In article <ff595e3b-8160-4655-a12a-641a304b86ff@y4g2000yqy.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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.
Show me a list of endless binary sequences or real numbers that you think contains what may reasonably be called an antidiagonal for it. I do not believe that any such list can exist.
Cantor's argument satisfies me that for lists of binary infinite sequences, no antidiagonal can ever be a member of it. > > Thise prodedure is abbreviated by > (..., An, ... A2, A1, A0, L0) > but of course there cannot appear at any stage a list without first > line.
But that list can be rearranged into a more standard form, for example with the An and the members of L0 listed alternatingly, and from such a rearrangement a non-member anti-diagonal can be constructed. > > > 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.
Which Cantor's argument shows cannot happen.
> 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.
They can easily be listed, as I have several times shown. Wm has to get over the notion that the elements to be listed cannot be rearranged. > > 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".
In, for example, FOL+ZFC, there is no assumption that every n in N "can be constructed". What is assumed is that there exists a set, along with all its elements, having the properties that we want for N and its elements , which we then call N.
This is wrong, because for every > n, there is an infinite set of unconstructed elements of N.
But we do not "construct" any of them. Though we do construct various naming conventions for elements of N. > > This is so for *every* n in N. Hence, N cannot be constructed
No one claims it can be. But it can be, and usually is, assumed without being constructed.
There is no reason to despise any axiom system, however much it may seem to diverge from physical reality, unless it allows proof of a statement of the form "Both P and not P".
WM, worshipping physics, does not understand that there are areas of mathematics completely outside of physics, so would hamper mathematics by forcing it not to look outside of physics.
If WM could have forced this, internet commerce, for example, would be impossible.
