Re: Does the number of nines increase?

On 6/30/2024 5:05 PM, Ross Finlayson wrote:

On 06/30/2024 08:44 AM, Jim Burns wrote:

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.

>
Well, iota-values are defined and
satisfy making for the IVT
which results the FTC's,
Fundamental Theorems of Calculus.

If I use the usual definitions for
the limit of a sequence of sets
for your iota.values,
they do not satisfy the Intermediate Value Theorem.
I understand your iota.values to be the limit
n/d: 0≤n≤d: d → ∞
For n/d: 0≤n≤d  I read {0/d,1/d,...,d/d}
For lim[d → ∞] I read ⋂[0<dᵢ<∞] ⋃[dᵢ<d<∞]
Is that what you mean? You (RF) don't say.
⋂[0<dᵢ<∞] ⋃[dᵢ<d<∞] {0/d,1/d,...,d/d}
does not satisfy the Intermediate Value Theorem.

For the past several years
a Mikhail Katz has been working on
rehabilitating infinitesimals,

I wish Mikhail Katz all the best in his endeavors.
I don't know what those endeavors are, but
I expect Mikhail Katz to know that,
from the Intermediate Value Theorem, one can prove that
the real numbers don't have infinitesimals.
My _guess_ is that Mikhail Katz is
_talking about something else_

and it reminds
me of a story where an educator surveyed an introductory
class whether .999... was the same, or different, than 1.0,
and at least according to their thought processes,
it was about 50/50.

They're the same, whatever the thought processes of
introductory students or Field medalists are,
_unless we're talking about something else_