Re: Incompleteness of Cantor's enumeration of the rational numbers

On 11/25/2024 7:11 AM, WM wrote:

On 24.11.2024 19:16, Jim Burns wrote:

On 11/24/2024 6:06 AM, WM wrote:

On 24.11.2024 06:37, Jim Burns wrote:

Consider the set of natural numbers for which
there are NOT enough hats.

>
It is dark.

>
It is not dark what we mean by 'natural number'.
A natural number is countable.to from.0

>
That is a definable number.

Thus
it is not dark that
what we mean by 'natural number' is
countable.to from.0  and is
what you call a definable number.
----

Therefore,
the number G^G^G is not.first for which
there are NOT enough hats.

>
We do not disagree.
Therefore you need not
prove a difference for G and G^G^G.
>

A similar argument can be made for
  each natural number.

>
No,
it can be made for each definable natural number,

We agree.
A similar argument can be made for each
definable  countable.to.from.0  what.we.call
natural.number.
⎛ For each
⎜ definable  countable.to.from.0  what.we.call
⎜ natural.number k
⎜ k is countable.past (k+1 exists)
⎜ k is countable.to (k-1 exists or k=0)
⎜ Each split of sequence ⟦0,k⟧ is countable.across
⎝  (i is last.before, j is first.after, i+1=j)

it can be made for each definable natural number,
i.e., for a number belonging to
a tiny finite initial segment which is followed bay
almost all numbers.

No. The
definable  countable.to.from.0  what.we.call
natural.numbers
can each be counted.past (k+1 exists)
are each not a second end
are not two.ended
are infinitely.many.
That is 'infinitely.many' which can be ordered
  with any non.empty.subset less.than.two.ended.
It is 'infinitely.many' which has
  fundamentally different behavior.
It is 'infinitely.many' which Bob to disappear
  with enough room.swapping _inside_ the Hotel.
It is not your 'infinitely.many' which is merely
  exceedingly.many.but.finite.

a tiny finite initial segment which is followed bay
almost all numbers.

Each
definable  countable.to.from.0  what.we.call
natural.number
is followed by almost all
definable  countable.to.from.0  what.we.call
natural.numbers.
They are not exceedingly.many.but.finite.