On 10/28/2024 01:15 PM, Jim Burns wrote:
On 10/26/2024 12:22 PM, Ross Finlayson wrote:
On 10/25/2024 12:44 PM, Jim Burns wrote:
On 10/23/2024 1:38 PM, Ross Finlayson wrote:
>
[...] that the most direct mapping between
discrete domain and continuous range is
this totally simple continuum limit of n/d
for natural integers as only d is not finite
and furthermore
is constant monotone strictly increasing
with a bounded range in [a,b], an infinite domain.
>
The continuum limit is not the continuum.
I know:
it sounds like it should be, but it isn't.
>
The continuum limit is
the spacing of a lattice approaching 0.
>
If we are _already_ working in the continuum,
the lattice points _in the limit_
are sufficient to
uniquely determine a _continuous_ function.
For many purposes,
uniquely determining a continuous function
is sufficient for that purpose.
>
But that isn't the continuum.
In a continuum,
each split has a point at the split,
either one which ends the foresplit
or one which begins the hindsplit
_which is different_
>
Do you yet recall that these properties:
extent density completeness measure,
would establish that ran(f) that being ran(EF)
is a continuous domain?
>
I still recall
you claiming that
EF(ℕ) is Dedekind.complete [0,1]ᴿ
You establishing that, not so much.
>
Do you recall that the continuum limit
is not the continuum?
>
The continuum limit is
letting the spacing of a lattice approach 0.
>
Then that completeness is as simply trivial
that it's defined that
the least-upper-bound of the set is
an element of the set, that
for f(...m) that f(m+1) is this?
>
Consider your
n/d n->d d->oo
>
Is that complete real interval [0,1]ᴿ ?
>
If
n/d n->d d->oo
means
limᵈ⁻ᐣⁱⁿᶠlimⁿ⁻ᐣᵈn/d
then no.
>
limᵈ⁻ᐣⁱⁿᶠlimⁿ⁻ᐣᵈn/d = limᵈ⁻ᐣⁱⁿᶠd/d = 1
>
>
If
n/d n->d d->oo
means
limᵈ⁻ᐣⁱⁿᶠ[0,d]ᴺ/d
(integer.interval [0,d]ᴺ ᵉᵃᶜʰ/d)
then also no.
>
⟨ [0,d]ᴺ/d ⟩ᵈ⁼¹ᐧᐧᐧⁱⁿᶠ is
the infinite sequence of sets [0,d]ᴺ/d
>
E([0,c]ᴺ/c) is an end.segment of ⟨ [0,d]ᴺ/d ⟩ᵈ⁼¹ᐧᐧᐧⁱⁿᶠ
E([0,c]ᴺ/c) = { [0,c]ᴺ/c [0,c+1]ᴺ/(c+1) [0,c+2]ᴺ/(c+2) ... }
>
⋃E([0,c]ᴺ/c) is the supremum of end.segment E([0,c]ᴺ/c)
>
Each end.segment.supremum ⋃E([0,c]ᴺ/c) is
a superset of any set.limit of E([0,c]ᴺ/c)
-- if that set.limit exists.
>
⟨ ⋃E([0,c]ᴺ/c) ⟩ᶜ⁼¹ᐧᐧᐧⁱⁿᶠ is
an infinite sequence of supersets of
any set.limit of ⟨ [0,d]ᴺ/d ⟩ᵈ⁼¹ᐧᐧᐧⁱⁿᶠ
-- if that set.limit exists.
>
⋂⁰ᑉᶜ⋃E([0,c]ᴺ/c) is
also a superset of
any set.limit of ⟨ [0,d]ᴺ/d ⟩ᵈ⁼¹ᐧᐧᐧⁱⁿᶠ
-- if that set.limit exists.
>
However,
⋂⁰ᑉᶜ⋃E([0,c]ᴺ/c) = rational interval [0,1]ꟴ
[0,1]ꟴ is not Dedekind.complete.
Each subset of [0,1]ꟴ is not Dedekind.complete.
