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

On 10/10/2024 09:22 PM, Jim Burns wrote:

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.
>
>

Halmos wrote a book "Infinite Dimensional Vector Spaces",
and in it he _has_ a survey and outline of results
in the objects of mathematical theory in the
"infinite dimensional vector spaces",
Halmos _has_ "infinite dimensional vector spaces",
in Halmos' communications, of Halmos' theory,
in both the literals of the common words
"infinite dimensional vector spaces", and,
in all the relations to the language of mathematics,
the words, the objects, Halmos _has_ them.
I have them too, Halmos' infinite-dimensional vector spaces,
then also I have my own infinite-dimensional vector spaces,
there are objects of mathematics infinite-dimensional vector spaces,
and for strong mathematical platonists these are the words
we use for "infinite-dimensional vector spaces":
infinite-dimensional vector spaces.
The mathematical objects of a strong mathematical platonist's
mathematical universe are _discovered_, not _invented_.
Then, a "spiral space-filling curve", is something I've
written up here since about twenty or more years, with
hundreds and hundreds of essays about it, as, what it is,
an outward spiral from an origin of an infinite-dimensional
vector space, as about a simple a mathematical proto-type
as "drawing a line form 0 to 1 by putting pencil to paper,
marking the paper then lifting the pencil".
Anyways then that's quite perfectly ordinary and regular.
All I care is that those are real and more mathematical
objects in a real mathematical universe, more "replete", say.