Cantor Confusion
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From: Virgil on 23 Nov 2006 03:16 In article <456556CD.4030107(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > Virgil> While all sequences are sets ( as functions from N to some set of > > values) not all sets are sequences. > > So you could be correct if referring to sets that are no sequences? > Look above: I referred to something going on forever. Such as EB's nonsensical utterances about that of which he displays such great ignorance?
From: Virgil on 23 Nov 2006 03:27 In article <456557C7.3090104(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/23/2006 7:10 AM, Virgil wrote: > > In article <45647BF3.7090907(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/21/2006 10:25 PM, Virgil wrote: > >> > In article <4563067B.5050805(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > > >> >> On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: > >> >> > >> >> > Every set of natural numbers has a superset of natural numbers which > >> >> > is > >> >> > finite. Every! > >> >> > >> >> > >> >> I am only aware of the unique natural numbers. If one imagines them > >> >> altogether like a set, then this bag may also be the only lonly one. > >> > > >> > Is that "lonely"? > >> > >> I meant: There is only one natural sequence of numbers: 1, 2, 3, ... > >> For instance 0, 1, 2, ... or 3, 4, 5, ... or 2, 4, 6, ... are something > >> else. "Every" implies that there are more than one. > > > > There is only one sequence of all natural numbers taken in natural > > order. > > But there are uncountably many different sequences of natural numbers if > > one is not required to include all of them nor to take them in any > > particular order. > > Countably many, yes. As many as there are natural numbers. If one is not required to include all of them nor to take them in any particular order, then least as many of sequences as there are infinite subsets of the infinite set of finite naturals, N, "multiplied" by the number of bijections from N to N. > However, none of them is identical with the notion of _the_ natural > numbers. Whyever need it be? A sequence is a function whose DOMAIN is the set of naturals in standard order, The values of such a function, even if all naturals, need not cover all naturals nor appear in natural order.
From: Eckard Blumschein on 23 Nov 2006 06:18 On 11/23/2006 5:15 AM, William Hughes wrote: > Eckard Blumschein wrote: >> Actual infinity is not as handsome as it seems to be. > > Non sequitur Do you have any argument to support: assuming > that the set of all natural numbers exists precludes counting > and/or the ability to separate from each other all single numbers. Let me try to give the gist of what Wittgenstein wrote: Too many laws correspond to an absolute lack of any obedience. In more formal symbolic description (a=any number): oo * a = oo oo + a = oo oo + 0 = oo oo * 0 = a In other words: Infinity, i.e. the actual indefinitely large, must not be used like a number. In order to have an readable order a, one has to restrict the number of included elements to b if the smallest readable interval is c: b*c=a. For b=oo the only reasonable c equals 0. >> >> > The elements of the >> > potentially infinite set are exactly the same elements with >> > exactly the same properties as the elements of the >> > actually infinite set. >> >> I object to exactly this assumption. Actual infinity denotes a diffent >> quality, not a larger quality, not a quantity at all. > > Again a non sequitur. Of course. My objection does not follow from your belief. The definition of infinity like a quality, something that can neither be enlaged nor exhausted is very basic and does not have a reasonable alternative. >> >> And the elusive belief hidden within and conveyed by this name. >> >> Aren't I correct? >> > >> > No. You can develop "potential set theory". >> >> Quite a while ago, it was a surprize to me: Axiomatic set theory does >> not really require the actual infinity. >> >> > Just define >> > an element of a potential set to be an element of any one >> > of the arbitrary sets that can be produced. Now go through >> > set theory and add the word "potential" in front of each >> > occurence of the word set. There is no difference between >> > saying "a potentially infinite set exists" and saying "an actually >> > infinite set exists". >> >> Yes. This is indeed a clever method of obscuration. > > Obfuscation of what? Hiding of the categorical difference between discrete and contiuous, also expresses like genuine (i.e. rational) number and really real "number", also expressed like countable and uncountable. > Do you agree with the statement: > There is no difference between saying "a potentially infinite set exists" > and saying "an actually infinite set exists". or not? Before I may decide I would like to clarify the meaning of the term set. Cantor's definition of a set has been confessed invalid at least since 1923 with no possiblity of correction in case of infinite sets. Why? Fraenkel confessed paradoxa due to this definition. The reason behind them is ambiguity. Cantor's