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

On 11/20/2024 11:56 AM, Ross Finlayson wrote:

On 11/20/2024 11:19 AM, Jim Burns wrote:

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On 11/19/2024 4:38 PM, Ross Finlayson wrote:

On 11/19/2024 11:56 AM, Jim Burns wrote:
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⎛ Necessary and sufficient conditions for finiteness
⎜
⎜ 3. (Paul Stäckel)
⎜ S can be given a total ordering which is
⎜ well-ordered both forwards and backwards.
⎜ That is, every non-empty subset of S has both
⎝ a least and a greatest element in the subset.
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https://en.wikipedia.org/wiki/Finite_set

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Yeah we looked at that before also, and
I wrote another, different, definition of finite.

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On 11/19/2024 10:59 PM, Ross Finlayson wrote:

On 11/19/2024 07:45 PM, Ross Finlayson wrote:

On 11/19/2024 02:38 PM, FromTheRafters wrote:

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[...]

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In
"Replacement of Cardinality (infinite middle)", 8/19 2024,
this was:
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I mean it's a great definition that finite has that
there exists a normal ordering that's a well-ordering
and that all the orderings of the set are well-orderings.
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That's a great definition of finite and now it stands
for itself in enduring mathematical definition in defense.
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Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?

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No, I think that they agree,
except possibly.not where countable.choice is possibly.wrong.
because...
⎛ Necessary and sufficient conditions for finiteness
⎜
⎜ If the axiom of choice is also assumed
⎜ (the axiom of countable choice is sufficient),
⎜ then the following conditions are all equivalent:
⎜ 1. S is a finite set.
⎜ 2. (Richard Dedekind)
⎜ Every one-to-one function from S into itself is onto.
⎜ A set with this property is called Dedekind-finite.
⎝
https://en.wikipedia.org/wiki/Finite_set
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I am satisfied that using the other definition
which you mentioned isn't bait.and.switch.ing.
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----
Dedekind.finite with countable.choice is
equivalent to Stäckel.finite.
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⎛ Countable.choice:
⎜ ∃S: ℕ→Collection: ∀k∈ℕ:S(k)≠{}  ⇒
⎝ ∃ch: ℕ→⋃Collection: ∀k∈ℕ:ch(k)∈S(k)
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Are there non.well.ordered finite sequences, Ross?
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If 'finite' is 'Dedekind.finite'
and countable.choice is valid,
then
no finite sequence is non.well.ordered.
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⎛⎛ Assume that
⎜⎜ P is a Dedekind.finite sequence of claims.
⎜⎜ Countable.choice is not possibly wrong here.
⎜⎜ Each claim in P is
⎜⎜  either true
⎜⎜  or after a false claim in P
⎜⎝ (not.first.false)
⎜
⎜ The subset F of false claims in P is
⎜  either empty
⎜  or holds a first.false claim.
⎜ (well.ordered P)
⎜
⎜ Each claim in P is true.or.not.first.false.
⎜ (by assumption)
⎜
⎜ F cannot hold a first.false claim.
⎜
⎜ The subset F of false claims in P is
⎜ empty.
⎜
⎝ Each claim in P is true.
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Therefore,
if
P is a finite sequence of claims, each claim of which
  is true.or.not.first.false,
then
P is a finite sequence of claims, each claim of which
  is true.
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That conclusion is the telescope which
finite beings use to observe the infinite,
because
each claim in finite.length.P is true whether.or.not
it is a claim referring to one of infinitely.many.
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Are you, though?
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With regards to choice and countable choice,
the weaker form that goes without saying anyways,
the existence of a choice function being a bijection
any given set, and an ordinal's elements lesser ordinals,
making a well-ordering of the set, has that,
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well-foundedness
and
well-ordering
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sort of result dis-agreement.
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One's about ordinals, the other about cardinals,
two distinct kinds of things,
neither of which are sets.
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About "define finite" then that gets into also
what's called the extensions and collapses or
as with regards to Skolem. Skolem is pretty great
because what he gives is that any countable thing
has an un-countable model and any un-countable thing
has a countable model.
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So, this is sort of in accords with there being
no standard model, then also demands how to define,
"finite", from a countable model to an un-countable
model, for example as my definition gives and Stackel's
does not.
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Either way, some of the elements in Skolem's extension,
of an unbounded domain of finites, are infinitely-grand,
because, that's so. Then "hyper-finite" in a sense
here as "as of a definition of finite, vis-a-vis
a fixed point and limit ordinal and cardinal not-finite",
is one of those things.
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Of course, I have a whole course of things ongoing here
since quite a while, including specifically the slates
of "uncountability" and "paradox", showing that there
_is_ a counterexample to the anti-diagonal for the
equivalency function and ubiquitous ordinals,
and, no logical paradoxes at all, mostly considering
the extra-ordinariness of the ubiquitous ordinals.
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Then of course there was written at least three
continuous domains, and those couched in at least
two more, only one of which is "standard" field-reals.
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As a strong mathematical platonist who also claims
that reason arrives at all the requirements of a
stronger logicist positivism, then also there's
that nominalism is not allowed to ignore these
things, except insofar as it's an incomplete fragment,
of the theory.
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Which in terms of "completions", properties concomitant
their completions, of course sees also direct contradictions.
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So, "concomitant" is a great word, reflecting the ontological
commitment that our convictions in principles both assure
and provide, here with regards to completions like
"infinite limits" or "gaplessness", show that a wider,
fuller dialectic account dis-qualifies and de-certifies
lesser, incomplete, partial, stipulated, inductive accounts.
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Or, such is an account of a greater mathematics of our time.
And, I don't know any better, or, more true.
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Of course for these "extra-ordinary ordinals" are things
like "super-cardinals" like "the cardinal of '1', the cardinal".
As a structuralist there's a universe of sets that
have only expansion of comprehension,
and none inconsistent-multiplicities,
only first-class intensionality,
and these kinds of things.
Of course these mostly have about zero relevance
to most practical matters of mathematical application,
except insofar as the formalization for continuum
mechanics and the law(s) of large numbers, which
correspondingly are involved in the being well-defined
and having mathematical rigor, the most awe-inspiring
super-classical mathematics of today, and, tomorrow.
That is to say, one need not any set theory at all
to learn adequate calculus for most all matters
applied, or what's called "standard" real analysis,
and any matters that kind of arrive at set theory
for standard real analysis, are not, "standard".
That, and the "Factorial/Exponential Identity, Infinity"
that I wrote is a thing - and includes a formula for
factorial the defies, just like Stirling's does,
some naive convergence criteria. Which thus are wrong, ....
So, not only is 'material implication' not ignored here,
it's excluded and put in a tiny box of a fragment of
a finite element analysis of a snapshot of stipulated
facts at a snapshot of time to result a quick check-sum.
And that's all it is, ....