Re: how (Aristotle says "potential is actual and actual is potential")

On 06/09/2024 09:46 PM, Jim Burns wrote:

On 6/9/2024 8:11 PM, Ross Finlayson wrote:

On 06/08/2024 10:47 AM, Jim Burns wrote:

On 6/8/2024 11:28 AM, Ross Finlayson wrote:

On 06/07/2024 10:35 PM, Jim Burns wrote:

>

It looks to me like
what you define the line.reals to be is
{0/d!,1/d!,…,d!/d!}.limit  =  ℚ∩[0,1]
>
Yes,
{0/d!,1/d!,…,d!/d!}.limit is the continuum limit.
>
"Continuum limit" means the distance between
nearest neighbors approaches 0, as it does in ℚ
>
"Continuum limit" does not mean "continuum".

>
I'm delighted that you note that the set ran(f)
by its values, "doesn't not" meet the definition
of least-upper-bound, then insofar as that it does.

>
Whether ran(f) meets the definition depends upon
what ran(f) is.
>
ℚ∩[0,1]  my (JB's) best guess at your (RF's) ran(f)
_does not_ meet the definition.
>
Is √½ = 0.70710678118... in ran(f) ?
Please explain.

>
What ratios?

>
I refer to the ratios in my best guess at ran(f)
>
| This putative f
| called EF the equivalency function,
| f(n) = n/d, 0 <= n <= d, d -> oo
| in the continuum limit,
|
Date: Fri, 31 May 2024 18:37:34 -0700
>
I guess
| 0 <= n <= d
means
{0,1,…,d}
>
I guess
| f(n) = n/d, 0 <= n <= d
means
f{0,1,…,d}  =
{0/d,1/d,…,d/d}
>
I guess
| f(n) = n/d, 0 <= n <= d, d -> oo
means
lim(d → ∞) f(n) = n/d, 0 <= n <= d  =
lim(d → ∞) {0/d,1/d,…,d/d}
>
My guess is that
ran(f)  =  lim(d → ∞) {0/d,1/d,…,d/d}
>
Ross Finlayson,
is  ran(f)  =  lim(d → ∞) {0/d,1/d,…,d/d}  ?
>
>
While you (RF) ponder whether to perform
a simple, easy task which would greatly reduce
the uncertainty around what you are talking about,
I will carry on as though you had answered "yes".
>
>
I know what  lim(d → ∞) {0/d,1/d,…,d/d}  is,
forc reasonable values of "set limit":
lim(d → ∞) {0/d,1/d,…,d/d}  =
⋂(0<dᵢ<∞) U(dᵢ<dᵤ<∞) {0/dᵤ,1/dᵤ,…,dᵤ/dᵤ)  =
ℚ∩[0,1]
>
It is the same as the limit of
the sub.sequence of factorial.denominators,
which is a nested.set sequence, so
lim(d → ∞) {0/d!,1/d!,…,d!/d!}  =
U(0<dᵤ<∞) {0/dᵤ!,1/dᵤ!,…,dᵤ!/dᵤ!)  =
ℚ∩[0,1]
>
√½ ∉ ℚ∩[0,1]
>

Is √½ = 0.70710678118... in ran(f) ?
Please explain.

>

I'm delighted that you note that
the set ran(f) by its values, "doesn't not"
meet the definition of least-upper-bound,
then insofar as that it does.

>
If ran(f) = ℚ∩[0,1]
⎛ which, if it isn't, I'd like you to say it isn't,
⎝ and, if it is, I'd like you to say it is,
then ran(f) "doesn't not not" meet the definition
of having  the least upper bound property.
>

These are only integer fractions
in the continuum limit, so the ordered field
doesn't even exist yet.

>
| With the order defined above,
| ℚ is an ordered field that has no subfield
| other than itself,
| and is the smallest ordered field,
| in the sense that every ordered field contains
| a unique subfield isomorphic to ℚ
|
https://en.wikipedia.org/wiki/Rational_number
>
ℚ isn't the compete ordered field.
ℚ is the smallest ordered field.
>
ℚ is the continuum limit, but
the continuum limit is not the continuum.
>

So, that
elements of the complete ordered field in [0,1],
like root two over two,
have values that are real values that
happen to equate to a value in ran(EF) in [0,1],
a unique value, and that,
there is no real value in [0,1] that
is not an element of ran(EF),
just has an existence result that
of the infinitely many distinct integers, and
the infinitely many distinct reals in [0,1],
they're 1 to 1.