definition claims to provide each singe element of the set as well as simultaneously the infinite set as an entity. So it is a chimera. If not, do you have anything even remotely relevent to say? Definitely. >> Nonetheless, if one >> prefers to distinguish between rational and real numbers, then the reals >> are only distinguished by being fictitious and therefore uncountable. >> Genuine reals being really real are only required for theoretical >> considerations. They do however, have properties quite different from >> the properties of genuine numbers. This is my original concern. After I >> already settled the somewhat puzzling matter by means of really real >> numbers, I took me some time to understand the relationship to the still >> thaught naive set theory, axiomatic set theory and what a comprehensive >> understanding of mathematics really needs. >> > > This semi-coherent ramble Semi-understandable to you. What did you not understand? > has nothing whatsoever to do with > the question of whether there are any real (as opposed to cosmetic) > differences between potentially infinite and actually infinite sets. As long as the mathematical notion of a set lacks a valid definition, I regrett: One cannot utter something reliable concerning infinite sets. While Fraenkel 1923 clearly explained the notions potential and actual infinity, he did neither use "potentially infinite set" nor "actually infinite set". Set theory is de facto based on the illusion that there is no difference between number and real "number". > If you do not wish to discuss this question fine, but a simple > statement > to that effect would have been far preferable to the above. > (Even simply not replying would have been preferable.) I cannot warrant that I am able to always reply since the discussion has been grown to an extent exceeding my time recources. Therefore do not interprete a lacking reply of mine a sign of surrender. Eckard Blumschein
From: mueckenh on 23 Nov 2006 06:26 Ralf Bader schrieb: > >> For example if one removes all the points in the Cartesian plane which > >> have either coordinate non-integral, what is left is about as > >> disconnected as one can imagine. > > > > Cantor wanted to suggest that the removal of a countable set leaves the > > plane connected while the removal of an uncountable set does not. As an > > example he chose the algebraic numbers (obviously in order to > > distinguish them from the transcendental numbers). > > This is the point I alluded to when I wrote "Mückenheim took that example > for the main issue, ascribing > some other weird opinions to Cantor on the way." You wrote that before I posted the above sentence. So you allude in advance? You look into the future? What a wretched excuse! > That Cantor wanted to > suggest that the removal of an uncountable set does not leave the plane > connected is nothing but your phantasy. Cantor emphasized: Was die abzählbaren Punktmengen betrifft, so bieten sie eine merkwürdige Erscheinung dar ... Concerning the countable sets of points we observe a strange phenomenon ... This means, that he did *not* consider uncountable sets. > It is trivial to give examples of > uncountable sets whose removal does not disconnect the plane. In fact, after I showed you such sets. But with Cantor's original proof this is impossible to see, because just the countability is the tool Cantor needs for his proof. Therefore I wrote in my paper that transfinite set theory veils even most simple structures. > E.g. the > points inside a circle or rectangle, and I'm pretty sure that Cantor was > aware of this. Because uncountability has nothing to do with connectedness > in this way, it is remarkable that countability has - it seems to be much > more plausible to me that this is how Cantor thought. Cantor wrote: "It is remarkable that removing a countable set leaves the plaine *being connected*." But it is not remarkable that removing an uncountable set leaves the plaine being connected? What a foolish assertion. > > > I proved that there is by far a simpler proof for the algebraic > > numbers. > > I proved that this proof can also be applied to the transcendental > > numbers. > > Yes, and this proof even seems to be correct, but it is trivial, that is: on > the level of an easy exercise in a beginner's topology course. Why then do you say "it seems" correct? It seems you are not quite sure, because you do not even understand this admittedly simple piece of mathematics. And that you missed the clue of my paper completely is obvious from your last posting. > > > See the appendix of http://arxiv.org/pdf/math.GM/0306200 > > It is telling that you consider such a crumb of dust worth to be published. > You have no idea about the current state of the art. Which art are you dreaming of? The art to slander and slaver? > > BTW, I am not obliged to explain your errors to your satisfaction. and not even able to explain simple mathematics your own satisfaction. You should have recognized at least that the coordinates considered by Cantor and me are the algebraic coordinates, and not (as you wrote) the rational coordinates. Yes, there is a difference! Regards, WM
