Re: Ultimate debunking of Cantor's Theory

On Jul 19, 11:27 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 18 Jul., 01:13, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

Consider the list
0.0
0.1
0.11
0.111
...

(1) Is the diagonal of the list an entry in the list? No, of course
it's not, since the diagonal is a denumerable sequence and each entry
in the list is a finite sequence.

That is one statement. The other claim is that the infinite diagonal
cannot exist without an infinite number of list entries and an
infinite number of 0's adjacent to 1's. This second claim proves the
existence of a sequence of infinitely many 1's. Why do you think this
second claim is wrong?

Whose claim?

The claim of those who see how the diagonal is constructed. The
diagonal cannot exist without lines which are as long as each initial
segment of the diagonal including the complete diagonal.

Let's look at this simple case:

1
11
111
1111
......

What I mean by that is this:

Let f be the denumerable sequence of finite sequences into {1} such
that dom(f(n))=n+1.

And I discuss what is provable about f as 'provable' means provable in
some given theory, which, unless stated otherwise, may be presumed to
be Z set theory.

Thus diagonal(f) = {<n y> | new & y=f(n)(n)}

And it is trivial to prove that that set exists.

Now you may mean whatever you want by such illustrations, but none of
your own interpretations refute that it is a theorem of Z that the
diagonal of f exists.

The list, more precisely the following matrix,
1
11
111
...

shows (by simple bijection of initial segments of the diagonal and
lines) that an infinite diagonal cannot exist unless there were also
an infinite line. Hence actual infinity is disproved. With it
uncountable infinity is disproved also.

You keep arguing with regard to you own informal notions with no
axioms, no inference rules, and no system of definitions. As to all
matters in regard to your own informal notions, indeed I defer to you
as the world's foremost, and indeed, single authority on them.

MoeBlee

.

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