Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 6 Nov 2006 15:55 MoeBlee schrieb: > > Here is my translation: It is allowed to understand the new number > > omega as limit to which the (natural) numbers n grow, if by that we > > understand nothing else than: omega shall be the first whole number > > which follows upon all numbers n, i.e., which is to be called larger > > than each of the numbers n. > > Okay. I don't see any problem with that. Would you please refresh the > context by saying what point it is that you draw from that quote? Cantor's first book (general paper): Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Leipzig 1883). Foundations of a general set theory. § 11, showing how one is lead to the *definition* of the new numbers... (Es soll nun gezeigt werden, wie man zu den Definitionen der neuen Zahlen geführt wird und auf welche Weise sich die natürlichen Abschnitte in der absolut-unendlichen realen ganzen Zahlenfolge, welche ich Zahlenklassen nenne, ergeben.) Regards, WM
From: mueckenh on 6 Nov 2006 16:00 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > > > For ordinals, > > > > > > x<y <-> xey > > > > > > where 'e' is the epsilon membership symbol. > > > > That is identical with my definition. Take for instance Zermelo's > > definition of the naturals or Cantor's own (Collected works, p. > > 289-290), then you can see it. Your definition has been simply > > translated from Cantor's. Please don't conclude from your own ignorance > > on mine. > > You wrote: > > "My question is : Do you maintain omega > n for all n e N? I know that > modern set theory says so. If something can be larger than a number, > then it must be a number." > > Given that "n < omega" stands for "n e omega", and given that N is > omega, what point is there in asking anyone whether they maintain that > n < omega for every n e N? You're asking someone whether he or she > maintains that n e omega for every n e omega. What is the point of > asking that? If omega > n then we cannot have a diagonal with omega digits in a matrix the lines of which have only n (= less than omega) digits, because the digits of the diagonal are digits of the lines. > > > > > If it cannot be a fraction because ZF does > > > > not yet know how to divide elements, > > > > > > In ZF we define various operations of division. As far as I know, there > > > is not a dvision operation for sets in general. > > By the way, I do see that there are definitions of subtraction, > division, and logarithms for ordinals. But they are not anything like > your non-definition of ordinal division. > > > Why then do you not understand how an edge of the binary tree can be > > divided? > > Haven't I asked you a few times already not to effectively put words in > mouth? I never said I don't understand how an edge can be divided. I > never said anything about "divided edges". So do you understand it now, how an edge can be divided in two pieces and each one can be related to a path? So do you understand now, how half an edge can be divided in two pieces and each one can be related to a path? So do you understand now, how a fraction of an edge can be divided in two pieces and each one can be related to a path? > > You are ignorant of the basics of set theory. I know its basics. That does not mean that I have to accept them. Regards, WM
From: Randy Poe on 6 Nov 2006 16:01 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > > If it cannot be a fraction because ZF does > > > > > not yet know how to divide elements, > > > > > > > > In ZF we define various operations of division. As far as I know, there > > > > is not a dvision operation for sets in general. > > > > > > Why then do you not understand how an edge of the binary tree can be > > > divided? > > > > But once divided, the parts are no longer edges at all, so are useless. > > Once divided the parts of a cake are useless? An edge of a graph is not a cake. It is a connection between two nodes. How do you divide a connection? What does half a connection mean? Here is a simple graph: A--B. In this graph, A is adjacent to B. If I divide the connecting edge between A and B into thirds, what is my resulting graph? What does that division mean? Is A a neighbor of B or not? - Randy
From: mueckenh on 6 Nov 2006 16:04 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: > >"Foundations of Set Theory", North Holland, Amsterdam (1984). > > Here's a post by mueckenh(a)rz.fh-augsburg.de from a thread that is now > not open to reply: > > [begin post] > > David R Tribble schrieb: > > > Yes, I can see now that these are all finite sets. > > > And which are proper subsets of infinite sets. The set of all naturals > > that have been written now, for example. Obviously it's an ever > > growing set as time goes on, and will never contain the entire set > > of naturals that are possible. So it's simply a finite subset of N, > > and always will be. > > > Somehow you are using this fact to "prove" that N can't exist, perhaps > > employing some marvelous mathematical logic that has not been > > tainted by mainstream teachings. You show several finite sets. > > How do they prove anything about infinite sets? > > "a reasonable way to make this conform to a platonistic point of view > is to look at the universe of all sets not as a fixed entity but as an > entity capable of 'growing', i.e. we are able to 'produce' bigger and > bigger sets" [A.A. FRAENKEL, Y. BAR-HILLEL, A. Levy: Foundations of Set > > Theory, 2nd ed., North Holland, Amsterdam (1984) p. 118]. > > Why should simple infinite sets exist in another way? Just because > there is an axiom which cannot be satisfied like the axiom that there > be a straight bent line? > > Regards, WM > > [end post] > > If I undestand WM correctly, he's arguing that infinite sets can only > exist as described - capable of "growing". Correct. That is the meaning of potential infinity. > If that is not his argument > and if that is not the purpose of his adducing this quote by Fraenkel, > Bar-Hillel, and Levy, then I welcome him correcting any > misunderstanding I have about what WM is trying to say. > > I admit that I do not fully master the philosophical remarks by > Fraenkel, Bar-Hillel, and Levy, but I think I get the general idea, and > it is not suggestive that the sizes of sets themselves are variable. > The context of the quote is a discussion of the pros and cons of an > axiom of restriction, which would, roughly speaking, stipulate that > there are no sets except those whose existence follows from the axioms. > And the context includes discussions of various proposed axioms to > "restrict" the universe of sets. So different axioms give us various > possible larger or smaller universes, or some universes that include > some sets and other universes that exclude certain of those sets but > include certain others. So that is the sense of "growing" or "producing > larger and larger", which is to say that we can decide which axioms we > are to adopt and such decisions will determine whether or not the > universe includes or excludes certain sets. Wrong.. You have not understood the meaning. We have: The set of all sets does not exist. But if all mathematical entities including all sets do exist in a Platonist universe, how can it be that the set of all sets does not exist? > That does not suggest that > Fraenkel, Bar-Hillel, and Levy consider that a set itself can grow or > have different members at different times or anything like that. They do not consider it for themselves but they report that some mathematicians could adhere to that point of view. As far as I remember, they do *not* say hat this point of view is wrong or illogical or silly. > For > such axiomatic set theories, membership in a set is definite and sets > are determined strictly by membership. And Fraenkel, Bar-Hillel, and > Levy do not dispute that. If sets are strictly determined by membership, and if all sets do exist, why then doesn't just that set exist the members of which are sets with no further specification. Regards, WM
From: David Marcus on 6 Nov 2006 16:04
mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > Here is my translation: It is allowed to understand the new number > > > omega as limit to which the (natural) numbers n grow, if by that we > > > understand nothing else than: omega shall be the first whole number > > > which follows upon all numbers n, i.e., which is to be called larger > > > than each of the numbers n. > > > > That's not a definition. It is just a remark. > > It is a definition. You should not try to judge about topics which you > must reject because you do not understand them. Please learn: A > definition is an explanation in other words. Nothing more can be done. That's a wrong definition of the word "definition", at least in Mathematics. Are you discussing Mathematics or some other subject? > Would you understand: "The new number omega is the limit to which the > (natural) numbers n grow"? Or is even hat too incomprehensible to you, > because you don't know the meanings of "number", "limit", and "grow"? Sure, I understand it as a remark. But, it still isn't a definition or theorem unless the other words are defined. -- David Marcus |