Re: Does the number of nines increase?

On 06/30/2024 09:22 PM, Jim Burns wrote:

On 6/30/2024 11:32 PM, Ross Finlayson wrote:

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

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

>

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.

>

Yes it does, the iota-values result that they do
make for the IVT,

>
Tell me what you are talking about.
>
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.
>
>

There must be satisfied "extent density completeness
measure" to satisfy the IVT, though one may aver that
"extent density completeness" would suffice.
So, iota-values or ran(EF) of the natural/unit equivalency function,
or sweep, has "extent density completeness measure",
thus the IVT follows.
It's sort of irrelevant what I intend as I don't see
value in nominalist fictionalism, what it is is what it is,
what it is.
It's not really any of the initial approximations,
this limit, this infinite limit, this continuum limit.
It's an _infinite_ limit.
In some places the only known non-standard function
with real analytical character is the Dirac delta,
the unit impulse function. Anywhere besides at zero:
its value is zero. It's not so that it's all the
things that go to it, it's where it goes.
Unless you'd care to see that most all the modern structure
of analysis in infinite series and function theory is, "wrong",
it's only so in the _infinite_ case, and it is so.