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

On 11/7/2024 2:33 PM, WM wrote:

On 07.11.2024 20:06, Jim Burns wrote:

On 11/7/2024 3:46 AM, WM wrote:

On 07.11.2024 16:29, Jim Burns wrote:

and shuffle them in a clever way,
then all rational numbers are midpoints of intervals
and no irrational number is outside of all intervals.

 

That means by clever reordering them
you can cover the whole positive axis
except "boundaries".

"Except boundaries" is the key phrase.

Yes.
In that clever re.ordering, not scrunched together,
the whole positive axis
is in the ε.cover or
in the boundary of the ε.cover.

>
It is impossible however to cover
the real axis (even many times) by
the intervals
J(n) = [n - 1/10, n + 1/10].

Those are not the cleverly.re.ordered intervals.
These are:

⎛ ε.cover = {[x⁽ᵋₖ,xᵋ⁾ₖ]:k∈ℕ⁺}
⎜  x⁽ᵋₖ = iₖ/jₖ-2¹ᐟ²⋅10⁻ᵏ
⎝  xᵋ⁾ₖ = iₖ/jₖ+2¹ᐟ²⋅10⁻ᵏ

There is an enumeration of ℚ⁺
the set of ratios of ℕ⁺ countable.to from.1
⎛ i/j ↦ kᵢⱼ = (i+j-1)(i+j-2)/2+i
⎜ k ↦ iₖ/jₖ
⎜  iₖ+jₖ = ⌈(2⋅k+¼)¹ᐟ²+½⌉
⎜  iₖ = k-((iₖ+jₖ)-1)((iₖ+jₖ)-2)/2
⎜  jₖ = (iₖ+jₖ)-iₖ
⎝  (iₖ+jₖ-1)(iₖ+jₖ-2)/2+iₖ = k

d is a Cantorian anti.diagonal of
the Cantorian rational.list ⟨iₖ/jₖ⟩
#d#ₖ = (#iₖ/jₖ#ₖ+5) mod 10

#x#ₖ is the -k.th digit of
the decimal representation of real number x

For each interval [x⁽ᵋₖ,xᵋ⁾ₖ] in ε.cover
d is "not in contact with" [x⁽ᵋₖ,xᵋ⁾ₖ]
However,
d is "in contact with" ⋃(ε.cover)
d is in the closure of ⋃(ε.cover)
and not.in the closure of any of its intervals.

No boundaries are involved because
every interval of length 1/5 contains infinitely many rationals and
therefore is essentially covered by infinitely many intervals of length 1/5
- if Cantor is right.

I haven't claimed anything at all about
your all.1/5.length intervals.