Re: abundance of irrationals!)

"r.e.s." <r.s@xxxxxxxxxxxxxxxx> wrote in message news:<Ym88e.7115$lP1.4604@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>...

> >> That produces a *string* of infinite length, namely s = 0.111...,
> >> (with a 1 in the nth position to the right of the radix point for
> >> every n in N = {1,2,3,...}. Since no limit is to be taken, s is
> >> merely an infinite string with no numerical value yet defined.
> >>
> >>
> >> > Take Sum b_k * 10^-k without letting k become actually infinite.
> >> Rather,
> >> "Take Sum 2^-k without letting k become actually infinite."
> >>
> >> How, exactly?
> >> Please *define* what you mean by that, without taking a limit.

I am not obliged to give that definition. This sum is used in
Cantor-lists as diagonal number. The limit is not contained in any
partialsum indexed with a natural number, because the limit is indexed
by oo. (Compare the terms of the sequence (1/n) and its limit 0.
Series can be considered sequences.)

> Your reply does NOT provide a definition ...
> We're still waiting for your definition of SUM[k in N] 1/2^k.

I hope, you will either accept my explanation or acknowledge that
Cantor's diagonal argument does not prove anything. (I would prefer
the latter.)

Regards, WM
.

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