From: mueckenh on 23 Nov 2006 06:46
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > The countability of the set W of all the words of a finite alphabet is > > > > proved by other means than the usual countability proof. The latter > > > > consists in constructing a bijection with N, i.e., in constructing a > > > > list. But it is impossible to construct a list of all words. It is > > > > impossible to construct a bijection between W and N. > > > > > > WRONG. It is a theorem that from the set of finite sequences of members > > > of a finite alphabet there is an injection into N. > > > > Nevertheless it is impossible to construct such an injection > > WRONG. Do it. > > > as it s > > impossible to construct a well order of the reals). > > Showing an injection from the set of finite sequences of members of a > finite alphabet into N is not analogous to well ordering the reals. I know that. Why do you stress it? > The > former can be explicit; Do it. while the later cannot be. > > > > > Why should it be > > > > possible to construct a bijection between N and N? > > > > > > Because ANY x is 1-1 with x, by the identity function on x. > > > > It is an unfounded assumption to believe that all x of the set appear > > in this bijection, only by writing x = x. Ridiculous! > > We don't prove the existence of an identity function just by writing 'x > = x'. Then demonstrate what you think is a proof. > > > > You are wrong. If a set cannot be well-ordered, it cannot be assigned a > > > > definite cardinal number. In order to assign a cardinal, at least one > > > > bijection to an ordinal is required. This requires at least one > > > > well-ordering. > > > > > > WRONG. By using Scott's method (which requires the axiom of > > > reqularity), we don't need the well ordering theorem to prove that > > > every set has a cardinality. > > > > No. But we need it to determine this cardinality. Again you think > > erroneously that an existence proof is of any value. > > Constructivity is a separate matter. My point stands that we do not > need the well ordering theorem for every set to have a cardinality. Your point is false. In order to have a cardinality, you must know the number class. In order to know the number class, you must have a bijection to an ordinal. In order to have a bijection to an ordinal, you need a well order. It was Cantor's most important assertion hat all sets can be well ordered, *in order to ascribe them a cardinal number.* Maybe that you can ascribe some symbol to some set and call this symbol a cardinal. That is the same deceit as your bijections to N which are none. I think if one swells to explosion about his knowledge of set theory, he should at least know the very foundation. But I know, that you do not even understand the simple texts of Fraenkel et al. > > PS: What about the binary tree in ZFC? Any results? Any idea why it > > can't be done? > > I already mentioned what I would require to discuss your tree argument. > If you want me to talk about your tree, then please answer my previous > posts from about a week ago on the subject of even discussing your > argument. I put it to you very clearly just what you need to tell me > before we proceed. If you won't do so, then I'm not inclined to waste > my time on your argument. What I can tell you is the following: The binary tree Consider a binary tree which has (no finite paths but only) infinite paths representing the real numbers between 0 and 1 as binary strings. The edges (like a, b, and c below) connect the nodes, i.e., the binary digits 0 or 1. 0. /a \ 0 1 /b \c / \ 0 1 0 1 .......................... The set of edges is countable, because we can enumerate them. Now we set up a relation between paths and edges. Relate edge a to all paths which begin with 0.0. Relate edge b to all paths which begin with 0.00 and relate edge c to all paths which begin with 0.01. Half of edge a is inherited by all paths which begin with 0.00, the other half of edge a is inherited by all paths which begin with 0.01. Continuing in this manner in infinity, we see by the infinite recursion f(n+1) = 1 + f(n)/2 with f(1) = 1 that for n --> oo 1 + 1/2 + 1/ 4 + ... = 2 edges are related to every single infinite path which are not related to any other path. (By the way, the recursion would yield the limit value 2 for any starting value f(1).) The load of 2 edges is only related to infinite paths because any finite segment of a path with n edges will carry a load of (1 - 1/2^n)/(1 - 1/2) < 2 edges. The set of paths is uncountable, but as we have seen, it contains less elements than the set of edges. Cantor's diagonal argument does not apply in this case, because the tree contains all binary representations of real numbers within [0, 1], some of them even twice, like 1.000... and 0.111... . Therefore we have a contradiction: |IR| > |IN| || || |{paths}| =< |{edges}| If you don't understand this simple and clear exposition, then there is no hope that you will be able to think any further. I really don't know what further information could be required. Regards, WM |