Re: Replacement of Cardinality

Am 26.08.2024 um 02:15 schrieb Moebius:

Am 25.08.2024 um 23:18 schrieb Jim Burns:

On 8/25/2024 3:35 PM, WM wrote:

Each non.empty set of ordinals holds a first.

>
Like each set of unit fractions.

 Fascinating.
 On the other hand, if u is a unit fraction, then 1/(1/s + 1) is a smaller one.
 

Ordered largest.first, yes.
Each non.empty set of unit.fractions holds a largest.
>
For each unit.fraction u
the claim that u is smallest is proved false
by counter.example ⅟(1+⅟u)

 Indeed! :-)

Btw. To show this we don't refer to "the smallest unit fraction" WM (<- a constant denoting "the smallest unit fraction").
I mean, we don't introduce (define) WM ("the smallest unit fraction") by a "definition" just to show (afterwards) that it does not exist: ~Ex e SB: s = WM.
Of course in a proof by contradiction we may proceed the following way:
Assume that there IS _a_ smallest unit fractions. i.e.
      Es e SB: As' e SB\{s}: s < s' .
Let wm (<- an arbitrary name or parameter or ...) be such a unit fraction. With other words:
      wm e SB & As' e SB\{wm}: wm < s'
and hence
      As' e SB\{wm}: wm < s'.
Now from wm e SB we get that 1/(1/wm + 1) e SB and 1/(1/wm + 1) < wm, contradicting
      As' e SB\{wm}: wm < s' .
qed.