On 10/29/2024 07:38 AM, Jim Burns wrote:
On 10/28/2024 5:45 PM, Ross Finlayson wrote:
On 10/28/2024 01:15 PM, Jim Burns wrote:
>
[...]
>
Of course
you can keep in mind that "continuum limit"
>
...AKA letting lattice spacing approach 0...
>
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.
>
I thought that I at least knew what it was
which you are talking about. Not so much, now.
>
Note, on the Dirac delta not.a.function:
It does not have a single non.0 value
such that it integrates to 1
>
That is an informal gloss of the not.a.function.
A useful intuition.builder, but
the existence of the Dirac delta doesn't prove
that a thick point like that exists.
>
----
Well, the property that
"a set is complete
if it contains each of its least-upper-bounds",
>
...if it contains,
for each of its bounded nonempty subsets,
a least upper bound...
>
is completeness,
is the one ascribed to line-reals,
ran(EF).
>
It just clicked with me that
you (RF) have been saying "range" and
I (JB) have been thinking "image".
My mistake.
>
Perhaps you should be saying "image".
>
It is the image EF(N) which is countable,
assuming a single.valued ("Cartesian"?) function.
>
The range is a superset of the image.
It might or might not also be countable.
>
You can't use the range to argue against
Cantor's uncountability proof.
Having a countable subset doesn't prove
countability,
>
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.
>
You have completely lost me.
>
as either a finite set contains its upper-bound,
or, a finite set has a next-greater upper-bound,
>
A finite nonempty set contains its upper bound.
So, yes,
a finite set has the least.upper.bound property.
Was it finite sets you've been talking about?
>
A finite set has the LUB property, yes,
but a finite set isn't a superset of Q
The smallest superset of Q with the LUB property
is R, which is what I've been talking about.
>
>
That ran(EF) is complete is rather simple, i.e. it's trivial,
in as to where, as you for whatever reason chose to simply
ignore: the LUB can be simply either max(n), or, max(n)+1,
being or making a critical point.
You'll notice it's also not the usual open topology,
though it's a "continuous topology" as has been defined
here "a topology its own initial and final topology".
Given function theory and topology, then arithmetic,
it's pretty great.
The Dirac delta, for its usual use in usual establishments
like all matters of transform theory and with regards to
establishing the Fourier-style analysis, is _nowhere non-zero_,
except at zero. I.e. of course it has _no analytical character
at all_, except at zero. (Of course the usual notion of the
unit impulse function being translated left or right translates
the area of it, the real analytical character.) As a "limit
of a family of standard functions, not-a-real-function", indeed
it's been variously called "not a function, some kind of functional",
yet it's a relation.
It's like I'm reading Cassirer the other day and he's doing
the breakdown about terms and functions and relations and
functions, and it's like, I wonder what he might find later,
where d'Espagnat at least is like "I get what Cassirer says
and it's simple to live in an inductive universe, yet,
for sure the de Broglie established continuous waves".
That is to say, as a matter of relation, the function,
for function theory, the Dirac delta, IS, a function.
And: it's analytical character, its content, is nowhere
non-zero except at zero, and, its area equals one.
Those are its only properties of analytical import in
all the regular and quite thoroughly successful applications.
The Equivalency Function has also analytical character,
which is rather unique among discrete functions, functions
with discrete domain, and particularly it exhibits a
doubling space, and also its area equals one.
So, much like Dirac delta is a pdf and CDF of a trivial sample,
the Equivalency Function also is, of naturals at uniform random.
That though is merely part of its special and unique character.
About "domain and range" or "image and co-image" and this kind
of thing, usage, here is that the range is not the "support set"
concept, that is to say, the range includes _only_ elements that have
inverses in the domain, there are no unmapped elements in the range.
Your fact about R and Q, is not relevant to this N and ran(EF),
except that the one is about the uncountability of the
complete-ordered-field, while the other is about the completeness of an
infinitude.
How do you feel when the logic's
primitives are relations instead of terms?
It can be either way - sort of like part or set theory.
How many different unit fractions are lessorequal than all unit fractions?
Re: How many different unit fractions are lessorequal than all unit fractions?