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

On 12/24/2024 5:53 AM, WM wrote:

On 23.12.2024 19:20, Jim Burns wrote:

[...]

>
There is a bijection {n} <--> E(n).
No cardinal number ℵ₀ is involved.

There is a bijection for all n < ℵ₀: n ∈ ⟦0,ℵ₀⦆
Which n are those?
Writing 'ℵ₀' gives a less.verbose description of
which n we are talking about,
each of which < ℵ₀
However,
if we don't want 'ℵ₀' involved,
not.writing 'ℵ₀' isn't a problem.
There are more.verbose descriptions of
which n we are talking about,
each of which < ℵ₀
n < ℵ₀  ⇔  n < n+1
Each n < ℵ₀ is
smaller.than fuller.by.one sets.
n = #⟦0,n⦆ < #(⟦0,n⦆∪{n}) = n+1
Anywhere we write 'n < ℵ₀',
we can write '#⟦0,n⦆ < #(⟦0,n⦆∪{n})'
-- in which 'ℵ₀' is not.written.
ℕ  =  ⟦0,ℵ₀⦆  =
the set of cardinals of
sets smaller.than fuller.by.one sets,
⦃i: #⟦0,i⦆<#(⟦0,i⦆∪{i}) ⦄
A less.verbose way to say
"set smaller.than fuller.by.one sets"
is "finite set".
ℕ is the set of finite.cardinals,
all the finite.cardinals,
the visibleᵂᴹ and the darkᵂᴹ.
I gather that we don't know the name of
any darkᵂᴹ finite.cardinal δ
Still, we know (to start with)
one fact about δ
#⟦0,δ⦆ < #(⟦0,δ⦆∪{δ})
because, being a finite.cardinal, δ ∈ ℕ
or, more.verbosely, we can write
δ  ∈  ⦃i: #⟦0,i⦆<#(⟦0,i⦆∪{i}) ⦄
----
For finite sets,
emptier.by.one sets are smaller.by.one, and
fuller.by.one sets are larger.by one.
That can serve as what.we.mean.by 'finite'
However,
that doesn't describe all sets.
In particular, it doesn't describe ℕ
For each finite.cardinal,
even for each darkᵂᴹ finite.cardinal δ
#ℕ  ≥  #(⟦0,δ⦆∪{δ})  >  δ
#ℕ  ≠  δ
¬(#ℕ ∈ ⦃i: #⟦0,i⦆<#(⟦0,i⦆∪{i}) ⦄)
¬(#ℕ < #(ℕ∪{ℕ})
For ℕ
emptier.by.one sets are NOT.smaller.by.one, and
fuller.by.one sets are NOT.larger.by one.
We have two concepts of relative set.size,
being.a.subset.of and fitting.into.
For finite.sets,
the two concepts agree nicely.
They do not agree in all circumstances.