Re: Replacement of Cardinality

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.
A point not.largest and not.smallest
is not an end.
 No point in the positive axis is
an upper.end or a lower.end.
 If an end of the positive exists,
it is in the positive axis.
A bound not.in is not an end.
 No point not.in the positive axis is
an upper.end or a lower.end.
 No point is
an upper.end or a lower.end.
 The end of the positive axis
does not exist.
 ----
In the land of rationals only with
countable.to numerators and denominators
and with each split situated
⎛ a last point in the foresplit or
⎝ a first point in the hindsplit,
no point in the positive axis is
largest in the positive axis or
smallest in the positive axis.
 for each positive rational p/q
p > 0, q > 0
(p+1)/q > p/q > p/(q+1) > 0
 for each positive rational p/q
p/q is not the upper.end or lower.end
of the positive rationals.
 for each point x situating a split F,H
of the positive axis,
there is a rational < x in F
and a rational > x in H
and x is not the upper.end or lower.end
of the positive rationals.

No other points are in the positive axis.
No points not.in are ends.