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

On 01/06/2025 05:36 PM, Ross Finlayson wrote:

On 01/06/2025 02:43 PM, Jim Burns wrote:

On 1/5/2025 1:14 PM, WM wrote:

On 05.01.2025 19:03, joes wrote:

Am Sun, 05 Jan 2025 12:14:47 +0100 schrieb WM:

On 04.01.2025 21:38, Chris M. Thomasson wrote:

>

For me,
there are infinitely many natural numbers, period...
Do you totally disagree?

>
No.
There are actually infinitely many natural numbers.
All can be removed from ℕ, but only collectively
ℕ \ {1, 2, 3, ...} = { }.
It is impossible to remove the numbers individually
∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

>
Well yes,
the size of N is itself
not a natural number.
Big surprise.

>
ℕ cannot be covered by FISONs,
neither by many nor by their union.
If ℕ could be covered by FISONs
then one would be sufficient.

>
ℕ is the set of finite.ordinals.
ℕ holds each finite ordinal.
ℕ holds only finite.ordinals.
>
⎛ A FISON is a set of finite.ordinals
⎝ up to that FISON's maximum (finite.ordinal) element.
>
A finite.ordinal is an ordinal
smaller.than fuller.by.one sets.
>
Lemma 1.
⎛ For sets A∪{a} ≠ A and B∪{b} ≠ B
⎜⎛ if A is smaller.than B
⎜⎝ then A∪{a} is smaller.than B∪{b}
⎝ #A < #B  ⇒  #(A∪{a}) < #(B∪{b})
>
Lemma 1
is true for both the darkᵂᴹ and the visibleᵂᴹ.
>
Consider finite.ordinal k.
Finite: ⟦0,k⦆ is smaller.than ⟦0,k⦆∪⦃k⦄
>
A = ⟦0,k⦆
A∪{a} = ⟦0,k⦆∪⦃k⦄
B = ⟦0,k⦆∪⦃k⦄ = ⟦0,k+1⦆
B∪{b} = (⟦0,k⦆∪⦃k⦄)∪⦃k+1⦄ = ⟦0,k+1⦆∪⦃k+1⦄
>
⎛ By lemma 1
⎜ if ⟦0,k⦆ is smaller.than ⟦0,k+1⦆
⎜ then ⟦0,k⦆∪⦃k⦄ is smaller.than ⟦0,k+1⦆∪⦃k+1⦄
⎜
⎜ If
⎜ k is in ℕ and
⎜ k is finite and
⎜ ⟦0,k⦆ is smaller.than ⟦0,k⦆∪⦃k⦄
⎜ then
⎜ ⟦0,k+1⦆ is smaller.than ⟦0,k+1⦆∪⦃k+1⦄ and
⎜ k+1 is finite and
⎝ k+1 is in ℕ.
>
k ∈ ℕ  ⇒  k+1 ∈ ℕ
is true for both the darkᵂᴹ and the visibleᵂᴹ.
>

If ℕ could be covered by FISONs
then one would be sufficient.

>
ℕ is the set of finite.ordinals.
>
A FISON is a set of finite.ordinals
up to that FISON's maximum (finite.ordinal) element.
>
If one FISON covered ℕ,
that FISON.cover would equal ℕ,
and the maximum of that FISON.cover
would be the maximum.of.all finite.ordinal.
>
However,
no finite.ordinal k is the maximum.of.all.
k ∈ ℕ  ⇒  k+1 ∈ ℕ
That is true for both the darkᵂᴹ and the visibleᵂᴹ.
>
Contradiction.
No one FISON covers ℕ.
>

ℕ cannot be covered by FISONs,
neither by many nor by their union.

>
No.
>
ℕ is the set of finite ordinals.
>
Each finite.ordinal k is in
at least one FISON: ⟦0,k⟧
>
Each finite.ordinal is in
the union of FISONs
>
The union of FISONs covers
the set ℕ of finite.ordinals
>

But for all we have:
Extension by 100 is insufficient.

>
Correct.
Which is weird, but accurate.
>
The source of that weird result is lemma 1.
⎛ For sets A∪{a} ≠ A and B∪{b} ≠ B
⎜⎛ if A is smaller.than B
⎜⎝ then A∪{a} is smaller.than B∪{b}
⎝ #A < #B  ⇒  #(A∪{a}) < #(B∪{b})
>
It would be great if you (WM) did NOT
find lemma 1 weird,
but it is what it is.
>
>

>
But, if I said it was a waste of time,
wouldn't that be a waste of time?
>
>
The inductive set being covered by
initial segments is an _axiom_ of ZF.
>
There are lesser theories where it's not
so, of course, why they added something
like "Infinity" as an _axiom_, vis-a-vis
the illative or univalent or infinite-union
which is _not_ an axiom, and furthermore
not by itself a theorem.
>
So, ..., I suppose that's part of the
idea of the "Reverse Mathematics" program,
which is about theories with less axioms,
about what's so, and what's not so.
>
Then, of course one can show that according
to pair-wise union is the _un-bounded_, then
as with regards to whether comprehension
brings the Russell Paradox on, on the way
from going from _fragments_ to _extensions_,
that is a simple result in, "set theory".
>
... That it's either not infinite or,
you know, not finite.
>
>
>

>
It's pretty simple,
you've invoked Russell as your ruler,
others don't, and, there's a land
where Russell never rules,
so, what you thought was an
at best and at worst incomplete theory,
has there's one where it's not even so.
>
(... That models of integers are fragments
or extensions is the idea, the extra-ordinary
and the super-cardinal.)
>
Don't get me wrong, it's a great way to
simplify ordinary cardinals.
>
And, there is a _function_ between the
integers and even integers, and half of
the integers are even, yet swapping out
the odds, never completes.
>
There's a limit, of course, though, standardly
you don't say that's ever, reached, say.
>
>
>
Boring, I know, ....
>
>