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

On 1/12/2025 10:54 AM, WM wrote:

On 12.01.2025 15:40, Jim Burns wrote:

On 1/10/2025 4:48 PM, WM wrote:

You are inconsistent.
You claim that
all natural numbers are an invariable set.

Objects do not vary in their
natural.number.ness or not.natural.number.ness.
One consequence of this invariance is that
we can refer to an indefinite natural number
and make claims which we know are true,
even where we don't know which natural number
we refer to.
By doing this, we finite beings explore infinity.
On the other hand,
your (WM's) slippery, 'variant' references
descend into gibberish,
which you apparently consider success.

But when all elements are doubled
then your set grows, showing it is not invariable.
That is nonsense.

>
Perhaps this argument won't look like nonsense.
>
It features an utterly.familiar property,
being.finite,

>
No, it depends on completeness.

It is completely true
that each natural number is a natural number   and
that only natural numbers are natural numbers.
Is that something your students object to?

If all natural numbers are there
such than none can be added,

For each finite set,
there is an ordinal larger and finite.
A finite set (other than {}) is
larger.than emptier.by.one sets.
A natural number is a finite ordinal
(finitely.many priors, well.ordered).

If all natural numbers are there
such than none can be added,
then doubling all of them
deletes odd numbers and
must create new even numbers which
cannot be natural numbers.

ℕ is the set of finite ordinals.
There is no finite ordinal larger than ℕ
ℕ isn't a finite set.
ℕ isn't larger.than emptier.by.one sets.
Related:
There is no DOUBLED finite ordinal larger than ℕ

⎜ ℕ is the set of finite ordinals.

>
Of all.
None can be added.

Each natural number stays a natural number.
Each not.a.natural.number stays not.a.natural.number.
A finite set is larger.than emptier.by.one sets.
For each finite set,
some ordinal is finite and larger.

If all are doubled,
then 50 % odd nubers are deleted,
50 % even numbers are added.

ℕ remains ℕ
The evens 𝔼 remains 𝔼
#𝔼 = #ℕ
𝔼\ℕ = {}

Because the total sum remains constant.
x = x/2 + y.

Because
no finite ordinal is larger than ℕ,
no finite ordinal is larger than 𝔼.