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

On 1/10/2025 10:32 AM, WM wrote:

On 10.01.2025 13:41, Richard Damon wrote:

On 1/9/25 11:48 AM, WM wrote:

It is true that
{1, 2, 3, ...} is a set and
{0, 1, 2, 3, ...} is a greater set.

>

No,
one may be the proper subset of the other,
but it turns out that
due to the way that infinity works,
they are both are the same size.

>
This has nothing to do with
"how infinity works".

Yes.
That has nothing to do with
how YOUR (WM's) infiniteᵂᴹ works.
On the other hand,
YOUR infinityᵂᴹ doesn't work.
YOUR infinityᵂᴹ is really.big.but.not.OUR.infinityⁿᵒᵗᐧᵂᴹ.
Where YOUR infinityᵂᴹ doesn't work,
you (WM) cover it up,
permitting impossibilities in your darkᵂᴹ numbers,
obscuring the difference in meaningᵂᴹ you give
to our wordsⁿᵒᵗᐧᵂᴹ,
or just labeling your problemsᵂᴹ "matheology".
Where OUR infinityⁿᵒᵗᐧᵂᴹ "doesn't work",
it's you who's saying it doesn't work,
which boils down to: it's not YOUR infinityᵂᴹ.
OUR infinityⁿᵒᵗᐧᵂᴹ isn't YOUR infinityᵂᴹ,
and that is the only thing you (WM) ever prove.
Is it matheology to want one's infinity to work?
Then we (not.WM) are and will remain matheologists.

It simply is a result of an insufficient method
to measure infinite sets.

Elsethread:
<WM>

Do you (WM) disagree with
'finite' meaning
'smaller.than fuller.by.one sets'?

>
That is also true for infinite sets.
>

</WM>[1]
No,
'smaller.than fuller.by.one sets' is
OUR finiteⁿᵒᵗᐧᵂᴹ::infiniteⁿᵒᵗᐧᵂᴹ distinction,
instead of the moving, darkᵂᴹ boundary you imagine
as YOUR finiteᵂᴹ:infiniteᵂᴹ distinction.
⎛ Other descriptions of our distinctionⁿᵒᵗᐧᵂᴹ
⎝ exist which serve equally well.
ω is the first infiniteⁿᵒᵗᐧᵂᴹ ordinal.
ω is the first ordinal not.smaller.than fuller.by.one.
Define
k < ω  :⇔  #⟦0,k⦆ < #⟦0,k+1⦆
ω ≤ k  :⇔  #⟦0,k+1⦆ ≤ #⟦0,k⦆
⎛ Lemma:
⎝ ¬( #A < #B  ∧  #(B∪{b}) ≤ #(A∪{a})) )
⎛ ¬( #⟦0,k⦆ < #⟦0,k+1⦆  ∧  #(⟦0,k+1⦆∪⦃k+1⦄) ≤ #(⟦0,k⦆∪⦃k⦄) )
⎜
⎜ #A = #⟦0,k⦆
⎜ #B = #⟦0,k+1⦆
⎜ #(B∪{b}) = ⟦0,k+1⦆∪⦃k+1⦄
⎜ #(A∪{a}) = ⟦0,k⦆∪⦃k⦄
⎜
⎜ ¬( #⟦0,k⦆ < #⟦0,k+1⦆  ∧  #⟦0,k+2⦆ ≤ #⟦0,k+1⦆ )
⎜
⎜ ¬( k < ω  ∧  ω ≤ k+1 )
⎜
⎝ ⦃i: i < ω ≤ i+1⦄ = ⦃⦄
Anything before.ω steps.to, sums.to, multiplies.to,
or exponentiates.to something before.ω.
That is not.weird.
After.ω is different from before.ω
Some things are weird about after.ω.
For example, after.ω,
sets are not.smaller.than fuller.by.one sets.
Weird or not,
that's how it is after.ω,
because otherwise it's before.ω
[1]
Date: Thu, 9 Jan 2025 21:23:15 +0100
Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers
  (extra-ordinary)
Message-ID: <vlpb7k$3fug2$2@dont-email.me>