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

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

On 21.11.2024 22:46, Jim Burns wrote:

On 11/21/2024 2:21 PM, WM wrote:

On 21.11.2024 19:54, Jim Burns wrote:

On 11/21/2024 11:24 AM, WM wrote:

By what is it covered,
after all n have been proved unable?

>
⎛ n ↦ i/j ↦ n
⎜
⎜ (i+j) := ⌈(2⋅n+¼)¹ᐟ²+½⌉
⎜ i := n-((i+j)-1)⋅((i+j)-2)/2
⎜ j := (i+j)-i
⎜
⎝ (i+j-1)⋅(i+j-2)/2+i = n

>
That is not an answer.

Further it is only valid for
the first numbers which are followed by
almost all numbers.

Never completed.

 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.
Therefore, no,
the description is an answer.
----
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.
More narrowly,
for what.we.are.considering,
no claim in that sequence of
true.or.not.first.false claims
is first.false,
so,
by the finiteness of the sequence,
no claim in that sequence of
true.or.not.first.false claims
is false.
More narrowly,
for what.we.are.considering,
the following not.first.false claims
ARE an answer.