Another AC anomaly?
in [Logic]
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From: Virgil on 8 Dec 2009 16:15 In article <fd82ac52-8f71-4930-8f8e-415187ae832b(a)g26g2000yqe.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Dez., 16:07, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > �> You have seen the axiom of infinity. It say that an infinite set > > �> exists and that implies that infinitely many elements of that set > > �> exist. That is actual infinity. > > > > Oh, so actual infinity means that a set with infinitely many elements > > exists? > > Yes. > > > In that case you should reject the axiom of infinity. �You are allowed to > > do > > that, and you will get different mathematics. �But you can not claim that > > mathematics with the axiom of infinity is nonsense just because you do not > > like it. �But go ahead without the axiom of infinity, I think you have to > > redo quite a bit of mathematics. > > Before 1908 there was quite a lot of mathematics possible. > There was quite a lot of possible mathematics. Most of which has been extended considerably since. Even if WM wants to restrict himself to only those parts of mathematics which precede 1908, he has not the right, or the power, to impose such restrictions on anyone else, except possibly his poor misled students. > > > �> The definition of an actually infinite set is given in set theory by > > �> the axiom of infinity. > > > > You are wrong, the axiom of infinity says nothing about "actually infinite > > set". �Actually the axiom of infinity does not define anything. �It just > > states that a particular set with a particular property does exist. > > That is just the definition of actual infinity. That may be WM's definition of "actual infinity", but his definitions carry weight only in Wolkenmuekenheim, and are of no importance anywhere else. > > �> The definition of a potentially infinite set is given by > > �> 1 in N > > �> n in N then n+1 in N. > > > > That does not make sense. �Without the axiom of infinity the set N does not > > necessarily exist, so stating 1 in N is wrong unless you can prove that N > > does exist or have some other means to have the existence of N, but that > > would be equivalent to the axiom of infinity. > > N need not exist as a set. If n is a natural number, then n + 1 is a > natural numbers too. Why should sets be needed? If WM chooses to work in a mathematics without sets, he is quite free to do so, but has no power to impose such limits on anyone else, except possibly his poor captive students. > > > > �> The complete infinite binary tree can be constructed using countably > > �> many finite paths (each one connecting a node to the root node), such > > �> that every node is there and no node is missing and every finite path > > �> is there and no finite path is missing. > > > > Right. > > > > �> Nevertheless set theory says that there is something missing in a tree > > �> thus constructed. What do you think is missing? (If nothing is > > �> missing, there are only countably many paths.) > > > > And here again you are wrong. �There are countably many finite paths. > > �There > > are not countably many infinite paths, and although you have tried many > > times you never did show that there were countably many infinite paths. > > There is not even one single infinite path! Then one does not have a COMPLETE infinite binary tree, but only a non-set of (finite) nodes. But there is every path > which you believe to be an infinite path!! Which one is missing in > your opinion? Do you see that 1/3 is there? In your model I do not see any infinite paths, since every infinite path requires an infinite set of nodes and your model disallows all infinite sets. > > What node of pi is missing in the tree constructed by a countable > number of finite paths (not even as a limit but by the axiom of > infinity)? Unless one is allowed to have a SET OF INFINITELY MANY nodes, one can not have even the fractional part of pi. Similarly for any proper fraction whose denominator is not a power of 2. A sequence which is not (actually) infinite (an image of the SET N), must have a last term, and thus be incapable of converging.
From: Virgil on 8 Dec 2009 16:26 In article <9f4850a5-268c-4b3d-a84b-34713cfaedd7(a)c34g2000yqn.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Dez., 16:13, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > �> Nevertheless it is a limit ordinal. > > > > Yes, that does not mean that necessarily a limit is involved. �It is a limit > > in the sense that you do not get there by continuously getting at the > > successor, in that case it is a limiting process. �But when you define > > N as an infinite union you do *not* go there by continuously getting at > > a successor. �The union of a collection (finite, countably infinite or > > some other infinity) is defined whithout resorting to successor operations. > > Moreover, they would even not make sens if the collection is infinite but > > not countably infinite. > > That does not make sense in either respect, so or so. What does not seem to make sense to WM makes very good sense to those not constrained by the artificial rules governing Wolkenmuekenheim > > > > �> As a starting point, we use the fact hat each natural number is > > �> identified with the set of all smaller natural numbers: n = {m in N : > > �> m < n}. > > > > Note that here N is apparently already defined, without using a limit. > > Natural numbers can be defined without using a set. As soon as one wants to talk about any property of ALL NATURALS, one inevitably gets the set of them. > > > > �> � � � � Thus we let w, the least transfinite number, to be the set N > > �> of all natural numbers: w = N = {0, 1, 2, 3, ...}. > > �> � It is easy to continue the process after this 'limit' step is made: > > �> The operation of successor can be used to produce numbers following w > > �> in the same way we used it to produce numbers following 0. > > > > Yes. �So what? �That you can define things using a limit does *not* > > imply that it is necessarily defined as a limit. > > O I see. No, you do not see at all. > That's like cardinality. The limit cardinality is not the > cardinality of the limit (because the limit is not a limit). The limit of a sequence of cardinalities of sets need not be the cardinality of the limit of a sequence of sets when the definitions of those two limit processes differ, as they do. For the definition of a limit of a sequence of sets, as given by Dik, and any reasonable definition of the limit of a sequence of cardinalities, there are examples for which the two limits differ. That WM does not like that result, does not make it false anymore than WM liking something makes it true.
