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

    From Jim Burns@21:1/5 to Ross Finlayson on Tue Jul 2 20:06:34 2024
    On 7/2/2024 4:32 PM, Ross Finlayson wrote:
    On 07/02/2024 05:07 AM, Jim Burns wrote:

    [...]

    Anyways,
    this putative countable domain

    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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  • From Jim Burns@21:1/5 to Ross Finlayson on Tue Jul 2 21:06:55 2024
    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 → ∞ ?

    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]

    [1]
    https://en.wikipedia.org/wiki/Continuum_limit

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

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  • From Jim Burns@21:1/5 to Ross Finlayson on Wed Jul 3 00:02:00 2024
    On 7/2/2024 9:49 PM, Ross Finlayson wrote:
    On 07/02/2024 06:06 PM, Jim Burns wrote:

    [...]

    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?

    For each nonempty bounded.below set S of reals
    there is nonempty bounded.above -1⋅ᴬS
    with a least.upper.bound -1⋅σ
    σ is the greatest.lower.bound of
    nonempty bounded.below S

    re: lub glb
    You pays yer money and you takes yer choice.

    That is to say,
    isn't any real number defined both ways?
    Aren't they, neighbors? No different?

    No different. No problem.

    Reading more from Hermann in that podcast, gets into
    that mathematicians and physicists sort of need to
    get together, and, mathematics _owes_ physics.

    Even more obviously,
    physics _owes_ mathematics, too.

    There's enough owing to go around.
    See also: shoulders of giants

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  • From Jim Burns@21:1/5 to Jim Burns on Wed Jul 3 12:12:16 2024
    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)

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