Re: Replacement of Cardinality

Am 25.08.2024 um 23:18 schrieb Jim Burns:

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

Dark numbers are not any bounds.

Fascinating: So we have that there are "dark numbers" which are smaller (larger) than each and every s e SB (n e IN), but they are no bounds? Huh?!
Das Mückenheim even know the definition of /bounds/ (of a set)?
https://en.wikipedia.org/wiki/Upper_and_lower_bounds

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! :-)

Not all sets of unit.fractions hold a smallest.
For example, the set of all unit.fractions doesn't.

Right. :-)

ω - 1 is the greates natural number.

Fascinating. This implies that there is a natural number (namely WM := ω - 1) such that WM + 1 is not a natural number.
On the other hand, this is excluded by the Peano-Axioms (and the definition of + on IN) as well as by the axioms and (usual definitions) in the context of set theory. (Hint: We can prove there that for each and every n e IN n u {n} is in IN too.)
It seems that this law only holds in mückenmath.