Re: Replacement of Cardinality

On 8/14/2024 8:34 AM, WM wrote:

Le 13/08/2024 à 20:34, Jim Burns a écrit :

On 8/13/2024 10:17 AM, WM wrote:

Le 12/08/2024 à 19:23, Richard Damon a écrit :

On 8/12/24 9:50 AM, WM wrote:

Le 11/08/2024 à 19:56, Jim Burns a écrit :

What causes an exception: nₓ ∈ ℕ:
⅟nₓ > 0 without ⅟(nₓ+1) > 0 ?

>
The end of the positive axis.

>
Which,
by the definition of the Natural Numbers,
doesn't exist.

>
The end of the positive axis exists.

>
No point in the positive axis is
largest in the positive axis or
smallest in the positive axis.

>
What is smaller than every positive x
is smaller than the interval (0, oo).

That which is not.in the interval (0,∞)
is not an end of (0,∞).
----
No point in the positive axis is
largest in the positive axis or
smallest in the positive axis.
No countable.to number is largest countable.to.
No rat. > 0 with countable.to num. and den.
is largest rat. > 0 with countable.to num. and den.
or smallest rat. > 0 with countable.to num. and den.
∀p/q ∈ ℚ⁺:  ℚ⁺ ∋ p/(q+1) < p/q < (p+1)/q ∈ ℚ⁺
No (0,∞).split F,H has a situating.point x
(x last.in.F or x first.in.H) such that
x is first in F∪H = (0,∞) or
x is last in F∪H = (0,∞)
No rational > 0 with countable.to num. and den.
and no situating point of any (0,∞).split
is an end of (0,∞)
There are no other points in (0,∞)