Re: Does the number of nines increase? (axiomatizing completeness)

On 7/2/2024 9:06 PM, Jim Burns wrote:

On 7/2/2024 8:47 PM, Ross Finlayson wrote:

On 07/02/2024 05:06 PM, Jim Burns wrote:

On 7/2/2024 4:32 PM, Ross Finlayson wrote:

Anyways,
this putative countable domain

>
Do you refer to  n/d: 0≤n≤d: d → ∞  ?

>
Do you refer to  n/d: 0≤n≤d: d → ∞  ?

Do you refer to  n/d: 0≤n≤d: d → ∞  ?

via its construction as
a range of continuum limit of functions
isn't contradicted by the anti-diagonal and so on,
nor by being a Cartesian function,
as a model of a unit line segment of
the linear continuum.

>
continuum limit
greatest.lower.bound of inter.point distances is 0
>
continuum
for each split, either
its foresplit holds a last  or
its hindsplit holds a first

>
Isn't that, ..., contiguum?

>
| In mathematical physics and mathematics,
| the continuum limit or scaling limit of a lattice model
| characterizes its behaviour in the limit
| as the lattice spacing goes to zero.
[1]
>
| [...]
| If [A,B] is a cut of C,
| then either A has a last element or B has a first element.
| [...]
[2]

A short way to say [2]:
[A,B] is _situated_
What we mean by continuum is that
all cuts/splits are situated ==
there's a point it's at we're discussing.
What
the nested all.after interval.sequence
shows is that
countably.many points can be all
one side or the other of a split,
which leaves that split unsituated ==
there isn't a point it's at we're discussing.
...which is why
the continuum limit isn't the continuum,
despite its name.

[1]
https://en.wikipedia.org/wiki/Continuum_limit
 [2]
https://en.wikipedia.org/wiki/Continuum_(set_theory)