Re: abundance of irrationals!)

In article <fb701d3c.0504160218.7078010d@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> "r.e.s." <r.s@xxxxxxxxxxxxxxxx> wrote in message
> news:<lWB7e.6258$lP1.6049@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.
> > (You do know that taking a limit with k -> oo does *not* involve
> > "letting k become actually infinite", right?)
>
> It does involve oo. Because we have for any finite natural number n:
>
> n
> SUM 1/2^k = 1 - 1/2^n < 1
> k = 1
>
> independent of the question how many terms contribute to the sum. The
> criterion for obtaining the Sum = 1 is not the infinitude of terms to
> add but that in the limit 1/2^n = 0 which is impossible for any
> natural number.

Wrong! That is not how limits are defined.

A series is defined in mathematics as a sequence of the partial sums of
another sequence. Given sequence f:N -> R, we construct series
g:N -> R by defining g(n) = SUM_{1<=k<=n} f(n).

Thus every series is a special type of sequence, and no series can exist
which are not sequences of partial sums, at least in mathematics. That
in Meucken's mind there may be some other meaning is mathematically
irrelevant.

Then the "sum", if it exists in any sense, of a series is defined as the
limit of the sequence of partial sums. The "sum" or "value" or whatever,
of a series has no other mathemetical meaning. That in Meucken's mind
there may be some other meaning is mathematically irrelevant.

Meucken, you have no power to impose your vision of what mathematics
should be on mathematics. Mathematics has progressed nicely without your
help for millennia, and will continue to do so without you nicely.
.