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

On 12/17/2024 4:00 AM, WM wrote:

On 16.12.2024 18:08, Jim Burns wrote:

On 12/15/2024 2:16 PM, WM wrote:

On 15.12.2024 19:53, Jim Burns wrote:

On 12/15/2024 7:00 AM, WM wrote:

(1) E(1), E(1)∩E(2), E(1)∩E(2)∩E(3), ...
loses all content.

>
the set of common finite.ordinals is empty.

>
Fine.

>
Each finite.cardinal
leaves after that.many steps,
with further steps to follow,
more steps than any other finite.cardinal

>
which is following upom it.

This is our foundation: a calculus of
sheep in pastures and pebbles in pockets.
Each finite.cardinal i after 0
is countable.to from.0
∃⟦0,i⦆ smaller than Bobbed ⟦0,i⦆∪{Bob}
Each finite.cardinal j after i
is countable.to from.i
∃⟦i,j⦆ smaller than Bobbed ⟦i,j⦆∪{Bob}
Each finite.cardinal k after j
is countable.to from.j
∃⟦j,k⦆ smaller than Bobbed ⟦j,k⦆∪{Bob}
And so on.
For each smaller.than.Bobbed ⟦0,i⦆, ⟦i,j⦆
∃⟦0,j⦆  =  ⟦0,i⦆∪⟦i,j⦆
#⟦0,j⦆  =  #⟦0,i⦆ + #⟦i,j⦆
⟦0,j⦆ is smaller than Bobbed ⟦0,j⦆∪{Bob}
For each smaller.than.Bobbed ⟦0,i⦆, ⟦i,j⦆, ⟦j,k⦆
∃⟦0,k⦆  =  ⟦0,j⦆∪⟦j,k⦆  =  ⟦0,i⦆∪⟦i,j⦆∪⟦j,k⦆
#⟦0,k⦆  =  #⟦0,j⦆ + #⟦j,k⦆  =  #⟦0,i⦆ + #⟦i,j⦆ + #⟦j,k⦆
⟦0,k⦆ is smaller than Bobbed ⟦0,k⦆∪{Bob}
And so on, and so on.
Our foundation of sheep and pebbles considers
that which is smaller.than.Bobbed: the finite.
We don't need to always restrict our considerations
to sheep and pebbles, to the finite.
The set {⟦0,i⦆:smaller.than.Bobbed}
cannot itself be smaller.than.Bobbed.
⎛ Each ⟦0,j⦆ ∈ {⟦0,i⦆:smaller.than.Bobbed}
⎜ is smaller than {⟦0,i⦆:smaller.than.Bobbed}
⎜⎛ because
⎝⎝ #⟦0,j⦆ < #⟦0,j+1⦆ ≤ #{⟦0,i⦆:smaller.than.Bobbed}
ℕ = ⋃{⟦0,i⦆:smaller.than.Bobbed} = {0,1,2,3...}
{⟦0,i⦆:smaller.than.Bobbed} is not.smaller.than.Bobbed
#ℕ = #{⟦0,i⦆:smaller.than.Bobbed}
ℕ is not.smaller.than.Bobbed
However,
∀j ∈ ℕ:
⟦0,j⦆ ∈ {⟦0,i⦆:smaller.than.Bobbed}
and ⟦0,j⦆ is smaller.than.Bobbed.
If I understand you (WM), you have been arguing that,
because ∀j ∈ ℕ: ⟦0,j⦆ is smaller.than.Bobbed,
ℕ must be smaller.than.Bobbed, and thus
there must be darkᵂᴹ numbers to account for the discrepancy.

Explain your vision of the problem:

(1)
When we consider all finite sequences ⟦0,j⦆,
meaning sequences smaller.than.Bobbed,
what we should consider are
the finite initial segments of ⋃{⟦0,i⦆:smaller.than.Bobbed}
That is, ℕ should be ⋃{⟦0,i⦆:smaller.than.Bobbed}
Inserting an epilogue makes our considerations about
things different from sequences smaller.than.Bobbed.
(1')
Our considerations indicate that,
unlike the finite initial segments of ℕ
ℕ is not.smaller.than.Bobbed.
That isn't a problem to be solved.
It is a possibly.interesting fact we have uncovered.
(2)
Inserting an epilogue does not perform as advertised.
ℕ is not.smaller.than.Bobbed.
For any epilogue 𝔻,  ∀d ∈ 𝔻: g(d) = d
ℕ∪𝔻 is not.smaller.than.Bobbed.
If we followed your directions,
we'd give up considering the smaller.than.Bobbed sequences
for no benefit,
not even a benefit we didn't much want,
such as the elimination of the not.smaller.than.Bobbed.