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

On 1/17/2025 4:08 AM, WM wrote:

On 16.01.2025 23:22, Jim Burns wrote:

Nowhere,
among what appears and
among what doesn't appear,
is there finite ω-1 and infinite (ω-1)+1

>
So it appears because ω and ω-1 are dark.

We never see ω and ω-1
We see descriptions of ω and ω-1
That is sufficient for knowledge of ω and ω-1
⎛ Unless it's false that
⎜ what is intended by 'ω' is 'the first infinite ordinal',
⎜ "what is intended by 'ω' is 'the first infinite ordinal'"
⎜ is a true claim, and
⎜ we don't need to see ω in order to know that.
⎜
⎜ For any claims P and Q
⎜ Q is not.first.false in ⟨ P P⇒Q Q ⟩  and
⎜ we don't need to see ω in order to know that.
⎜
⎜ For any finite sequence of claims,
⎜ if any claim is false,
⎜ then some claim is first.false, and
⎜ we don't need to see ω in order to know that.
⎜
⎜ For any finite sequence of claims,
⎜ if each claim is true or not.first.false,
⎜ then no claim is first.false, and
⎜ we don't need to see ω in order to know that.
⎜
⎜ For any finite sequence of claims,
⎜ if each claim is true or not.first.false,
⎜ then each claim is true, and
⎝ we don't need to see ω in order to know that.

So it appears because ω and ω-1 are dark.

We never see ω and ω-1
We see descriptions of ω and ω-1
That is sufficient for knowledge of ω and ω-1
I am able to show to you
a finite sequence of claims in two parts.
One part is claims about intentions.
What ω is intended to mean.
What ω-1 is intended to mean.
What ordinal is intended to mean.
What finite is intended to mean.
What those are among which λ is intended to refer.
These claims can be known by
people who know what is intended.
There might be people who don't know.
You (WM), for example.
But, for people who know, it is knowledge,
if not awe.inspiring knowledge.
Seeing ω plays no part in that knowledge.
Another part is claims which are not.first.false,
and which can be seen to be not.first.false
by examining them on a printed page or
scratched in the sand.
Patterns like ⟨ P P⇒Q Q ⟩ can be perceived
and known to be not.first.false
as certainly as 7×11×13 = 1001
Seeing ω plays no part in that knowledge.

But if ω is assumed to exist,
then there is a set cotaining ω elements.

...then there is a set of all finite ordinals.
ω = ω is true, but not helpful.

From this set one element can be subtracted.

For ⟦0,ω⦆,  there is
emptier.by.one ⟦1,ω⦆

Also, more generally,
there is no infiniteˢᵉᵗ smaller than ℕ

>
Therefore ℕ \ {1} is finite.

No.
If ℕ\{1} was finite, then
a larger finite ordinal k would exist.
However,
all finite ordinals are in ℕ
Ordinals 0 to k and k+1 and k+2 are in ℕ
⟦0,k+1⦆ is alleged to be
a subset of ℕ which is larger than ℕ
You (WM) introduce
negative cardinality (darkᵂᴹ numbers)
in an attempt to fit these claims together.
Matheologians don't use anything so fancy,
only sets larger.than.any.finite.

But it appears infinite like all sets which cannot be counted by FISONs.

"cannot be counted by FISONs"
and
"larger.than.any.finite"
sound similar.
However, matheologians don't have darkᵂᴹ sets
allegedly capable of backing a larger.than.any.finite set
down to a not.larger.than.any.finite set,
by _inserting_ elements.
That is why you (WM) think we're crazy or stupid,
negative cardinality.