Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 11/2/2024 6:15 AM, WM wrote:

On 02.11.2024 01:55, Richard Damon wrote:

Infinite sets do not need to have both ends in them.

>
Between 0 and 1 there are unit fractions.
Hence there must be a first one.

A.
⎛ S can be given a total ordering which is
⎜ well-ordered both forwards and backwards.
⎜ That is, every non-empty subset of S has
⎝ both a least and a greatest element in the subset.
[1]
A. isn't true of all sets.
If, at some point in time,
you _define_ A. to be true of all sets,
then
for a set S of which A. is false,
  that set S does not change.
What changes is
  which sets are in our discourse:
Claims about "a set" (after A. is claimed)
  are not claims about S.
If you "change" S so that A. is true,
the "changed" S _is not S_
[1]
https://en.wikipedia.org/wiki/Finite_set
Necessary and sufficient conditions for finiteness
3. (Paul Stäckel)

The interval without unit fractions and
the interval containing unit fractions
are separated by the smallest unit fraction.

The set ⅟ℕ of all unit fractions
⎛ the set of reciprocals to ℕ⁺
⎜ where ℕ⁺ is the minimal set of those
⎝ holding 1 and closed under n↦n+1
does not hold a smallest element.
A. is not true of ⅟ℕ
If you augment ⅟ℕ with more elements
that is a different set.
A. is still not true of ⅟ℕ
If you, nonetheless, assert A.
A. is true of _all the sets in the discourse_
Because A. is not true of ⅟ℕ,
⅟ℕ is not in the discourse.
After A., claims about "sets" are not
claims about ⅟ℕ