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

On 10/03/2024 11:10 AM, Jim Burns wrote:

On 10/3/2024 12:49 PM, Ross Finlayson wrote:

On 10/03/2024 09:14 AM, Jim Burns wrote:

On 10/2/2024 10:01 PM, Ross Finlayson wrote:

>

Such casual extensionality as
"these Q's are those Q's"
is certainly usual,

>
Since we aren't being lax here,
you should say
⎛ Such casual isomorphistry as
⎜ "these Q's are those Q's"
⎝ is certainly usual,

>

The "up to isomorphism" like "the reals are
the complete ordered field 'up to isomorphism'",
does _not_ have that R = C, and indeed it is
so that R =/= C, so,
no, we do not say "isomorphistry".

>
ℝ is not isomorphic to ℂ
ℚ₀ is isomorphic to ℚₛ
>
The casual isomorphistry of
ℚ₀ is (implicitly.isomorphic.to) ℚₛ
is usual.
>

Having the properties of a complete ordered field,
and, being analytic under continuous functions,
then, maintaining analyticity under
transforms of continuous functions,
is _not_ something that R and C both have,

>
Yes. "not".
>

so, the usual idea that
"the reals R are unique up to isomorphism
the complete ordered field",
gets broken

>
The usual idea of ℝ unique.up.to.isomorphism
does not involve ℂ
because ℂ is not the complete ordered field.
>
( ℝ is unique.up.to.isomorphism
  ==
⎛ each two models M M′ of
⎜ the axioms for the complete ordered field (ℝ)
⎝ have a '+-×÷<'.preserving bijection.
>
Map identities to identities′.
Map integers to integers′.
Map rationals map to rationals′.
Map least.upper.bounds to least.upper.bounds′.
_Because_ M and M′ are both complete ordered fields,
there exists an isomorphism between them.
>

and is not considered thorough.

>
Who is it who is doing this considering?
Are they unaware of the proofs?
Do they consider proofs not thorough?
>

The extensionality is
the term of model theory with regards to that
a) it's assumed that models are faithful, and
b) it's assumed that models are equivalent
in all interpretations, thusly
c) model Q_1 and Q_2 each a, b are
extensionally equivalent and considered equals,
while of course
no properties of their structure that isn't
equi-interpretable and bi-relatable
is considered true for both, and
such aspects of their consideration are
with regards to a different theoretical object.
>
About "isomorphistry", getting into stuff
like "equals, 'almost' everywhere", has
that: "equals, a.e., is _not_, 'equals'".

>
And "equals everywhere" _is_ "equals".
>

And R, or R^2, and C, "up to isomorphism",
are _not_ equals.

>
ℚ₀ from which ℝₛ is constructed and
ℚₛ a subset of ℝₛ
_are_ equals.
>
How does someone make a claim to you (RF) that
is _not_ subject to being carried off and
put to work in foreign contexts?
>
>

>
>
Well it's pretty simple that in mathematics
that things that are distinct are things
that are unique,