Re: The non-existence of "dark numbers"

On 3/25/2025 12:23 PM, WM wrote:

On 25.03.2025 00:03, Jim Burns wrote:

On 3/24/2025 3:34 PM, WM wrote:

On 24.03.2025 01:46, Jim Burns wrote:

On 3/23/2025 3:41 PM, WM wrote:

On 23.03.2025 19:23, Jim Burns wrote:

The reason that we know about these pairs
is that,
if a finite sequence has a false claim,
then it has a first.false claim,

>
But in case of darkness this cannot be seen.

 Perhaps you (WM) haven't noticed that
that is a finite sequence of claims,
each claim of which can, in principle, be printed.
 Perhaps you (WM) haven't noticed that
that is a finite sequence of claims,
each claim of which can, in principle, be printed.

>
That is a wrong claim!
*In fact* at most 10^80 claims can be printed.

>
The claims I'm referring to, the claims upon which
my claims to knowledge of the infinite stand,
are many orders of magnitude from 10⁸⁰.many.
>
How many claims do you (WM) see here?
[ k is in the emptiest inductive set. ]
>
I see one claim.

>
If n ∈ ℕ_def then n+1 ∈ ℕ_def
is one claim.

Is it a darkᵂᴹ claim?
That seems like a claim I can see.
The reason that we know about these pairs
⎛ of cells in
⎜ the matrix, its.own.only.inductive.subset  and
⎝ the first.column, its.own.only.inductive.subset
is that,
if a finite sequence (of claims) has a false claim,
then it has a first.false claim,
There are infinitely many cells.of.concern.
There are finitely.many claims.of.concern.
Here, I am discussing claims.of concern,
claims which are about cells.of.concern.
If
we can _see the claims_ and we see
that they're finitely.many
(easy but important)
and that each of them is not.first.false,
as Q is in <P P>Q Q>,
(easy to check, usually difficult to accomplish)
then
we _know those claims_
Even if
what.the.claims.are.about is darkᵂᴹ.

In fact
it claims infinitely many steps.
Not all steps are true.

One claim. Infinitely.many steps.
Consider Boolos' theory.fragment ST
⎛ {} exists.
⎜ If X and y exist, then Xu{y} exists
⎝ Equal.membered sets are equal sets.
How many claims?
How many darkᵂᴹ claims?
Each of those claims is true of
the sets in the ST domain.
We know it because
that's what we mean by the ST domain:
the sets for which those claims are true.
It's the same way in which we know that
a square has four corners.
-- NOT by counting the corners of squares.