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

Op 02.apr.2024 om 00:21 schreef olcott:

On 4/1/2024 3:59 PM, André G. Isaak wrote:

On 2024-04-01 14:30, olcott wrote:

On 4/1/2024 2:37 PM, Fred. Zwarts wrote:

Op 01.apr.2024 om 20:54 schreef olcott:

>

Since PI is represented by a single geometric point on the number line
then 0.999... would be correctly represented by the geometric point
immediately to the left of 1.0 on the number line or the RHS of this
interval [0,0, 1.0).

>
In the real number system it is incorrect to talk about a number immediately next to another number. So, this is not about real numbers.
>

>
PI is a real number.
If there is no real number that represents 0.999...
that does not provide a reason to say 0.999... = 1.0.

>
I'm a bit unclear why you keep bringing pi into this. pi isn't a repeating decimal, unlike 0.999... which is.
>
But if you want to talk about pi, that also can be construed as the limit of an infinite series:
>
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, not to some

When olcott says:

To says that 0.999... = 1.0 means that after the never ending
sequence ends (a contradiction) then we reach exactly 1.0.
 

he is again changing the meaning of 0.999... = 1.0 from what is defined for real numbers. It is clear that his interpretation then brings him to a contradiction. This is more evidence that this interpretation is incorrect for the real number system.