On 10/03/2024 02:27 PM, Ross Finlayson wrote:
On 10/03/2024 02:09 PM, FromTheRafters wrote:
Ross Finlayson pretended :
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:
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Such casual extensionality as
"these Q's are those Q's"
is certainly usual,
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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".
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ℝ is not isomorphic to ℂ
ℚ₀ is isomorphic to ℚₛ
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The casual isomorphistry of
ℚ₀ is (implicitly.isomorphic.to) ℚₛ
is usual.
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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,
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Yes. "not".
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so, the usual idea that
"the reals R are unique up to isomorphism
the complete ordered field",
gets broken
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The usual idea of ℝ unique.up.to.isomorphism
does not involve ℂ
because ℂ is not the complete ordered field.
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( 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.
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and is not considered thorough.
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Who is it who is doing this considering?
Are they unaware of the proofs?
Do they consider proofs not thorough?
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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".
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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?
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Well it's pretty simple that in mathematics
that things that are distinct are things
that are unique,
>
There can be many distinct results from a dataset inquiry yet only one
unique result.
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[...]
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Well, it's "discernibles" and "definables",
from sort of the "entirety", with regards
to for example: "there is a space it is of
numbers, in it is a number 1, it is unique",
yet it's as well that "the usual model of
cardinality as the equivalence class of sets
equal cardinally has that singletons is the
largest class of sets a cardinal", that,
"1 is the largest cardinal, if you'd consider
the cardinal a set, of course cardinals aren't
sets in set theory".
>
Set theory's a great theory and of course
all matters of the rulial and regularity are
in it, and ordinary, thanks to Goedel we know
it's at best incomplete.
>
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Otherwise it is indeed a sort of "strong mathematical
platonist monist's" view of things, that there's a
universe of mathematical objects, that we're in it,
this kind of thing.
>
Then most always in application or practical or
ad-hoc theories, most people aren't formalists of
the entirely fundamental sort, yet here it's arrived
at that due there being a consistent and complete
theory, of truth, it can actually be a thing. I.e.,
most people are just not needing to care for the
universally fundamental when their practical theory
suffices and quite well, and most people who get into
foundations quite directly get thrown into incompleteness,
when hopefully they aren't lacking a true intuition.
>
So, in that way "what's unique is distinct",
all other things.
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I started reading this Bateson's "The Ecology of the Mind"
the other day, and he was like "most students think inductive
while I think deductive, having an idea way things should be",
it was like that.
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Then, "sweep" is what it's called sometimes,
the line-drawing, the iota-values, the thusly
a linear continuum or segment thereof, where
much like "iota" for standard infinitesimals,
then "sweep" is a quite reasonably descriptive word,
then also as where it means "win, place, and SHOW".
Reading Bateson then has he's sort of a Britisher so
he tends to frame a bunch of things in contrast to
British "order" as he snobbishly puts it, for example
to American "individuality", then has mostly he has
a great vocabulary and it's a pleasure to read,
yet he's all about his schismogenic his schizophrenic,
then he inevitably wanders into a discussion about
the Balinese tribe where I imagine he drank a great
bowl of ayahuasca in about 1925. Or, reading about
his 1942 bit about morale and what's about propaganda,
is that most massively-monopolized media today,
is very polarizing and polemical, when as with regards
to something like a notion of "national culture",
for example "we are a country of individual-freedom
loving responsible hard-working friends", and the
ideas that "well, it's gonna vary", has that what
overall I found was the greatest error in the
approach of anthropology was to consider that
its subjects weren't aware of its approach,
the "free will" bit, and it's like "scientist:
you're subject, too".
Anyways the massively-monopolized media has
had it way too easy for a long time, at least
as with regards to having a cheap, giant funnel
into eyeballs, if there was even a micro-fee,
it would invert most the economies.
Anyways "sweep" is a great concept, there are basically
the concepts of "the slates of sweep for uncountability
and the extra for paradox", in case the entire mathematics
is consistent and complete, both.
How many different unit fractions are lessorequal than all unit fractions?
Re: How many different unit fractions are lessorequal than all unit fractions?