Re: The non-existence of "dark numbers"

On 3/23/2025 1:23 PM, WM wrote:

On 22.03.2025 20:33, Jim Burns wrote:

On 3/22/2025 5:19 AM, WM wrote:

Of course.
The reason is that
all pairs of
the bijection proving same cardinality
have
infinitely many dark successors
which cannot be bijected.

>
A bijection which does not biject everything
is not a bijection,

>
So it is!
But it is impossible to prove everything
in Cantor's bijections.

Cantor's bijections are between sets which
are their.own.only.inductive.subsets
  (for certain values of 'inductive').
To prove everything,
it is enough to prove inductivity
(a first element and a closed successor)
because,
for such sets,
there is no other inductive subset.

Almost all elements follow upon every defined pair.

Yes,
that is a property of sets which
are their.own.only.inductive.subsets.
The reason that we know about these pairs
is not
that we come to their end, not in any sense.
The reason that we know about these pairs
is
that we have an indefinite description,
true of each one of infinitely.many pairs,
and
that we can follow with finitely.many claims
which we can _see_ are not.first.false,
with or without understanding what they _say_
⎛ I don't know what Q in ⟨ P P⇒Q Q ⟩ _says_
⎝ I can _see_ Q is not.first.false there.
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,
and,
if it doesn't have a first.false claim
(something we _see_ in some sequences),
then it doesn't have a false claim.
We are able to reason about infinitely.many
without performing infinite tasks.
Our ability is the product of long centuries
of intense effort by our best people.
If someone wakes up one morning and announces
"I'm going to repeat that work of centuries today"
and they _fail_ at their project,
they should not conclude, from that,
that all that previous work was in error.