Re: Does the number of nines increase?

On 7/1/2024 7:42 PM, Ross Finlayson wrote:

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.

The intermediate value theorem
is equivalent to
the least upper bound property.
Either one implies the other.

So, iota-values or
ran(EF) of the natural/unit equivalency function,
or sweep, has "extent density completeness measure",
thus the IVT follows.

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.

It's sort of irrelevant what I intend

I disagree about the irrelevance of
what you (RF) mean by
your (RF's) notation.
However,
let's assume for argument's sake
that what you mean is irrelevant.
What does it cost you to tell me?

as I don't see value in nominalist fictionalism,
what it is is what it is, what it is.

In some possible worlds,
what n/d: 0≤n≤d is is what {0/d,1/d,...,d/d} is
In other possible worlds, it isn't.
In some possible worlds,
what lim[d → ∞] is is what ⋂[0<dᵢ<∞] ⋃[dᵢ<d<∞] is
In other possible worlds, it isn't.
Knowing which possible world the actual world is
is relevant to discussing n/d: 0≤n≤d: d → ∞

It's not really any of the initial approximations,
this limit, this infinite limit, this continuum limit.
>
It's an _infinite_ limit.

Is it
⋂[0<dᵢ<∞] ⋃[dᵢ<d<∞] {0/d,1/d,...,d/d} ?