On 24 Jun., 09:34, Virgil <Vir...@home.esc> wrote:
> > The symbols above abbreviate the sequence of lists > > > An > > ... > > A0 > > L0 > > > Each of them has an antidiagonal that either is not in the list (then > > the set of all of them is unlistable) or is in a list. Then Cantors > > argument is wrong. > > If one has only a list of lists,
There is always, at every stage of the construction, one single list like Ln above, that either contains or not its antidiagonal.
Thise prodedure is abbreviated by (..., An, ... A2, A1, A0, L0) but of course there cannot appear at any stage a list without first line.
> then its union can be listed and > therefore has many "antidiagonals
Before doing so, you first need the elements. There are two alternatives: 1) There is a list in the construction that contains its antidiagonal. Then Cantor's proof is wrong. 2) There is no list in the construction that contaiuns its antidiagonal, then the antidiagonal cannot be placed at position 0 at the previous list without generating another list and another antidiagonal. Therefore, the set of the lines of L0 and the set of all antidiagonals occuring during construction cannot be listed.
It is as simple as that.
It is not a big deal. Set theory shows many contradictions arising from the idea, that from "every n in N can be constructed" it is implied that "N can be constructed". This is wrong, because for every n, there is an infinite set of unconstructed elements of N.
This is so for *every* n in N. Hence, N cannot be constructed, and no proof for all n in N has ever been valid.
... classical logic was abstracted from the mathematics of finite sets and their subsets .... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. ... As Brouwer pointed out this is a fallacy, the Fall and Original sin of set theory even if no paradoxes result from it. [Weyl, Hermann (1946), "Mathematics and logic: A brief survey serving as a preface to a review of The Philosophy of Bertrand Russell", American Mathematical Monthly 53: 2?13.]