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

On 10/30/2024 1:27 AM, Ross Finlayson wrote:

On 10/29/2024 07:38 AM, Jim Burns wrote:

A finite set has the LUB property, yes,
but a finite set isn't a superset of Q
The smallest superset of Q with the LUB property
is R, which is what I've been talking about.

>
That ran(EF)

Do you still define  ran(EF)  as  n/d n->d d->oo  ?

That ran(EF) is complete is rather simple,
i.e. it's trivial, in as to where,
as you for whatever reason chose to simply ignore:
the LUB can be simply either max(n), or, max(n)+1,
being or making a critical point.

I think that the reason that it seems that
I'm ignoring you is that, for some reason,
you can't see me telling you that
you are describing ran(EF) to be finite.
Whatever you mean by  n/d n->d d->oo
if what you mean by it is finite,
you aren't saying what you think you're saying.
⎛ Necessary and sufficient conditions for finiteness
⎜
⎜ 3. (Paul Stäckel) 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.
⎝
https://en.wikipedia.org/wiki/Finite_set
Consider a set ran(EF) such that min.ran(EF) = 0 and,
for each non-empty subset B of ran(EF),
max.B and 1+max.B are in ran(EF)
Both a least and a greatest element is in each non-empty B
That makes ran(EF) finite.
max.B is the greatest element in B
B ᵉᵃᶜʰ≤ max.B
If 0 ∈ B, 0 is the least element in B
If 0 ∉ B
consider the non-empty set L of
strict.lower.bounds of B
L = {x ∈ ran(EF): x <ᵉᵃᶜʰ B}
max.L is a strict.lower.bound of B
1+max.L isn't a strict.lower.bound of B
max.L <ᵉᵃᶜʰ B
¬(1+max.L <ᵉᵃᶜʰ B)
∃x ∈ B: max.L < x  ∧  ¬(1+max.L < x)
∃x ∈ B: 1+max.L = x
1+max.L ∈ B
1+max.L ≤ᵉᵃᶜʰ B
1+max.L is the least element in B
Both a least and a greatest element is in B
Both a least and a greatest element is in each non-empty B
That makes ran(EF) finite.