Musatov's infinity

[Julio Di Egidio]

On Saturday, 30 April 2022 at 09:37:41 UTC+2, Ross A. Finlayson wrote:
> On Friday, April 29, 2022 at 12:55:55 PM UTC-7, Mostowski Collapse wrote:
> > 3 category mistakes in one MSE question.
> Shut up Musatov / answer the paradoxes.

You give me an idea:

Take the real interval [0,1], which is of course
continuous, compact, closed and indeed all
that we want!

Now, for m > 0, map it to the interval [0,m] by
f(x) = m*x, the inverse map being g(y) = y/m.

Now let m go to infinity! By the Mostowski
collapse lemma... (by isomorphism...) this
interval stays continuous, compact, closed
and indeed all that we want!

No? And why not?

Julio

[Ross A. Finlayson]

It's rather direct the relation [0,1)~[0,1] as that of [0,oo)~[0,oo].

The compactification of the naturals (a point at infinity) and
that it's included or "the naturals are compact", helps to illustrate
that more than "finite" they're "infinite".

The very notion of a "scalar" infinity, is more than less the "modular".

These are of course plainly "only numbers", where though whatever
models the continuous results the same fine-ness of the "smooth".

That as "only numbers as only models of only objects", when the
objects are finite, is about the "continuous and discrete".

It's rather as once I wrote "field operations for [-1,1]".

[Julio Di Egidio]

On Sunday, 1 May 2022 at 18:50:22 UTC+2, Ross A. Finlayson wrote:

> It's rather direct the relation [0,1)~[0,1] as that of [0,oo)~[0,oo].

That's rather and quite precisely the "standard problem".
Indeed, that completely mystifies what we were saying,
which rather looks like [0,1)~[0,oo) vs [0,1]~[0,oo].

> The very notion of a "scalar" infinity, is more than less the "modular".

That notion for sure.

> These are of course plainly "only numbers", where though whatever
> models the continuous results the same fine-ness of the "smooth".

That's what needs *proving*. Just repeating it won't do.

> It's rather as once I wrote "field operations for [-1,1]".

Which adds exactly nothing to our problem, but at least
is closed...

Put up or shit up, Ross.

Julio

[Julio Di Egidio]

> Put up or sh[*]t up, Ross.

Eh, that was a typo, sorry about that.

Julio

[Ross A. Finlayson]

Why no worries, except here the matter:
"there isn't a continuous function between [0,1) and [0,1]",
is that
"there's a function only discontinuous at one point, that is".

I.e., "what transform results [0,1) <-> ? <-> [0,1]", is
here related exactly to "... [0,oo) <-> ? <-> [0,oo]",
that this "SCALE" the scalar is that measure zero defines
a scale, but really that's "defined" by the infinity, first.

Id est it's a "scalar infinity".

Then, the field operations I wrote, basically are as after
having a linear functions, that though unbounded,
results in the asymptotic, yet a finite value.
(That it achieves as a limit.)

Of course the usual notation for intervals by their
endpoints and inclusion is written the usual square-bracket-[]
for included or closed endpoints and parenthetical-bracket-()
for not-included and open "endpoints", makes for that what
is a matter of the _topology_, i.e. definition of open and not-open,
definition of closed and not-closed, and "when open and closed
are considered complements and each dense", basically is for
a simpler topology on a [0,1) and [0,1] that results as because
there are multiple models of real numbers, there are arranged
relations the functions by the endpoints and middles instead of
each the point-set. (For and after topology's sake.)

[Jim Burns]

On 5/3/2022 4:25 PM, Ross A. Finlayson wrote:

> Why no worries, except here the matter:
> "there isn't a continuous function between [0,1) and [0,1]",

(i)
Who are you quoting?

(ii)
Why do they think f: [0,1) --> [0,1] isn't continuous?

f(x) = 1 - (1 - x/(3/4))^2

> is that
> "there's a function only discontinuous at one point,
> that is".

