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, this is for example for
a mathematical platonist that there is a
"the" Archimedean rationals and "the" Archimedean
reals, then that models of those that can be
considered equal, can not be considered except
via properties of those.
Now, I want to comment that isomorphism is
a very, very genial concept, and when learning
model theory, one of the first things that happens,
is that anything like or alike, has an abstract
isomorphism, that it's very much so, that loosely
there are consideration of sameness and difference,
and equality and inequality, and then matters of
shape and so on, with regards to morphisms, and
homo- and diffeo- and iso-morphisms, not-different
and different and "same", that here where you've
written above the properties or qualities as so
as what we'd all be familiar with from abstract
algebra, has that then there's more about an
example deconstructive account in model theory
of
arithmetic
algebra
number theory
geometry
function theory
topology
as with regards to
set theory / class theory
set theory / part theory
set theory / ordering theory
ordering theory
as with regards to
integer continuum
line-reals \
field-reals ----- linear continuum
signal-reals /
long-line continuum
that there's one theory overall,
that has all the theories in it,
there's only one theory over all after all,
then as with regards to morphisms,
for the shape the morphology and the state the morpheme,
that then when talking about
R
and
C
there is for example that division can be be deconstructed
in C and it's not the same as the usual way.
Then with regards to Q_Cauchy and Q_Dedekind,
I suppose those are things being subsets of Q_Cauchy
and partitions of rationals, yet I won't yet agree
that partitions of rationals by reals are except as
by R_Cauchy.
Then those are mostly all R_Archimedes, yet as well
there are notions like R_Conway or R_Robinson,
non-archimedean.
Then for example I wrote field axioms for [-1, 1] before.
Anyways isomorphism is truly a great conflater, and while
it's at once one of the most facile relations, "not much
different" or "at least similar", it's also let out about
first when other ideals or axes are not invariants.
There is though that when learning model theory, and
that "a model is as a model does as a model is",
and all it is, it's pretty usual.
Anyways in my podcasts and here also I discuss a lot
the deconstructive accounts of the usual theories,
when they all have to be a single theory of the
single universe of all the mathematical, logical
objects, in terms, predicates/propositions, and relations,
then that there's sole definition
and that's what defines definition.