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

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.