(iii)
Functions discontinuous at one point is not the same.
g is discontinuous at one point
g(x) =
{ 0 for x < 1/2
{ 1 for x >= 1/2

[Julio Di Egidio]

On Tuesday, 3 May 2022 at 22:25:57 UTC+2, Ross A. Finlayson wrote:
> On Monday, May 2, 2022 at 5:08:15 AM UTC-7, ju...@diegidio.name wrote:
> > On Monday, 2 May 2022 at 13:37:49 UTC+2, Julio Di Egidio wrote:
> > > On Sunday, 1 May 2022 at 18:50:22 UTC+2, Ross A. Finlayson wrote:
> > >
> > > > It's rather direct the relation [0,1)~[0,1] as that of [0,oo)~[0,oo].
> > > That's rather and quite precisely the "standard problem".
> > > Indeed, that completely mystifies what we were saying,
> > > which rather looks like [0,1)~[0,oo) vs [0,1]~[0,oo].

<snip>

> Why no worries, except here the matter:
> "there isn't a continuous function between [0,1) and [0,1]",
> is that
> "there's a function only discontinuous at one point, that is".
>
> I.e., "what transform results [0,1) <-> ? <-> [0,1]", is
> here related exactly to "... [0,oo) <-> ? <-> [0,oo]",
> that this "SCALE" the scalar is that measure zero defines
> a scale, but really that's "defined" by the infinity, first.

There is a gap there that wishful thinking won't fill. And there is even more the matter of not conflating potential with the actual. All you keep doing is simply mangling the point.

Never mind, thanks for replying.

Julio

[Ross A. Finlayson]

Just collecting the words, ....

What is continuous is a function or domain, at a point,
that "C^\infty are the most usual continuous functions,
that being the space of those containing all them, and
besides a space in the usual sense of through all them",
continuous domains are "all continuous functions defined
on this continuous domain are continuous throughout it".

[sergio]

On 5/3/2022 3:25 PM, Ross A. Finlayson wrote:
> On Monday, May 2, 2022 at 5:08:15 AM UTC-7, ju...@diegidio.name wrote:
>> On Monday, 2 May 2022 at 13:37:49 UTC+2, Julio Di Egidio wrote:
>>> On Sunday, 1 May 2022 at 18:50:22 UTC+2, Ross A. Finlayson wrote:
>>>

>
>
> Why no worries, except here the matter:
> "there isn't a continuous function between [0,1) and [0,1]",

Wrong.

[Jeff Barnett]

On 4/30/2022 2:07 AM, Julio Di Egidio wrote:
> On Saturday, 30 April 2022 at 09:37:41 UTC+2, Ross A. Finlayson wrote:
>> On Friday, April 29, 2022 at 12:55:55 PM UTC-7, Mostowski Collapse wrote:
>>> 3 category mistakes in one MSE question.
>> Shut up Musatov / answer the paradoxes.
>
> You give me an idea:
>
> Take the real interval [0,1], which is of course
> continuous, compact, closed and indeed all
> that we want!

An interval is continuous?? I'm not familiar with that terminology. I am
familiar with calling it connected though. So please define a continuous
interval for me. Thank you.

> Now, for m > 0, map it to the interval [0,m] by
> f(x) = m*x, the inverse map being g(y) = y/m.
>
> Now let m go to infinity! By the Mostowski
> collapse lemma... (by isomorphism...) this
> interval stays continuous, compact, closed
> and indeed all that we want!
>

> No? And why not?--
Jeff Barnett

[Ross A. Finlayson]

Domains where identity is a continuous function?

