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

On 07/02/2024 05:07 AM, Jim Burns wrote:

On 7/1/2024 11:19 PM, Ross Finlayson wrote:

On 07/01/2024 06:10 PM, Jim Burns wrote:

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What that means is that
I think
theory is a strong mathematical platonism,
it matters what is _attained_ to,
or that to which we _attain_,
the "true" objects of
a universe of mathematical objects,
"a" universe, then with regards to descriptions

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"descriptions"
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there's that
the "extent density completeness measure" provide
"extent density completeness"
which you would agree that
"extent density completeness" makes for
satisfying the IVT.
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Then, as I mentioned,
there's a theory,
in all the universe of theories,
all the abstract and contingent and fanciful and
practical and otherwise,
one of which is "the true theory",
that among those,
there's one where it appears that
"it is so" is an axiom.
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So, given that
you won't accept that via inspection,

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"inspection"
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that a least-upper-bound is given and
also a sigma algebra is given,
given that extent and density are givens,
then,
given that it's axiomatic,

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"axiomatic"
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and, doesn't contradict the ordinary
because
it just makes for the "only-diagonal" contra
the "anti-diagonal",
then, how's that.
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Good sir, ....

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Do I understand you correctly?
You have declined my invitation to say
what your symbol.string  n/d: 0≤n≤d: d → ∞  means
because
you consider what you've told me to have answered me:
n/d: 0≤n≤d: d → ∞  satisfies IVT
n/d: 0≤n≤d: d → ∞  is countable
etc.
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You have defined it so.
Do you realize that?
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Definitions are two.edged swords.
They grant you unrestricted power,
but only inside the area of what.you.mean
and outside of that, no power.
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If what.you.mean by  n/d: 0≤n≤d: d → ∞  is that
n/d: 0≤n≤d: d → ∞  satisfies IVT
n/d: 0≤n≤d: d → ∞  is countable
etc.
then, okay, you can define it so, but
defining it so doesn't mean it exists.
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Those proofs which
  you think  n/d: 0≤n≤d: d → ∞  disproves
prove that  n/d: 0≤n≤d: d → ∞  doesn't exist.
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>

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These things are demonstrated, then there's at least
one putative theory in the theory of all theories
where "axiom of iota-value truth" or "IVT axiom",
is so.
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I don't claim any disproofs at all, except insofar as
the integers are non-standard, rather, it's another
proof in usual set theory's usual descriptive milieu
that there are "non-Cartesian" functions at least
this "only-diagonal" and it's arrived at about
the same way as there are "non-countable" domains
for what's the "anti-diagonal". Pick one: get both.
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Cantor-Schroeder-Bernstein theorem gets built a little barrier
about the transitive quality of cardinality, and a bit
of book-keeping results that quite standard looking results
combine to abound.
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I'm glad you've arrived at "exists", though,
mathematical objects, defined by relation.
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Definitions are only as good as they're sound.
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And axioms: that they're not.ultimately.untrue.
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You know, some have that least-upper-bound is
only provided to Eudoxus/Dedekind/Cauchy real field
via axiom.
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And, that "non-countable" is a non-constructivist result.
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Anyways, this putative countable domain 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.
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"Real-Valued"
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