Re: Does the number of nines increase?

On 06/30/2024 06:38 PM, Jim Burns wrote:

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.
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There are others, ....

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...but not in that discussion.
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Yes,
we CAN discuss things other than the real numbers.
>
However,
if we ARE discussing the real numbers,
then we AREN'T doing that.
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Define the Dedekind.complete real numbers.
Prove the Intermediate Value Theorem.
>
Counter.propose(?) the rational numbers,
for which the Intermediate Value Theorem is false.
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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.
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I understand your iota.values to be the limit
n/d: 0≤n≤d: d → ∞
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For n/d: 0≤n≤d  I read {0/d,1/d,...,d/d}
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For lim[d → ∞] I read ⋂[0<dᵢ<∞] ⋃[dᵢ<d<∞]
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Is that what you mean? You (RF) don't say.
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⋂[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_
>
>

Yes it does, the iota-values result that they do
make for the IVT, and, aren't contradicted by
un-countability, and are the usual notion since
Aristotle, of the at least both kinds of continuity,
by establishing "extent density completeness measure",
the properties each you've seen here.
You can find them in the line elements of the line integral,
or path integral, and about the Jordan measure that's not
the Lebesgue measure, which on the Wiki last year, was
renamed something else in a great shaming of it from presuming
to carry the name of "measure" when it contradicted resultingly
another usual thing many call "measure", yet is currently in
a great state of being organized together, on the "measure theory"
page, because, it got around to cooler heads that it was
making mathematicians look bad by playing giving things
wrong definitions.
Katz' infinitesimals are rather conservative yet harken
to Bishop and Cheng, who make a model of infinitesimals
as about a partially ordered ring, with a, as it's said,
a "rather restricted transfer principle", between the
models of Bishop and Cheng's constructible infinitesimals,
and Robinson's which are so conservative they say nothing.
Of course mine are direct and called "iota-values",
and fit together in the theory and make another result
rather similar exactly in form to the anti-diagonal
argument, this only-diagonal argument, which if you
recall has simply the same criteria for acceptance
or rejection as does the anti-diagonal argument.
This was, "ha, I put them together, you get both or none".