Cantor Confusion
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From: mueckenh on 6 Nov 2006 15:40 David Marcus schrieb: > > Why then do you not understand how an edge of the binary tree can be > > divided? > > Because you don't state clearly how you wish to divide the edges and how > the division is used in the "relation" between edges and paths that you > are defining. It is completely irrelevant how an edge is divided. If you divide a cake of five pounds into 10 equal pieces then you can calculate the weight of each piece without knowing the details of the division. > > > > > then it can only be a whole number, I would guess. > > > > > > These problems you're having are of fitting set theory to your own > > > system of terminology. To work in set theory, we don't need to care > > > about your own system of terminology. > > > > "Whole number" is not my terminology. > > Regardless of whose terminology it is, if you use it, you should be > willing to define it when asked. So, please define what you mean by > "whole number". > If you knew what it means to divide a number, then I could explain "whole number" as the opposite, namely an undivided number 1, like a whole edge is an undivided edge. But, alas, I am afraid you cannot understand. > > Please don't conclude from your own ignorance on mine. > > Wouldn't dream of it. I am sure. But I know what a whole number is. Regards, WM
From: mueckenh on 6 Nov 2006 15:42 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > > In article <1162472271.664965.315880(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > > Virgil schrieb: > > > > > Does WM choose to claim that a set of naturals which has no maximum must > > > > still have a finite upper bound? > > > > 1 > > 11 > > 111 > > ... > > > > The length of each column is omega. > > I assume that we are using ZFC as our logical system. > > Assuming by the "length of each column" that you mean the number of 1's > in each column, then the number of 1's in each column has cardinality > aleph_0. So, what you wrote is essentially correct. > > > The length of each line is less than omega. > > Each line has a finite number of 1's. Finite cardinality is less than > aleph_0. So, what you wrote is essentially correct. > > > The length o the diagonal is less than omega. > > The number of 1's in the diagonal is aleph_0. So, what you wrote is > false. In fact, I have no clue what you could possibly be thinking that > would lead you to make such an obviously incorrect statement. The length > of the diagonal is clearly the same as the length of the first column, > and you just wrote above that the length of each column is omega. The length of the diagonal is clearly not more than the length of any line. > > > Set theory is based upon intermingling maximum and supremum. > > If what you wrote above is supposed to demonstrate this, then you have > failed for two reasons: You made a false statement, and you didn't use > the words "maximum" or "supremum" in what you wrote. Nevertheless a mathematician would understand where maximum and supremum occur in the above reasoning. Regards, WM
From: mueckenh on 6 Nov 2006 15:44 Virgil schrieb: > > Whatever your point, you won't be able to show that merely dropping the > > axiom of rabbithood from the Z axioms entails that there are only > > non-rabbit > > sets. > > Which axiom of ZF or NBG does WM connect with rabbits? Which axiom of ZF - INF do you connect with infinity? Regards, WM
From: mueckenh on 6 Nov 2006 15:47 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. Regards, WM
From: mueckenh on 6 Nov 2006 15:53
MoeBlee schrieb: > > Whatever your point, you won't be able to show that merely dropping the > > > > axiom of rabbithood from the Z axioms entails that there are only > > non-rabbit > > sets. > > Your analogy just agrees with what I said. You don't seem to have a > point. > So you would not exclude sets with rabbithood in ZFC? > > To put it in other words: It simply an imbecile nonsense to talk about > > finished infinity without explicitly stating that it was not. > > I don't know what you mean by "stating that it was not". It simply an imbecile nonsense to talk about finished infinity without explicitly stating that it was not an imbecile nonsense to do so. > > Neither the axiom of infinity nor its negation are theorems of ZF > without the axiom of infinity. Get it? > And infinity without an axiom stating its existence is as meaningful as the rabbithood of sets within ZFC. You need not declare or prove that a triangle is faster than a matrix when there is no axiom stating the contrary. Regards, WM |