Any set.limit of ⟨ [0,d]ᴺ/d ⟩ᵈ⁼¹ᐧᐧᐧⁱⁿᶠ
is not Dedekind.complete
-- if that set.limit exists.
>
Either
limᵈ⁻ᐣⁱⁿᶠ[0,d]ᴺ/d ≠ [0,1]ᴿ
because complete [0,1]ᴿ ⊈ rational [0,1]ꟴ
or
limᵈ⁻ᐣⁱⁿᶠ[0,d]ᴺ/d ≠ [0,1]ᴿ
because limᵈ⁻ᐣⁱⁿᶠ[0,d]ᴺ/d isn't anything.
>
>
If
n/d n->d d->oo
means
[0,1]ᴿ
_by definition_
then who cares?
>
You have drawn a conclusion
no more sure.footed than
whatever that intuition was which
led you to make that definition.
And, anyway, a bare intuition is not shareable.
>
That's why we make proofs.
>
Then, about the "anti" and "only", and there being
this way that this ultimately tenuous continuum
limit (I'm glad at least we've arrived at that
being a word, "continuum-limit"),
[...]
makes for that
its range is a "continuous domain" itself
>
No.
That's not what the continuum limit is.
https://en.wikipedia.org/wiki/Continuum_limit
>
>
Well, the property that "a set is complete if
it contains each of its least-upper-bounds",
is completeness, is the one ascribed to line-reals,
ran(EF).
It's an upper-bound, it's least, rather trivially
as either a finite set contains its upper-bound,
or, a finite set has a next-greater upper-bound,
one or the other of those is discernible and
one or the other of those exists and
one or the other of those is an upper-bound and
one or the other of those is least,
thusly the least-upper-bound "LUB" property holds,
completeness.
Then, why you arrived at your first "no", or as with
regards to your usual principle of "not.first.false",
and I'm not so much averring a simply contradictory
scheme of induction, though there is one to break it down,
is that these are yet "not members of the complete ordered
field the field-reals" these "standard iota-value infinitesimals
the line-drawing of the line-reals".
Then, they both share the integer continuum or lattice,
then as well each embody the "non-integer part",
completely.
Also keep in mind there are signal-reals, a _third_
definition of continuity, that Dedekind's definition
(or, Eudoxus' definition since it's same up above
since thousands of years ago vis-a-vis the late 1900's),
... that Dedekind's definition is not unique and then
furthermore not sufficient: the "replete", complete.
It's as well then that "line-reals" such as they are
also are as of the Demokrites or Democritus the "atomism",
with as regards to the "infinitely-divisible", and with
regards to Aristotle's exposition(s) of BOTH of these,
as with regards to Aristotle's later mainstream "there
is no atomism", as with regards to both approaches.
So, the idea that FIRST that there's constructively:
"extent density completness measure",
establishes the requirements and desiderata of
constructively (constructivistically) a continuous
domain (simply "the real-valued"), then also
that there's established that
"it's not contradicted by the arguments for uncountability"
because via inspection it's not the type of function
that resulting properties of the infinitely-divisible
this a-tomic individua, then that
"the same sort of approach as anti-diagonal argument showing
there's a missing element otherwise is used to make
an only-diagonal argmument there's at least this non-Cartesian
function"
has it thusly exists this way, doesn't not exist this way,
and furtherwise co-exists with that way.
Of course you can keep in mind that "continuum limit"
was not admitted to mathematics since modern mathematics
since at least for the past few decades, since when for
example "everybody's favorite non-standard not-a-real-function
with standard analytical character, function, Dirac's delta
the unit impulse function", has of course that EF is a
non-standard and not-a-real-function, in the usual sense,
yet has real analytical character itself.
Of course it stands up for itself, exists, then
gets along, doesn't not exist, then furthermore
makes to co-exist, as part of the universe of
mathematical objects.
How many different unit fractions are lessorequal than all unit fractions?
Re: How many different unit fractions are lessorequal than all unit fractions?