Cantor Confusion
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From: MoeBlee on 6 Nov 2006 18:13 mueckenh(a)rz.fh-augsburg.de wrote: > 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 is NOT Fraenkel, Bar-Hillel, and Levy's point at all! So you have not made ANY point regarding your own views by quoting Fraenkel, Bar-Hillel, and Levy. > > 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. And they don't advocate it at all. If you want a theory in which there are growing sets, then, by all mean, go ahead and develop one. > > 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. Because no one who works with Z set theory claimed that for any possible description there is a set that has the properties described. And what do you mean "all sets exist"? Do you mean all sets that exist do exist? (Which is of course true.) Or do you mean that for any specification of properties, there exists a set having those properties? MoeBlee
From: Virgil on 6 Nov 2006 18:18 In article <1162846073.738566.241750(a)h48g2000cwc.googlegroups.com>, 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? Cut the cake at your > grandson's 6th birthday and look whether he will find the parts > useless. They would be useless now as my grandsons are all past 6. Besides, cakes are made to be subdivided, edges in a binary tree are not.
From: Virgil on 6 Nov 2006 18:20 In article <1162846423.657118.268250(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > You need not declare or prove that a > triangle is faster than a matrix when there is no axiom stating the > contrary. > > Regards, WM WM is going incoherent!
From: Virgil on 6 Nov 2006 18:33 In article <1162846826.373281.277770(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Maybe WM can't but even freshmen mathematicians can. If line n has length >= n, then Length(diagonal) >= number of lines. See how easy that was, WM? > So do you understand it now, how an edge can be divided in two pieces > and each one can be related to a path? No. As far as I can see, either an edge is entirely contained in a path or entirely disjoint for it. > So do you understand now, how > half an edge can be divided in two pieces and each one can be related > to a path? Still no. Which paths gets which parts of which edges? And how do you insure that some parts of some edges are not counted more than once? > 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? I neither understand it nor believe it. Until WM can show explicitly how which parts of which edges get matched to which parts of which paths, I will stick to matching whole edges with whole paths, in which I can show that the set of edges injects into the set of paths, but the set of paths cannot inject into the set of edges. > > > > > You are ignorant of the basics of set theory. > > I know its basics. That does not mean that I have to accept them. You do not even know them, as your silly postings have confirmed. It is what you think you know that ain't so that is hurting you.
From: Virgil on 6 Nov 2006 18:42
In article <1162847096.380167.135420(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > 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. Except that there is no set theory in which sets themselves are variable. One must construct set valued functions in order to achieve such "variability". > > 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? We are operating within some axiom system, and while in that system, what it says goes. In ZF, for example, there is no such thing as a set of all sets, because the axioms limit how sets may be "constructed". In NBG, there can be a class of all sets, but it cannot itself be a set. > > > > 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. If asked, I'll bet they would call it wrong and illogical and 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. Ask Russell. Among others. > > Regards, WM |