Re: how

Le 27/05/2024 à 19:16, Jim Burns a écrit :

On 5/27/2024 10:03 AM, WM wrote:
Note that
points on the positive real axis are fixed and
can be subdivided into two sets, namely
the set of unit fractions and
the set of positive non-unit fractions.
 

The [positive] real axis and all point sets in it
have an end at or before zero.

 The positive real axis ℝ⁺
does not hold a visibleᵂᴹ lower end.

So it is. Neither do the sets of unit fractions or the set of positive irrational numbers.
But there must be ends because below 0 no further positive real numbers exist.

 Not all non.∅.subsets of visibleᵂᴹ ℝ⁺ hold two ends.
ℝ⁺ doesn't.

So it is. But nevertheless all ℝ⁺ ends before zero.

 For each darkᵂᴹ.number.holding superset ℝ⁺∪𝔻ᵂᴹ
not all non.∅.subsets of ℝ⁺∪𝔻ᵂᴹ hold two ends.
ℝ⁺ doesn't.

All ℝ⁺ ends above zero. If you disagree look for an element below zero.

 Holding two ends is insufficient for
a set to be finite.

If a set has no elements below 0, then it ends above 0.
Regards, WM