[Julio Di Egidio]

On Wednesday, 4 May 2022 at 18:26:53 UTC+2, Ross A. Finlayson wrote:
> On Tuesday, May 3, 2022 at 9:43:50 PM UTC-7, Jeff Barnett wrote:
> > On 4/30/2022 2:07 AM, Julio Di Egidio wrote:
> > > On Saturday, 30 April 2022 at 09:37:41 UTC+2, Ross A. Finlayson wrote:
> > >> On Friday, April 29, 2022 at 12:55:55 PM UTC-7, Mostowski Collapse wrote:
> > >>> 3 category mistakes in one MSE question.
> > >> Shut up Musatov / answer the paradoxes.
> > >
> > > You give me an idea:
> > >
> > > Take the real interval [0,1], which is of course
> > > continuous, compact, closed and indeed all
> > > that we want!
> >
> > An interval is continuous?? I'm not familiar with
> > that terminology.

(Things foundational should be read in their broadest
possible sense unless noted otherwise. There is
a notion of "continuity" and that comes before any
specific incarnation of it.)

> > I am
> > familiar with calling it connected though. So please
> > define a continuous interval for me. Thank you.
>

> Domains where identity is a continuous function?

That's cute I sort of begs the question, doesn't it: let
our numbers be those functions... I cannot in fact
find it, but what I have in mind looks more like some
"topological" definition based on "never empty
arbitrary small neighbour-hoods": yet that's not
enough to distinguish a real interval from a rational
one...!?

For now, I won't deny I am still not sure I can even
see a solution here even if I/we bump into it.

Julio

[Ross A. Finlayson]

Connectedness is gaplessness.

Connections and density, makes rationals exhaust,
the rationals are not gapless, but are dense in their
real values in themselves.

This is where "a continuous function means delta-epsilonics
according to correspondingly small and neighborly
that a 'continuum function' generally applies".

This is of course all formal and , ..., modern.

[Jim Burns]

On 5/4/2022 12:26 PM, Ross A. Finlayson wrote:

> Domains where identity is a continuous function?

In any domain in which continuity makes any sense at all,
identity is a continuous function.

Even if domains 𝑋 and 𝑌 can't have a metric defined,
they may have collections 𝒪(𝑋) and 𝒪(𝑌) of _open sets_
defined.

A function 𝑓 from 𝑋 to 𝑌 is _continuous_ iff,
for each open set 𝑉 in 𝒪(𝑌),
its _inverse image_ 𝑈 = 𝑓⁻¹(𝑉) is an open set in 𝒪(𝑋)

Suppose 𝑖𝑑 is the identity from 𝑋 to 𝑋
If 𝑈 is open, then 𝑖𝑑⁻¹(𝑈) = 𝑈 is open.
and thus 𝑖𝑑 is continuous.

[Jeff Barnett]

Whether an ID function is continuous or not depends on the topology of
the space if I'm not mistaken. My question was simply "What is a
continuous interval?" Any references to either a peer reviewed and
published article or a text book published by other than the Vanity
Press that defines "continuous interval"?
--
Jeff Barnett

[Jim Burns]

On 5/4/2022 7:20 PM, Jeff Barnett wrote:
> On 5/4/2022 12:20 PM, Jim Burns wrote:
>> On 5/4/2022 12:26 PM, Ross A. Finlayson wrote:

>>> Domains where identity is a continuous function?

>> A function 𝑓 from 𝑋 to 𝑌 is _continuous_  iff,
>> for each open set 𝑉 in 𝒪(𝑌), its
>> _inverse image_ 𝑈 = 𝑓⁻¹(𝑉) is an open set in 𝒪(𝑋)
>>
>> Suppose 𝑖𝑑 is the identity from 𝑋 to 𝑋
>> If 𝑈 is open, then 𝑖𝑑⁻¹(𝑈) = 𝑈 is open.
>> and thus 𝑖𝑑 is continuous.
>
> Whether an ID function is continuous or not depends on
> the topology of the space if I'm not mistaken.

Under the definition I gave, 𝑖𝑑 is continuous.

I think that that is a standard definition for
topological spaces.
https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces

> My question was simply "What is a continuous interval?"

From context, it sounds to me as though
Julio Di Egidio means "connected".
You pays your money and you takes your choice.

