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

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

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

On 10/02/2024 11:57 AM, Jim Burns wrote:

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[...]

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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,
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It is _up to isomorphism_ that
ℚ₀ is the same as ℚₛ =
{ {q′∈ℚ₀:q′<q}: q∈ℚ₀ }
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ℝₛ = {open.foresplits of ℚ₀} =
{ S⊆ℚ₀: {}≠Sᵉᵃᶜʰ<ₑₓᵢₛₜₛSᵉᵃᶜʰ<ᵉᵃᶜʰℚ₀\S≠{} }
is the complete.ordered.field
ℚ₀ ⊈ ℝₛ
ℚₛ ⊆ ℝₛ
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Sorry for any confusion caused by
my not.using the i.word earlier.
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it's considered a lax sort of accommodation
that thereafter
that "all proofs reflecting on a model of Q
will be as entirely agnostic to
the actual set modeling Q
as any other may do,
only fulfilling the role of the model of Q
of representing structurally all the relations
of all the elements of Q
with all the elements
of all the elements
of other structures so related,
model theory".

>
When you say

it's considered a lax sort of accommodation

who is it who is doing this considering?
Are they unaware that
there are proofs of isomorphistry?
Do they consider proofs lax accommodation?
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Then about how you find
properties of the function _in_ the limit
_as_ a limit,
it's that:
those properties don't exist at all,
that function doesn't exist at all
except _in the limit_.
Now, you're talking about
a family of functions that model
this not-a-real-function, in the limit,
yet, they are not it, in the limit.

>
Are you still talking there about
what you earlier were talking about in this way
⎛ the range of the function n/d
⎜ with 0 <= n < d and as d -> oo,
⎝ i.e., only in the infinite limit,
<RF>
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Isn't
the range of n/d with 0≤n≤d
the set ⟨0/d,1/d,...,d/d⟩ ?
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Isn't
limᵈᐧᐧᐧ⟨0/d,1/d,...,d/d⟩
the limit of a sequence of sets?
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Is the limit you're talking about
some limᵈᐧᐧᐧ f(d) which _does not_ fall between
lim.infᵈᐧᐧᐧ f(d) and  lim.supᵈᐧᐧᐧ f(d) ?
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What "limit" is it you're talking about?
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This way the extent, density, completeness,
and measure, aren't from being the union
of ranges of functions that model it,
they're the infinite integers in it.

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Ah, "infinite integers" is you brainstorming:
tossing creative options into the discussion.
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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".
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, so, the usual idea that "the
reals R are unique up to isomorphism the complete
ordered field", gets broken and is not considered
thorough.
Then as well I've written here field axioms for [-1, 1].
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'".
Anyways I've written non-standard field axioms
for [-1, 1] before so when I recall sitting
Algebra 301 and getting instructed "the complete
ordered field is unique, 'up-to-isomorphism'",
it's like "congratulations I made yon uniqueness
result a distinctness result".
And R, or R^2, and C, "up to isomorphism",
are _not_ equals.