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

On 22.11.2024 19:51, Jim Burns wrote:

On 11/21/2024 4:57 PM, WM wrote:

The _description_ is completed.
It's right there.

>
The description of the set
not of all its elements.

 The description is sufficient in order to
finitely.investigate infinitely.many.

For instance, not all indices of the endsegments can be counted to.
Remember: The intersection of all endsegments is empty, but the intersection of endsegments which can be counted to is infinite.
Note that every endsegment loses only one number. Therefore there must exist infinitely many finite endsegments.

----
We have spent a lot of pixels discussing FISONs,
finite initial segments of naturals.
 However,
here I consider FISOCs,
finite initial segments of claims.
 FISOCs share a useful property with FISONs,
  they are well.ordered.
If any claim has a property,
  then some claim has that property first.
If any claim is written in Comic Sans,
  then some claim is in Comic Sans first.
If any claim is false,
  then some claim is false first.
 Consider a specific FISOC with
a description of what.we.are.considering, broadly.
⎛⎛ ℕ⁺ holds numbers countable.to from.1
⎜⎜ ℚ⁺ holds ratios of numbers in ℕ⁺
⎜⎝ ℝ⁺ holds points between splits of ℚ⁺
⎜ Further claims about elements of ℕ⁺ ℚ⁺ ℝ⁺
⎝ which are each true.or.not.first.false
 Broadly speaking,
claims can be true and can be false.
 Broadly speaking,
the initial ℕ⁺.ℚ⁺.ℝ⁺ claims can be false
about some three sets or other.
 In those broader after.false instances,
the following not.first.false claims
are not.first.false
whether they are true or they are false.
 In the broader after.false instances,
the following not.first.false claims
are NOT an answer.
 However,
more narrowly,
for what.we.are.considering,
the initial ℕ⁺.ℚ⁺.ℝ⁺ claims are true.

They are true for a potential infinity. Consider the claim: Every endsegment is infinite. This claim is true for the potentially infinite sequence of infinite endsegments. It is not true for finite endsegments. But without finite endsegments there is no empty intersection of endsegments possible.
Regards, WM