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

On 11/7/2024 9:59 AM, WM wrote:

On 07.11.2024 16:29, Jim Burns wrote:
 

⎛ The boundary of a set S holds
⎜ those points x′ such that
⎜ each interval [x,x″] with
⎜ x′ in its interior, x < x′ <  x″,
⎝ holds points in S and points not.in S
>

Do you think you need the boundary in my last example?
 When we cover the real axis by intervals
--------_1_--------_2_--------_3_--------_4_--------_5_--------_...
J(n) = [n - √2/10, n + √2/10]
and shuffle them in a clever way, then all rational numbers are midpoints of intervals and no irrational number is outside of all intervals.
Do you believe this???

Trying to relate it to a fractal wrt escape time. The fractal border is infinitely complex. So we can iterate a point using the fractal algorithm. If it stays within the boundary for how far it can go during said iteration, then we say its in the set, otherwise its out of the set. Some points can take billions of iterations to make that decision.
But this is a heck of a lot different that saying there must be a largest natural number type shit... :^)