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

On 12/23/2024 6:18 AM, WM wrote:

On 23.12.2024 12:02, Jim Burns wrote:

On 12/22/2024 5:24 PM, WM wrote:

Another approach is to call [prove] the empty set
the limit of the sequence E(n).
But note that a limit is a set [JB:]
⎡ in.which is each element in each set of
⎢ almost.all of infinitelyⁿᵒᵗᐧᵂᴹ.many sets
⎢ and
⎢ not.in.which is each element not.in each set of
⎣ almost.all of infinitelyⁿᵒᵗᐧᵂᴹ.many sets
between which [it] and all terms of the sequence
nothing fits.
Therefore the limit empty set causes
sets with few elements only.

>
The empty limit set is caused by
each finiteⁿᵒᵗᐧᵂᴹ.cardinal being not.in
almost all of the end.segments
(in only finitely.many end.segment.exceptions)

>
And why is this so?

Because
each of more.than any.finiteⁿᵒᵗᐧᵂᴹ.cardinal.many
finiteⁿᵒᵗᐧᵂᴹ.cardinals is finiteⁿᵒᵗᐧᵂᴹ.

Because
every endsegment removes exactly one number from the content.
There is a bijection between
endsegments and removed numbers.

There is a bijection between
more.than any.finiteⁿᵒᵗᐧᵂᴹ.cardinal.many
end.segments and
more.than any.finiteⁿᵒᵗᐧᵂᴹ.cardinal.many
removed numbers?
Have you told your students yet?
⟨i,j⟩ ↦ k ↦ ⟨i,j⟩
( k = (i+j)⋅(i+j+1)/2+j
⎛ kⱼ₌₀ = iⱼ₌₀⋅(iⱼ₌₀+1)/2
⎜ iⱼ₌₀ = (2⋅kⱼ₌₀+¼)¹ᐟ²-½ = ⌊(2⋅k+¼)¹ᐟ²-½⌋
⎜ j = k - kⱼ₌₀
⎝ i = iⱼ₌₀ - j
ℕ×ℕ ⇉ ℕ ⇉ ℕ×ℕ one.to.one