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

On 10/10/2024 8:54 PM, Ross Finlayson wrote:

On 10/10/2024 01:47 PM, Jim Burns wrote:

ω is the first (our) transfinite ordinal.
∀γ:  γ ∈ ⟦0,ω⦆  ⇔
∀β ∈ ⦅0,γ⟧: ∃α ∈ ⟦0,β⦆: α+1=β

>
Halmos has for "infinite-dimensional vector spaces",

What I _suspect_ is that
'has' used as it is just above here
is an idiom I'm not familiar with --
possibly transported from some non.English language.
Would you be able to provide some context to
this way in which you use 'has', Ross?

so not only is the Archimedean contrived
either "potential" or "un-bounded",
so is the matter of the count of dimensions
and the schema or quantification or
comprehension of the dimensions,
where there's a space like R^N in effect, or R^w,
then for a usual sort of idea that
"the first transfinite ordinal"
is only kind of after all those, ...,
like a "spiral space-filling curve".

My current best.understanding of your posts of this genre
is that you are conducting brainstorming exercises,
in which the most 'points' are awarded for _creativity_
and not as many for merely connecting ideas in
a narrative of some kind.
So, I will no longer try to decipher
how what you post connects to what I post.
If my understanding is close to the mark,
you (RF) might even prefer that they do not connect
-- more creativity that way..
What I mean by 'first transfinite ordinal' ω is that,
of all ordinals which are not.finite,
ω is the first such ordinal.
What I mean by 'ordinal' is that
each set of ordinals holds a minimum or is empty.
What I mean by 'finite ordinal' γ is that
it is first (ie, γ=0) or
its predecessor.ordinal γ-1 exists and,
for each non.0 prior ordinal β<γ
its predecessor.ordinal β-1 exists.