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

On 12/12/2024 4:07 AM, WM wrote:

On 11.12.2024 23:28, Jim Burns wrote:

On 12/11/2024 4:53 PM, WM wrote:

Is the complete removal of natural numbers
from the sequence of intersections
bound by the law
∀k ∈ ℕ :
∩{E(1), E(2), ..., E(k+1)} =
∩{E(1), E(2), ..., E(k)} \ {k}
or not?

Note that law 'E(k)\{k} = E(k+1)' is used
to show that  ⋂{E(i):i} = {}
∀k ∈ ℕ:
k ∉ E(k)\{k} = E(k+1) ⊇ ⋂{E(i):i} ∌ k
∀k ∈ ℕ:
⎛ k ∉ E(k)\{k}
⎜
⎜ E(k)\{k} = E(k+1)  [law]
⎜
⎜ E(k+1) ⊇ ⋂{E(i):i}
⎝ ⋂{E(i):i} ∌ k

Each "leaves" by
not.being.in.common.with.all.end.segments.

>
That means no natural number remains
in all endsegments.

Yes,
all end.segments of all natural numbers ==
all cardinals which can change by 1
None remain in all is what  ⋂{E(i):i} = {}  means.

But every endsegment has only
one number less than its predecessor.

Each end.segment is one.element.emptier
than its predecessor.
Each end.segment holds,
for each k which.can.change.by.1
more.than.k,
and so holds other.than.k numbers
Each end.segment is of a size which is
none of the sizes which.can.change.by.1

This closes the bridge between
infinite intersections of endsegments and
the empty intersection of endsegments.

Each end.segment is emptiest.so.far
but it is too large to have changed in size
from the set of all finite.cardinals.
The existence of a bridge implies that,
somewhere we can't see,
a size which cannot change by 1
changes by 1 to
a size which can change by 1
We know that  that doesn't happen,
not because we can see what we can't see,
but because we can see
true.or.not.first.false.ness
in claims which we can see.
Where all we see are
true.or.not.first.false claims,
we know they're all true claims.
Even claims about things we can't see.
We can see not.first.false.ness of claims.
What we see,
looking at cleverly arranged sequences of claims,
is the absence of bridges we can't see.

∀k ∈ ℕ:
k ∉ E(k)\{k} = E(k+1) ⊇ ⋂{E(i):i} ∌ k
>
⋂{E(i):i} = {}.

>
There must be
a continuous sequence of steps of height 1
from many elements to none.

The key word here is 'many'.
'Many' is not.what we mean by 'infinite'.
We see here two descriptions
many: countable.down.from to.none
infinity: after.all countable.down.from to.none
Each description is potentially a claim about
what the other claims here are about.
They are different claims.
They are in different finite
all true.or.not.first.false claim
sequences.
Their respective subsequent
not.first.false claims in different sequences
are different,
which is not concerning because
the different claims are about different things,
many vs infinity.

There must be
a continuous sequence of steps of height 1
from many elements to none.

What you describe is many, not infinity.

Can you confirm this?

For many, yes,
For infinity, never.
Having a continuous sequence of steps of height 1
from many elements to none
is what makes it finite.