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

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

On 10.01.2025 21:08, Jim Burns wrote:

<WM<JB>>

Do you (WM) disagree with
'finite' meaning
'smaller.than fuller.by.one sets'?

>
That is also true for infinite sets.
>

</WM<JB>>

Where OUR infinityⁿᵒᵗᐧᵂᴹ "doesn't work",
it's you who's saying it doesn't work,

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

I (JB) have a theory about
your (WM's) theory about infinity and
what is and isn't 'invariableᵂᴹ'.
Elsethread:
<WM>

(Losing all numbers but
keeping infinitely many
can only be possible if
new numbers are acquired.)

</WM>[1]
It sounds as though
the only explanation which you (WM) accept
for the constancy of end.segment.size is
that elements are inserted (at the darkᵂᴹ end?)
as other elements are deleted (at the visibleᵂᴹ end?)
(Somehow this happens. Perhaps ℕ has homeostasis.)
It seems to me that you are imagining
insertions and deletions which cause
infinitesⁿᵒᵗᐧᵂᴹ to be variableᵂᴹ and
finitesⁿᵒᵗᐧᵂᴹ to be invariableᵂᴹ.
If I am correct about what you mean.
Infinityⁿᵒᵗᐧᵂᴹ doesn't work that way.
Infiniteⁿᵒᵗᐧᵂᴹ sets do not change.
Change an infiniteⁿᵒᵗᐧᵂᴹ set,
and get a different set.
Changed set and unchanged set co.exist.
⎛ No sets exist different in SIZE by one
⎜ from an infiniteⁿᵒᵗᐧᵂᴹ set.
⎝ If there were, it would not be infiniteⁿᵒᵗᐧᵂᴹ
Compare to:
⎛ No fourth side exists
⎜ of any triangle.
⎝ If there were, it would not be triangular.

Where OUR infinityⁿᵒᵗᐧᵂᴹ "doesn't work",
it's you who's saying it doesn't work,

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

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, a property which
flocks of sheep and bags of pebbles have.
⎛⎛ For each set,
⎜⎜ if it is finite, then
⎜⎝ there is a finite ordinal larger than it.
⎜
⎜ ...which is equivalent to...
⎜
⎜⎛ For each set,
⎜⎜ if there is no finite ordinal larger than it,
⎜⎝ then it is not a finite set.
⎜
⎜ ℕ is the set of finite ordinals.
⎜
⎜ There is no finite ordinal larger than ℕ
⎜
⎝ ℕ is not a finite set.
What sets the cat among the pigeons,
scatters the flock, and tears the bag  is
to say what this 'finite' means
(instead of giving examples: sheep, pebbles)
The cat must be set among the pigeons
because,
however utterly.familiar, commonplace, humdrum
we describe flocks of sheep as being,
the argument proves ℕ is the opposite of that.
In short, ℕ is weird, and must be.
Bob cannot become absent from a finite set
by swapping, swapping, swapping inside a finite set.
Of course. To be otherwise would be weird.
For ℕ, it is otherwise.
And, yes, that is weird,
utterly unlike sheep and pebbles,
allowing sequences to grow emptier.not.smaller,
amongst other weirdnesses.
However,
the alternative to weirdness is contradiction,
which is incoherent, every claim becoming provable,
and every claim's negation provable , too.