<JDE>

>
> Take the real interval [0,1], which is of course
> continuous, compact, closed and indeed all
> that we want!
>

</JDE>

[Jeff Barnett]

On 5/4/2022 6:56 PM, Jim Burns wrote:
> On 5/4/2022 7:20 PM, Jeff Barnett wrote:
>> On 5/4/2022 12:20 PM, Jim Burns wrote:
>>> On 5/4/2022 12:26 PM, Ross A. Finlayson wrote:
>
>>>> Domains where identity is a continuous function?
>
>>> A function 𝑓 from 𝑋 to 𝑌 is _continuous_  iff,
>>> for each open set 𝑉 in 𝒪(𝑌), its
>>> _inverse image_ 𝑈 = 𝑓⁻¹(𝑉) is an open set in 𝒪(𝑋)
>>>
>>> Suppose 𝑖𝑑 is the identity from 𝑋 to 𝑋
>>> If 𝑈 is open, then 𝑖𝑑⁻¹(𝑈) = 𝑈 is open.
>>> and thus 𝑖𝑑 is continuous.
>>
>> Whether an ID function is continuous or not depends on
>> the topology of the space if I'm not mistaken.
>
> Under the definition I gave, 𝑖𝑑 is continuous.

Assuming 1) the domain and range are the same set and 2) the assumed
topology is the same for both, ID is continuous for every such set.

The inverse image (just to be pedantic) of every open set is open since
that set is the image and the topology is the same.

So we now have the continuous set theorem ready for publication: For all
sets (open, closed, neither, both) in a topology, the ID function with
that set as domain and range is continuous!!!!!!!!!!!

Corollary: All sets are continuous!! Here we put the above theorem to
work for the first time.

Since my original question was" what's a continuous interval?", we have
found out that every interval is continuous by your definition; as is
every other set!

I think at this time I really want to know what concept one was actually
trying to convey when the term "continuous interval" was introduced.
Please share with us. Somebody.

I could be wrong but I think this conversation has become a victim of a
brain fart.

> I think that that is a standard definition for
> topological spaces.
> https://en.wikipedia.org/wiki/Continuous_function#Continuous_functions_between_topological_spaces
>
>
>> My question was simply "What is a continuous interval?"
>
> From context, it sounds to me as though
> Julio Di Egidio means "connected".
> You pays your money and you takes your choice.
>
> <JDE>
>>
>> Take the real interval [0,1], which is of course continuous, compact,

>> closed and indeed all that we want!--
Jeff Barnett

[Julio Di Egidio]

On Thursday, 5 May 2022 at 08:41:56 UTC+2, Jeff Barnett wrote:

> I think at this time I really want to know what concept one was actually
> trying to convey when the term "continuous interval" was introduced.
> Please share with us. Somebody.
>
> I could be wrong but I think this conversation has become a victim of a
> brain fart.

I had replied to that, and you don't notice? And it was already quite a
clueless question of the Dunning Kruger kind. Maybe it's just you out
of your depth and then behaving like a perfect dick?

Julio

[Jim Burns]

On 5/5/2022 2:41 AM, Jeff Barnett wrote:
> On 5/4/2022 6:56 PM, Jim Burns wrote:
>> On 5/4/2022 7:20 PM, Jeff Barnett wrote:
>>> On 5/4/2022 12:20 PM, Jim Burns wrote:
>>>> On 5/4/2022 12:26 PM, Ross A. Finlayson wrote:
>>
>>>>> Domains where identity is a continuous function?

> Since my original question was" what's a continuous
> interval?",

I was responding to Ross's comment.

Apparently deleting everything but Ross's comment
was not enough of a clue.
Sorry for your confusion.

>
> So we now have the continuous set theorem ready
> for publication: For all sets (open, closed,
> neither, both) in a topology, the ID function
> with that set as domain and range is continuous
> !!!!!!!!!!!

Done.

https://proofwiki.org/wiki/Identity_Mapping_is_Continuous

> I could be wrong but I think this conversation has
> become a victim of a brain fart.

I will certainly bear this exchange in mind
in the future.

