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

On 10/17/2024 12:39 PM, FromTheRafters wrote:

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:

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[...]

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

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To describe an indefinite one of
an infinite domain is
an infinite force.multiplier.
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But it needs to be
only those in that domain,
of that description.
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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.
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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.
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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.
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So, if you don't recognize that as a fact,
then, you're not talking about numbers.
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Numbers the arithmetization, ...,
which is a most usual first thing in
all manners of descriptive set theory
as would-be relevant.
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 From an arithmetization:
all the properties
of integers so follow,
and one of them is "asymptotic density".
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Half of the integers are even.

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If that was what he was talking about, then he should say so. Otherwise
it is not what we were talking about.
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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.

The _cardinality_, among _size relations_, is same,
which is obvious, while other _size relations_,
particularly more relevant _size relations_,
in the _spaces_ their _elements_, make for that
of course notions like outpacing are _size relations_,
particularly those being of nested sets, and notions
like density are _size relations_, those being numbers.
... or any thing which so models those according to set theory.
Numbers are never bereft their relations as numbers,
and when a set theory _all its universe_ is arithmetized
like Goedel says, it also so happens that it's always so,
if not so obviously.
The cardinality: is not the sole size relation when any
other will do, and also it's usually mostly _not_ relevant.