you > > > > have > > > > non-empty Q for the null set, and the singletons {1} and {2}. So, what > > > > does > > > > that prove? > > > > > > The mapping from X to P(X) is not "natural", it is 'arbitrary', meaning > > > that it can be anything and cannot be assumed to have any special > > > properties. > > > > > > In particular, f(0) = f(1) = f(2) = X is possible, whenever {0,1,2} is > > > a subset of X, . > > > > > I defined the mapping and it's as natural as it gets. Each succesive bit > > represents membership of each successive element. It doesn't get any more > > natural than that.n Every unique infinite string of bits represents a unique > > subset of the naturals, so I don't know WHAT that last sentence means. > > > There is a 1-1 correspondence between infinite bit strings and > > subsets. > > But there is no 1-1 correspondence between naturals and INFINITE bit > strings (only with finite bit strings). This is just another instance of > that same delusion that TO has that there exist naturals with more than > finitely many naturals as predecessors. > If they have all 1's in finite positions, then there is indeed a bijection between infinite bit strings and finite naturals. If they have any 1's in positions infinitely to the left of the point, then they represent sets that include infinite integers, and also have infinite values as binary numbers. The bijection works perfectly, IF one allows infinite integers. This restriction on the whole numbers is the only thing standing in the way of bijection betweem [0,1) and N. -- Smiles, Tony
From: MoeBlee on 25 Jul 2005 15:35 >From a post by Tony Orlow (aeo6): > Well, Bill, I have come to realize that this is one of my most central > complaints about mathematics as it is today. There is great emphasis put on > axiomatic systems, and axioms are given this status as unquestionable atomic > fact without justification Such unquestioned status giving is not required for axiomatics. > Axioms are taken as fact within an > axiomatic system and used for proofs and arguments, but that doesn't mean every > axiom is universally true as stated, or true at all outside of the system where > they reside. Non-logical axioms, by definition, are not "universally true". Nor does axiomatics depend on claiming truth in any model that is not indeed a model of the theory. > our inductive step (f(n)->f(n+1)) involves an increment (n+ > 1 is finite), then we can see that the infinite loop There is not an infinite loop. > to > produce an infinite value. No infinite value is produced. > In my opinion, mathematics needs to be more fully integrated, and not further > decomposed into independent axiomatic systems without justification. Enter set theory, which integrates axiomatic systems. MoeBlee
From: Tony Orlow on 25 Jul 2005 15:41 W. Dale Hall said: > > > Tony Orlow (aeo6) wrote: > > The World Wide Wade said: > > > >>In article > >><MPG.1d472484f809e37c989f2c(a)newsstand.cit.cornell.edu>, > >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > >> > >> > >>>The idea of uncountability as being equivalent to "larger than the set of > >>>naturals" is unfounded. There is no reason to believe that larger sets cannot > >>>be enumerated. the power set of the naturals can be enumerated and bijected > >>>with the naturals, as I described in another post, as long as infinite > >>>natural > >>>numbers are allowed. > >> > >>Sort of like saying "if 0 = 1" is allowed. > >> > > > > No, more like saying "if infinite digits are allowed", which is what is > > required to have an infinite set of digital numbers using a finite base. That > > was a useless comment. I shouldn't even be responding. > > This is untrue, with the possible exception of whether it's worthwhile > to respond to these articles. It is definitely not necessary to have any > infinitely long numbers to accommodate an infinite set of numbers. It > is necessarily to have numbers of arbitrarily finite length, but no > integer requires infinitely many digits. Actually, whether you are talking about reals in [0,1) or n in N, or ANY infinite set of elements represented as strings of symbols, in order to have an infinite set, you either need an infinite set of symbols, or infinitely long strings. To the right of the point, these infinitely long digital strings represent finite values, whereas to the left, they represent infinite values (mostly). This is a fact of any symbolic systems. > > I'm sure I'm not the only one to suggest the following, but here goes: > > 1. Note that the successor function is a 1:1 function from the set of > natural numbers to a proper subset. This establishes N as an infinite > set, since it is in 1:1 correspondence with a proper subset. > > 2. Note that the induction axiom (one of the Peano axioms) states: > > If A is a subset of N that is closed under the successor > function, and which contains 0, then A = N. > > The set of all finite natural numbers satisfies the requirements of the > induction axiom. Therefore, the set of finite natural numbers is equal > to the set of all natural numbers. In other words, every natural number > is finite. Take a look at the rewrite of Peano's axioms I posted earlier today. > > 3. Note that the set of finite natural numbers is (without the > induction axiom) already an infinite set by virtue of (1) above: > the successor of every finite natural number is another finite natural > number, and the set of all successor numbers is a proper subset. > > Whatever set you take to be the set of natural numbers, if it contains > infinite natural numbers, is not the same set that mathematicians are > referring to when they use the term "natural numbers". Okay. Let's just call them integers. > > Perhaps you mean to be claiming that there is no model of Peano > arithmetic? I am claiming that after infinite succession, we have changed values by an infinite amount. > > I'll also note that your "infinite series" argument is not germane. The > issue is not whether one *can* formulate infinite series, but whether > one *must* formulate these. Arithmetic by itself does not mandate such, > nor does algebra. In fact, you nod to that point by raising the notion > of convergence, a notion that is decidedly non-algebraic, but instead > topological, in nature. The very act of defining convergence and of > assigning numerical values to the formal sums that can be written > involves an expansion of the notion of summation; if it were not so, > then there would be no choice in the matter of assigning the value > of sums. Instead, there is a great deal of choice in the matter, > viz. p-adic numbers, being completions of the rationals wrt norms > other than the usual one. In addition, note that the phenomenon of > conditional convergence carries with it the ability to assign the > value of a sum to any real value whatsoever by merely rearranging terms. That may all be true, but none of it makes it possible to increment a value an infinite number of times and get a finite value. You are adding 1 an infinite number of times to produce the set. You have infinite numbers in it. > > Dale. > -- Smiles, Tony
From: georgie on 25 Jul 2005 15:40
So what you are really saying is that averything is an axiom. "whfh a3r 23r237818n df er rdq2" is an axiom in some formal system that (probably) nobody had ever considered yet and never will beyond this thread. I would hardly call that rigorous, since i just pounded my fingers on my keyboard to develop it. |