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

On 9/24/2024 4:00 AM, joes wrote:

Am Sun, 22 Sep 2024 15:28:38 +0200 schrieb WM:

On 22.09.2024 01:25, Richard Damon wrote:

On 9/21/24 9:57 AM, WM wrote:

On 21.09.2024 01:06, Richard Damon wrote:

Its just when they become infinite that
there might be ends that don't exist.

>
In order to count a countable set,
you have to start at 1.

>
Right, so
the countable numbers have ONE end that
can be used to count from.

>
Really existing sets of real unit fractions
have two ends.

>
What is „an end”?

I have been using "end of S" to mean
"an element of S ≤ or ≥ each element of S"
as distinct from "bound of S" meaning
"anything  ≤ or ≥ each element of S"
I have been using those senses a lot.
I suspect that that is what WM means here.
WM's argument goes something like this:
🛇⎛ The set of unit.fractions exists.
🛇⎜ No _identifiable_ unit.fraction is its second end.
🛇⎜ (axiom) All sets have two ends.
🛇⎜ The second end of the unit.fractions exists
🛇⎝ but it is _not identifiable_
There is plenty to correct in that,
but I think WM's cornerstone.error is
how he thinks axioms work.
I think WM thinks that,
if he declares all sets two.ended,
all sets which they have been discussing
thereby become two.ended.
⎛ An obviously.false example of the technique:
⎜ If one declares all triangles right triangles,
⎜ all triangles become right triangles.
⎜ Maybe with 'dark' degrees in some angle,
⎝ bringing it up to 90°?
How axioms actually work is that,
if WM or I or you declare all sets two.ended,
and the set of unit.fractions isn't two.ended,
then
the set of unit.fractions is still the same,
but it is not one of the sets we are discussing.