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

On 10/30/2024 09:18 AM, Jim Burns wrote:

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.
>
>

Well, I think you're familiar with the language of limits:
we say that it's "unbounded".
Unless you'd care to exclude the "unbounded"
or introduce "the actually really infinite limit"
about the language of "infinite limit", ....
That that there's "0/d, ..., { ... infinitely-many ... }, ..., d/d",
has that the "infinitely-many" is a usual sort of word that
is among the few words that end up in "math mode", like "lim".
In fact " ... infinitely-many ... " is one of the few such
constructs that see in-line usage in "math mode".
Otherwise you've built a neat fixture that
inductive inference never gets anywhere, ....