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

On 13.11.2024 20:38, Jim Burns wrote:

On 11/13/2024 11:31 AM, WM wrote:

>
If infinite sets obey the rules sketched above,

 ... _and are finite_ ...

Not necessary to apply logic, geometry and analysis.

 

then set theorists must discard geometry
because
 by shifting intervals
 the relative covering 1/5 of ℝ+ becomes oo*ℝ,
and analysis
because
 the constant sequence 1/5, 1/5, 1/5, ...
 has limit oo,
and logic
because of
 Bob.

 ----

 by shifting intervals
 the relative covering 1/5 of ℝ+ becomes oo*ℝ,

 By definition,
the value of a measure is an extended real≥0

The value of the relative measure is 1/5. For every finite interval this is true. The limit can be calculated in analysis of real numbers without extension.

 An extended real≥0 is either
Archimedean == having a countable.to bound,  or
non.Archimedean == not.having a countable.to bound.
 The extended reals≥0 have only
the standard reals≥0, which are Archimedean, and
a single non.Archimedean point≥0  +∞

That is what I call ω. But the limit can be calculated from the reals alone, in particular when the sequence is constant.

 No,
the measure doesn't _become_ +∞
It has the same value +∞ before and after shifting.

Take the relative measure that can be obtained from every finite interval of the real line.

 ----

 the constant sequence 1/5, 1/5, 1/5, ...
 has limit oo,

 

Nonsense deleted.

 ----

 Bob.

 KING BOB!
https://www.youtube.com/watch?v=TjAg-8qqR3g
 If,
  in a set A which
  can match one of its proper subsets B,
>

That is nonsense too.
Regards, WM