Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-standard)

On 11/21/2024 01:10 PM, Ross Finlayson wrote:

On 11/21/2024 12:28 PM, Jim Burns wrote:

On 11/21/2024 2:46 PM, Ross Finlayson wrote:

On 11/21/2024 09:57 AM, Jim Burns wrote:

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[...]

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(the existence of a choice function,
i.e. a bijection between any set and some ordinal)

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A well.ordering of a set is
a bijection between that set and an ordinal.
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A choice function is a function 'choice',
  typically not a bijective function,
from a collection of non.empty sets S
to their elements, such that
for each set S, choice(S) ∈ S
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∀Collection:
∃choice: Collection\{∅} -> ⋃Collection:
  ∀S ∈ Collection\{∅}: choice(S) ∈ S
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Yeah, Well-Ordering and Choice (the existence of
a choice function, i.e. a bijection between any
set and some ordinal) are same.

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Well.Ordering and Choice are inter.provable.
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Countable-choice is weak and trivial.

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Because we prefer our assumptions weak and trivial,
that's a good thing.
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Countable.choice is sufficient to prove that
Well.Ordering and Choice are inter.provable.
Proving they are inter.provable with
weak and trivial assumptions is a good thing.
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https://www.youtube.com/watch?v=wmuxeHqF-Vw
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Lecture64||week8||Physico-Mathematical Foundations of the Dynamics of
Nonlinear Processes||by Harsh
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Guy mentions frequency-doubling, a mathematical feature
after continuum mechanics, which cannot be a thing for
those who take the easy way out that shoves itself off.
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Here it's related to doubling- and halving-spaces, and
measures, real things or rather about real analytical character,
and about models of continuous domains and Vitali and
Hausdorff, great geometers.
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So anyways one thing about that is line-reals and their
doubling-space with regards to taking their integral
and that it doubles itself up, the iota-values that
ran(EF), integrating EF, integrates and equals one.
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Then these are well-ordered, these real-valued members
of a continuous domain.
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Yet, you'll never find one anywhere else with regards
to the complete-ordered-field, because there can't be
an uncountable subset that relates to an uncountable ordinal
where, any subset of a well-ordering the tuples (set, ordinal)
is also a well-ordering a set the tuples (set, ordinal),
there can't be that with uncountably many in their normal
ordering, because, quite directly each pair of those as
read off from the choice function, which is merely the
first element existing according to the mapping of a set
to an ordinal, each pair would have a distinct rational
between them.
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So, "well-order the reals" arrives at "or, you know,
aver that it exists yet don't actually give one, ...",
because it would be contradictory either way.
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Anyways that's come up many times, that "well-order the
reals" never quite works out for retro-thesis hacks
of the quite fully the ordinals and cardinals as sets sort,
then though for example it's built up for line-reals
how a resulting, "set", of them, may be so.
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So anyways, you don't need any infinity at all for
such usual matters of induction you describe as so
simple, you're welcome to keep it that way, yet then
that's a sort of "finite combinatorics" not mathematics,
per se.
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In Cantor space there are duelling arguments where
according to Borel almost all and according to Combinatorics
almost none, of the members, are a given way, and
then also a third alternative where it's exactly one-half.
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These are a bit independent, say, either ZF minus Infinity
or ZF with Infinity and may even have that there's always
according to Skolem an extension, and according to Mirimanoff
an extra-ordinary, that Russell's retro-thesis an "ordinary",
well-founded infinity is rejected as not-a-thing, instead
that there are either unbounded fragments or extra-ordinary
extensions, in as regards to three definitions of continuous
domains and three definitions (or, perspectives) of Cantor space.
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Claiming to "make things simple" like "initial ordinal assignment,
a cardinal" or "Dedekind cut, a real", is actually sort of having
conflated separate notions that do not fulfill each other.
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Yeah, it's trivial that the existence of a choice function
and that a subset of the ordered-pairs the tuples a well-ordering
is also a well-ordering, establish each other.
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So, well-order the reals.
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https://www.youtube.com/watch?v=IldqDZklJCg
"Lowenheim Skolem Thereom is Explained By Referential Externalism"
Here a prototypical continuous domain arrives at a theory
with regards to the heno-theory, what's primary in the
theory, when continua are primary in the theory.