Re: Does the number of nines increase?

On 6/30/2024 10:48 AM, Ross Finlayson wrote:

On 06/30/2024 02:55 AM, FromTheRafters wrote:

This *complete* ordered field of reals guarantees
cauchy sequence convergence.
See how real numbers are defined.

>
It's axiomatic,
and about the usual open topology.
>
There are others, ....

...but not in that discussion.
Yes,
we CAN discuss things other than the real numbers.
However,
if we ARE discussing the real numbers,
then we AREN'T doing that.
Define the Dedekind.complete real numbers.
Prove the Intermediate Value Theorem.
Counter.propose(?) the rational numbers,
for which the Intermediate Value Theorem is false.
So what?
The rational numbers aren't the real numbers.
We can still apply the Intermediate Value Theorem
to the real numbers,
which is all anyone has claimed.