Re: Definition of real number ℝ --infinitesimal--

wij <wyniijj5@gmail.com> writes:

On Tue, 2024-04-02 at 05:46 +0800, wij wrote:

On Mon, 2024-04-01 at 15:03 -0600, André G. Isaak wrote:

On 2024-04-01 14:59, André G. Isaak wrote:
 
 

π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
 
For any *finite* number of terms, the above series never quite reaches
pi, but the LIMIT of this series is exactly equal to pi.

Obviously, I mean the limit is exactly equal to π/4.
 

 
Note that the limit of sequence about is π/4. But none of any number
in the sequence is π/4
 

>
To be more correctly:
>
  π/4 = lim 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...

No, that's no more correct because the limit is already implied and,
more importantly, the /correct/ limit is implied.  The limit is that of
a sequence of partial sums, so to write it without the implied limit one
should write either

  π/4 = Sigma_{n=0}^oo -1^n * 1/(2n+1)

or

  π/4 = lim_{k->oo} Sigma_{n=0}^k -1^n * 1/(2n+1).

Just sticking "lim" in front gains you nothing.

or π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... +non_zero_remainder

No, because the limit is implied.  That's what the informal "..." means.
The only correct equation using the rather informal ... is

  π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11... + 0.

You can say what you probably mean using the proper notation as above,
but I doubt you want to do that.

or π/4 ≒ 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...

No.  The ... implies the limit of the partial sums and, as you know,
that limit is π/4.  Exactly π/4.

André already said the no finite partial sum is equal to π/4.  What do
you gain by trying to say that again?

Have you got any further in defining the operations on your "numbers as
strings or TMs" so that (a+b)/2 is neither a nor b when a=1 and
b=0.999...?  That's a fun project, but I don't think you are able to do
it.

--
Ben.