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

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

On 11/19/2024 11:56 AM, Jim Burns wrote:
>

⎛ 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.
>
https://en.wikipedia.org/wiki/Finite_set

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

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:

[...]

>
In
"Replacement of Cardinality (infinite middle)", 8/19 2024,
this was:
>

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.
>
That's a great definition of finite and now it stands
for itself in enduring mathematical definition in defense.
>
Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?

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
I am satisfied that using the other definition
which you mentioned isn't bait.and.switch.ing.
----
Dedekind.finite with countable.choice is
equivalent to Stäckel.finite.
⎛ Countable.choice:
⎜ ∃S: ℕ→Collection: ∀k∈ℕ:S(k)≠{}  ⇒
⎝ ∃ch: ℕ→⋃Collection: ∀k∈ℕ:ch(k)∈S(k)
Are there non.well.ordered finite sequences, Ross?
If 'finite' is 'Dedekind.finite'
and countable.choice is valid,
then
no finite sequence is non.well.ordered.
⎛⎛ 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.
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.
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.