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

On 10/18/2024 02:05 AM, FromTheRafters wrote:

Ross Finlayson has brought this to us :

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.

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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.
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... or any thing which so models those according to set theory.
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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.
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The cardinality: is not the sole size relation when any
other will do, and also it's usually mostly _not_ relevant.

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When he states n+1, that one is a finite cardinal number. If he wants to
use some other notion of 'one' then he should stipulate exactly what
that notion is.

Sure, finite cardinals naturally model standard natural integers,
and, vice-versa.
See, here we're mathematicians with _both_ definitions in _one_
theory, there's only _one_ theory of mathematics and somehow
according to words and correspondingly defined symbols relating
to _natural_ numerical and geometrical and mathematical relations,
we are conscientious mathematicians include that the conscience
of mathematics is a bit different than the concept of conscience
in human mores, which sometimes demands a polite conscious
ignorance, here as conscientious mathematicians we are _not_
ignorant because all mathematical objects are mathematical objects.
Then, it was a bit of a stretch to say that "cardinals are usually
irrelevant", though for any given person with a usual well-rounded
education it's about 50/50 or less whether they've heard of them,
with regards to that everybody knows arithmetic and geometry.
I know that Burns is a platonist at least one day each week,
so, hope and a bottle of ketchup isn't a hamburger,
yet at least there's hope.
Same goes for the rest of you and us all.