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

Ross Finlayson explained :

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

On 10/16/2024 9:05 PM, Ross Finlayson wrote:

On 10/16/2024 11:06 AM, Chris M. Thomasson wrote:

>

[...]

>
and people who follow only one,
ignorant the other,
need to look up from their nose
because it's leading them.

>
To describe an indefinite one of
an infinite domain is
an infinite force.multiplier.
>
But it needs to be
only those in that domain,
of that description.
>
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.
>
>
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.

If that was what he was talking about, then he should say so. Otherwise it is not what we were talking about.
Your previous mention of outpacing is also what we are not talking about. Comparing the naturals and the doubled naturals does imply a certain regular two to one ratio between the two as finite subsequences of each get built/run in parallel, but the Prime Density (RH) is not that regular and predictable. Anyway, the size of these infinite sets are equal despite outpacing and asymtotic density considerations.