>
Please explain.
>
Does
| f(n) = n/d, 0 <= n <= , d -> oo
  mean something else other than
| lim(d → ∞) {0/d,1/d,…,d/d}
?
>
Is your use of "limit" in
| lim(d → ∞) {0/d,1/d,…,d/d}
  something else other than
| ⋂(0<dᵢ<∞) U(dᵢ<dᵤ<∞) {0/dᵤ,1/dᵤ,…,dᵤ/dᵤ)
?
>
If something else, then what else?
>
>

>
>
I like where you're going with this.
>
For a while, there was a sort of trend in mathematics
called "reverse mathematics". Now, what this is, is
the idea to make a deconstructive account for any
particular result, to suss out only what axioms it
employs in its derivation, to help reflect that the
axiomatic dependencies it has, are reduced, then
that sub-theories of grander theories like ZF set
theory, with the usual attachment of least-upper-bound
as an axiom and measure 1.0 as an axiom for real
analysis or for "Hilbert's geometry with a postulate
of continuity and also a unit measure", the idea is
that "reverse mathematics" is a sort of "deconstructive
account", which is a term usually from postmodern
criticism, that I've appropriated for mathematical
foundations.
>
So, the idea of a reverse mathematics, is for
"minimal models", for what makes things
so results, of a model theory, which of course
is where model theory and proof theory are
equi-interpretable, overall.
>
Then, here, it's that the fractions of the unit,
with denominator d, are a simpler construction,
in a model where the greater ordered field isn't
yet necessary to exist.
>
I.e., all the laws that result of equalities and
inequalities, for equi-partitioning and the
uniform, and the modular, and the law of
large numbers, are only "nascent" as ratios,
rationals, they would so model a modular
section of rationals, yet needn't.
>
Then, as the inferences carries along making
the unit interval, "infinitely-divided", it's a
particular form, in a world of book-keeping
numerical resources, where "the infinity",
of it, is only for it.
>
Then, in the continuum limit, it actually
loses the character of being "infinitely-divisible",
as after, that it's "infinitely-divided". Now,
I know this gets into things like "the limit is
the sum" and "the limit is _not_ the sum",
about here that the limit is no different
than the sum, and, deduction arrives at
that the limit of a process that goes from
0 to 1, must have reached, touched, and passed
1/2, with regards to a subprocess or derived
process of it that has a limit, 1/2. Thusly here
there is that "the limit is the sum or the limit
is no different than the sum".
>
So, this sort of "deconstructive account" is
rather necessary to let the natural/unit
equivalency function its properties, for
example as something like "Hilbert's
postulate of continuity for geometry"
is arrived at as, necessary, and what was
underdefined, taken for granted, implicitly
assumed, and otherwise not reflecting
all its nature.
>
So, these "iota-values" as "iota-cuts",
are not the same as the rationals Q[0,1].
>
Also, they don't necessarily have "addition
and multiplication" together like you'd
expect from the usual central notion of
trichotomy of real-valued numbers and
membership in a most usual algebra,
the field, with addition and multiplication
together. Instead, a deconstructive account
arrives at that "iota-sums" and "iota-multiples"
are different things, because and including
that the operations of arithmetic are broken
down to be increment and division not
addition and multiplication.
>
(Which they are, ....)
>
So, it's not yet the later algebras that share
all the relations about the integer lattice,
it's nascent, and is what it is where it may be,
and is what it is.
>
Then, that iota-sums and iota-multiplies
fall apart or don't have the expected
combined formalism that they later
do for arithmetic proper and centrally
and classically, is a feature of these things.
>
So, this "only-diagonal" for the "anti-diagonal"
gets in this manner firstly its own constructive
formalism, relating to integers and specifically
the modular character of the integers, then
later sees it makes arises for itself the simple
result in standard set theory from analysis
or the completions of these things the
limits which must be infinite limits which
here is a continuum limit, this thing.
>
>