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

On 07/02/2024 06:06 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

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Do you refer to  n/d: 0≤n≤d: d → ∞  ?

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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

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Isn't that, ..., contiguum?

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

https://en.wikipedia.org/wiki/Wu%E2%80%93Yang_monopole
Reading more from Hermann in that podcast, gets into
that mathematicians and physicists sort of need to
get together, and, mathematics _owes_ physics.
(More and better mathematics of continuity and infinity.)
With the least-upper-bound property for reals in their
_normal_ ordering and reals in their _reverse_ ordering,
doesn't that sort of confound just the usual partitioning
scheme?
That is to say, isn't any real number defined both ways?
Aren't they, neighbors? No different?