From: WM on 8 Dec 2009 16:31 On 8 Dez., 19:36, A <anonymous.rubbert...(a)yahoo.com> wrote: > > A function f is said to be continuous at a point x in its domain if > the limit of f(a), as a approaches x, is equal to f(x); in others > words, the limit of the values of f is equal to the value of f at the > limit, speaking loosely. Of course, not every function is continuous > at every point in its domain, and some functions are not even > continuous at any point in their domains at all. > > The situation for sets and cardinality is no more mysterious than > that. The cardinality of a limit of subsets of the integers is not > guaranteed to be the limit of the cardinalities of those subsets. You > don't expect an arbitrary function to always be continuous, so perhaps > it's unreasonable to expect the cardinality "function," defined on > subsets of the integers, to be continuous.- That depends on the circumstances. If infinite sets exist, then they have a cardinality. Then the limit cardinality is the cardinality of the limit set. If they do not exist but are merely an arbitrary, perhaps inconsistent delusion, then everything is possible. I argue, based on Cantor's claim, that infinite sets exist (in order to show that they do not). Regards, WM
From: K_h on 8 Dec 2009 17:03 "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote in message news:KuAGqH.FrI(a)cwi.nl... > In article <yvCdnW28VrXBqIXWnZ2dnUVZ_s-dnZ2d(a)giganews.com> > "K_h" <KHolmes(a)SX729.com> writes: > ... > > > > > When you mean with your statement about N: > > > > > N = union{n is natural} {n} > > > > > then that is not a limit. Check the definitions > > > > > about > > > > > it. > > > > > > > > It is a limit. That is independent from any > > > > definition. > > > > > > It is not a limit. Nowhere in the definition of that > > > union a limit is used > > > or mentioned. > > > > Question. Isn't this simply a question of language? > > Not at all. When you define N as an infinite union there > is no limit > involved, there is even no sequence involved. N follows > immediately > from the axioms. I disagree. Please note that I am not endorsing many of WM's claims. There are many equivalent ways of defining N. I have seen the definition that Rucker uses, in his infinity and mind book, in a number of books on mathematics and set theory: On page 240 of his book he defines: a_(n+1) = a_n Union {a_n} and then: a = limit a_n. He writes "...that is, lim a_n is obtained by taking the union of all the sets a_n". The text book I have on set theory defines N as the intersection of all inductive subsets of any inductive set. So clearly there are many equivalent approaches to defining N. In Rucker's approach, we could define N as a limit: a_0 = {} //Zeroth member is the empty set. a_(n+1) = a_n Union {a_n} and then: N = limit a_n. My text book on set theory also explicitly states that we can have a limit of a set of ordinals, for example: "...the phase successor ordinal for an ordinal which is a successor and limit ordinal for an ordinal which is a limit". In fact, one of the problem sets is to prove the bi-conditional: If X is a limit ordinal then UX=X (U is union) and if UX=X then X is a limit ordinal. > > My > > book on set theory defines omega, w, as follows: > > > > Define w to be the set N of natural numbers with its > > usual order > > < (given by membership in ZF). > > > > Now w is a limit ordinal so the ordered set N is, in the > > ordinal sense, a limit. Of course w is not a member of > > N > > becasuse then N would be a member of itself (not allowed > > by > > foundation). > > Note here that N (the set of natural numbers) is *not* > defined using a > limit at all. That w is called a limit ordinal is a > definition of the > term "limit ordinal". It does not mean that the > definition you use to > define it actually uses a limit. (And if I remember > right, a limit > ordinal is an ordinal that has no predecessor, see, again > no limit > involved.) N can be defined as a limit or not as a limit. These are really equivalent approaches. k
From: Virgil on 8 Dec 2009 20:23
In article <3501073d-9187-4242-b925-37824eb682e7(a)m26g2000yqb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 8 Dez., 19:36, A <anonymous.rubbert...(a)yahoo.com> wrote: > > > > > A function f is said to be continuous at a point x in its domain if > > the limit of f(a), as a approaches x, is equal to f(x); in others > > words, the limit of the values of f is equal to the value of f at the > > limit, speaking loosely. Of course, not every function is continuous > > at every point in its domain, and some functions are not even > > continuous at any point in their domains at all. > > > > The situation for sets and cardinality is no more mysterious than > > that. The cardinality of a limit of subsets of the integers is not > > guaranteed to be the limit of the cardinalities of those subsets. You > > don't expect an arbitrary function to always be continuous, so perhaps > > it's unreasonable to expect the cardinality "function," defined on > > subsets of the integers, to be continuous.- > > That depends on the circumstances. If infinite sets exist, then they > have a cardinality. No one says otherwise, but that in no way implies that the cardinality of the limit of a sequence of sets as defined by DIk need equal the limit, if it even exists, of the cardinalities of those sets. Such a claim requires proof, which WM has been unable to provide, and such a claim cannot stand in view of the counterexample which Dik provided, so that WM is doubly wrong, in (1) not having a proof of his claim and (2) having found no flaw in Dik's counterproof. > Then the limit cardinality is the cardinality of the limit set. Often claimed by WM but not proved by him nor has WM, or anyone else, refuted Dik's counterexample. > > If they do not exist but are merely an arbitrary, perhaps inconsistent > delusion, then everything is possible. In Wolkenmuekenheim, WM may declare to be possible whatever he likes and declare to be impossible whatever he likes, but outside of his private world, he is not master over what is and is not possible, however often he may claim to be. I argue, based on Cantor's > claim, that infinite sets exist (in order to show that they do not). At which argument, as with so many others, WM fails miserably. |