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

On 12/9/2024 4:41 PM, WM wrote:

On 12/9/2024 2:45 PM, WM wrote:

On 12/9/2024 4:04 AM, WM wrote:

On 12/8/2024 5:50 AM, WM wrote:

On 08.12.2024 00:38, Jim Burns wrote:

Each end.segment has, for each finite.cardinal,
a subset larger than that cardinal.

>
That is not true for the last dark endsegments.
It changes at the dark finite cardinal ω/2.

∀n ∈ ℕ: E(1)∩E(2)∩...∩E(n) = E(n).
The limit of the left-hand side is empty,
the limit of the right-hand side is full,
i.e. not empty.
I do not tolerate that.

The cardinality of the limit is
the cardinality of the limit set.
This set is defined
by the sequence of cardinalities from
∀k ∈ ℕ : E(k+1) = E(k) \ {k}
and by nothing else.

By the way, we need no cardinality.
We need only the sequence of sets with the empty set in the limit.

The limit set is the same
for both sequences.
(E(1)∩E(2)∩...∩E(n)) and (E(n))
In order to stop tricksters
we go without cardinality.

Define a finite.cardinal as one of
the (well.ordered) ordinals which
can grow by 1  and
can shrink by 1 or is zero.
A finite.cardinal is Original Cardinal.
⎛ When your sheep head out to the field to graze,
⎜ as each sheep passes, put a pebble in your pocket.
⎜ When they head in at end.of.day,
⎜ as each sheep passes, take a pebble out.
⎜ When your pocket is empty, all your sheep are in.
⎝ Original Cardinal.
What I mean by
the ability of set S to shrink by 1 and grow by 1
is
the non.existence of one.to.one functions
from S to an emptier.by.one set,
⎛ ∀x ∈ S:
⎜ ¬∃f: S ⇉ S\{x}: one.to.one
⎝ S ⇉| S\{x}
or to S from a fuller.by.one set.
⎛ ∀y ∉ S:
⎜ ¬∃g: S∪{y} ⇉ S: one.to.one
⎝ S∪{y} ⇉| S
⎛ The broader principle which I call upon is that
⎜ different.sized sets cannot be matched,  and that
⎝ sets which cannot be matched are different.sized.
Way, way back at the Original Cardinals, there
is where cardinality must somehow enter the discussion.
These whats.its are (or aren't) involved in matches
between themselves and other sets. Call that what you will.

It changes at the dark finite cardinal ω/2.

A finite.cardinal is Original Cardinal.
Finite.cardinals are (well.ordered) ordinals which
can grow by 1  and
can shrink by 1 or are zero.
Each finite.cardinal k is followed by
more.than.k finite.cardinals.
A finite.cardinal ω/2 midway between 0 and ω
is not.a.thing.

∀n ∈ ℕ: E(1)∩E(2)∩...∩E(n) = E(n).
The limit of the left-hand side is empty,
the limit of the right-hand side is full,
i.e. not empty.

For both set sequences,
the set of finite.cardinals in common with
each set in the sequence,
which is what the limit set is,
is the empty set.

By the way, we need no cardinality.
We need only the sequence of sets with the empty set in the limit.

The limit set is the same
for both sequences.
(E(1)∩E(2)∩...∩E(n)) and (E(n))
In order to stop tricksters
we go without cardinality.

What you (WM) mean by "going without cardinality"
is
pretending that
a sequence of sets
and
the sequence of the cardinalities of those sets
are the same thing.
Which, for end.segments of final.cardinals,
they are not the same.
#⋂{E(i):i} = #{} = 0
⋂{#E(i):i} = ⋂{ℵ₀:i} = ℵ₀