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

On 09/09/2024 11:23 AM, Jim Burns wrote:

On 9/9/2024 12:59 PM, Ross Finlayson wrote:

On 09/08/2024 09:59 PM, Jim Burns wrote:

>

2.
It is an essential aspect of these discussions
that
whatever is not being discussed
is not being discussed.
>
That essential aspect seems to be
what you (RF) consider
hypocritically ignoring
whatever is not being discussed.
>
Consider that it is possible for there to be
many discussions, with many different topics.
It is true at the same time
that
nonstandard analysis can be pursued
and
the complete ordered field doesn't hold
anything other than elements of
the complete ordered field.

>

As context expands there's nothing left out.
The objects of mathematics all live in
one mathematical universe,

>
We have the ability to _discuss_ only
irrational elements of the complete ordered field,
to pick one example.
>
If we choose to discuss _only_ them,
we can make claims about one of them which
we know are true, even if
we don't know _which_ one the claims are _about_
>
⎛ x is an irrational real.
⎜  x ∈ ℝ\ℚ
⎜ Some split Sₓ of ℚ exists with x between sides of Sₓ
⎜  ∃Sₓ ⊆ ℚ:  {} ≠ Sₓ ᵉᵃᶜʰ< x <ᵉᵃᶜʰ ℚ\Sₓ ≠ {}
⎝ ...
>
If we choose to discuss only irrational reals,
those are true claims.
⎛ A true claim is not.first.false.
⎜ A finite sequence of not.first.false claims
⎝ (AKA a proof) holds only true claims.
>
If we widen the discussion to include rationals,
claims 'x ∈ ℝ\ℚ'  etc.  aren't always true,  and
we can't reliably use an argument we can use in
the narrower discussion.
>
Not.widening the discussion does not
wipe rationals out of ℝ
Not.widening the discussion allows us to learn,
by means of assembling not.first.false claims,
things which might not be universally true
in a wider discussion.
>
Very little is true across all of the widest context,
and probably none of that very little is interesting.
>

1.
I wonder what your own conscientious response
will be to the infiniteness of the set of
all finite non.self.membered sets.

>
I still wonder.
>
>

To your question, "are the finites infinite",
well yeah. It's like, the other day, was introduced
another notion of finite distinct Staeckel's and Dedekind's.
Then, at some point I learned that some of what
are called the, "inconsistent multiplicities", results
as are of antinomies or resulting contradictory outcomes,
are as well called "extra-ordinary", this would have been
about 25 years ago when I was learning about anti-foundational
set theory, then it was quite later that I learned about
Mirimanoff, his defining "extra-ordinary", and the strong
influence it has on resulted ZF, where one imagines that
Zermelo felt ZF set theory with well-foundedness quite
lacking without well-ordering, which otherwise on its
own makes itself quite as strong.
So, what I've arrived at is that there isn't a standard
model, instead it's fragments and extensions, either of
which suffices for "a law of large numbers with respect
to the law of small numbers", that there are larger numbers,
while allowing that there are multiple laws of large numbers,
and for example infinite magnitudes if there are infinite
multitudes.
If this helps I grew rather territorial about it and
quite rather wrote up what is "the theory", it's a
study in foundations so things like "ubiquitous ordinals"
result quite total, while as well in a humble sort of
sense, axiomless and all.
Or, in my perception I do not a waffle make, and,
when I do, it's deliberate and explicit.
I got definitions of continuity three.
(Five, ....)