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

On 10/17/2024 2:20 PM, Ross Finlayson wrote:

On 10/17/2024 04:25 AM, Jim Burns wrote:

Loosen the restriction on the discussion,
lose the force.multiplier.
There isn't much useful to be said
about things which _might or might not_
be well.ordered. Etc.

>
It is a fact:
that in number theory,
according to number theorists,
that there's a descriptive aspect of numbers,
and it's "asymptotic density",
and according to "asymptotic density",
half of the integers are even.
>
Whether it's SETS of numbers
or the sets of NUMBERS, in
number theory, the set of integers,
has associated with the set of even integers,
a relative size relation, of: one half.
>
So, if you don't recognize that as a fact,
then, you're not talking about numbers.

It is a fact
that
ω is the first ordinal after all finites,
that
ordinals are well.ordered and successored,
that
finite ordinals k are predecessored or 0
  with each prior j < k predecessored or 0
There is no ω
such that the numbers are
evenly.spaced between 0 and ω
because
that describes a finite ordinal, not ω
So, if you don't recognize that as a fact,
then, you're not talking about ω

Numbers the arithmetization, ...,
which is a most usual first thing in
all manners of descriptive set theory
as would-be relevant.
>
 From an arithmetization:
all the properties
of integers so follow,
and one of them is "asymptotic density".
>
Half of the integers are even.