Re: abundance of irrationals!)

"r.e.s." <r.s@xxxxxxxxxxxxxxxx> wrote in message news:<iUU8e.10358$lP1.3161@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>...

> The first step of *your* putative definition of sum[k in N] 2^-k
> leads to the infinite string 0.111..., as you've quoted above.
>
> But you now refuse to follow through with the next step -- that of
> defining exactly what is *meant* by 0.111... -- without admitting
> that it is just the binary representation of unity (i.e. the limit
> of an infinite sequence of partial sums) as used standardly in
> diagonal arguments.

It is used standardly and erroneously in diagonal arguments. The limit
(of the partial sums) is not part of the bijection of the sequence (of
the partial sums) with N (as realized by the lines of Canor's list).
>
> Given that you do not accept the *standard* definition of 0.111...
> as a binary representation of unity, your "explanation" amounts to
> another refusal to say what is your non-standard definition.

I accept the agreement that binary 0.111... = 1. But I do not accept
that 0, as a term, belongs to the sequence (1/n). And it is similarly
wrong to believe that all the digits in 0.111... can be enumerated by
natural numbers. Hence, from the following list,

0.1
0.11
0.111
....

we obtain two theorems:

A) the nth partial sum of the diagonal can always be found in the nth
line.
B) The binary representation of 1 is not in a line.
Conclusion:
C) The binary representation of 1 is not in the diagonal.
Set theory denies C. So set theory is incompatible with
{limit-mathematics} u {logic}.

Regards, WM.
.

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