[Ross A. Finlayson]

... any of which are already continuous domains.

[Ross A. Finlayson]

Furthermore: up above set theory, and in set theory, i.e.
consistently, are all sorts what include most all set theory.

[Julio Di Egidio]

Or no thanks: you have deranged the whole thread, anyway as
soon as I say you are going off a tangent everybody pushes right
along that direction, so I suppose I should rather no-thanks all.

For a minimum of clarification, I had not in fact asked about
[0,1], I had asked about [0,oo], and that interval in standard
mathematics DOES NOT EVEN EXIST! Just to begin with: as
for the subtle nuances of a metaphoric cross metonymic
communication, indeed WTF are you even talking about...

I could rather try and narrow it down, but at this point I am
simply depressed.

Julio

[Ross A. Finlayson]

"Number Theory" has for example "infinity", in it, for itself,
for that the primary theory has primary elements.

sets
partitions
wholes

numbers

objects of geometry

words

that "Number Theory" and "[0,oo], the interval
in the numbers that includes an infinity", makes
that "Number Theory" and "point at infinity" have
that the very same "infinity" is in the numbers.

So, for category theory or a model of operators,
or fundamentally enquiry, theory, number theory
has its own place under formalism: primary.

So, where "point-at-infinity" basically doesn't exist
in "unbounded finite integers" but does exist from
"perspective: makes these angles and the point
at infinity look like the point at infinity", that ordinals
are so "primary", before and then counting numbers,
and so on.

Then, a real continuous framework, reflects what models
under theories establish both the ordinal, "locally, and totally"
where all that is number theory's and geometry's.

Thus, the spiral, space-filling curve, bla bla bla,
is Archimedes again, in a theory in time.

The spiral space-filling curve and "this is a continuum",
makes for most the friendliest and usual [0,1], one unit interval.

With all geometry on it of course ....

The spiral space filling curve from axiomless geometry:
it's most usual representation as a line segment.

I find it easiest to put _that_ in the theory first:
then I can care and think much less: from what
all "the theory" establishes.


Though, for having that in the theory first, it's much
later than the first theory "there exist straight lines".

It is as much a, "final", theory.

[Julio Di Egidio]

On Saturday, 7 May 2022 at 17:59:34 UTC+2, Ross A. Finlayson wrote:

> "Number Theory" has for example "infinity", in it, for itself,

> for that the primary theory has primary elements. [...]

> Then, a real continuous framework, reflects what models
> under theories establish both the ordinal, "locally, and totally"
> where all that is number theory's and geometry's.

You keep missing the point:
<< With these definitions, [the extended real number system]
is not even a semigroup, let alone a group, a ring or a field as
in the case of [the real numbers]. >>
<https://en.wikipedia.org/wiki/Extended_real_number_line#Algebraic_properties>

Julio

[Julio Di Egidio]

Eventually, I think the answer is yes: as with
something as simple as the affinely extended
reals (*) we actually already get benefits in
terms of topological properties:

<< Without allowing functions to take on infinite
values, such essential results as the monotone
convergence theorem and the dominated
convergence theorem would not make sense. >>

<< Moreover, with this topology, [the extended
reals] is homeomorphic to the unit interval [0,1].
Thus the topology is metrizable, corresponding
(for a given homeomorphism) to the ordinary
metric on this interval. >>

The problem to me remains that we miss most
of the algebraic properties, up to having a field
of these, so better than just the extended real
number line" is needed.

Julio

(*) <https://en.wikipedia.org/wiki/Extended_real_number_line>

[Ross A. Finlayson]

All its standard members fall in one, though.

(Properties of real numbers, scale.)

The extended real number system first includes (all) the real numbers.


Then that there are straight lines and right angles,
is for two objects 1 unit apart as (seen) from infinity.

Straight lines, right angles, and an infinite ray,
have Cantor space as a square in numbers
while column in words.

Or "Two to the Omega".

There is extending the real numbers with
"this is the one that doesn't exist" and
"this is the one that does exist".

That the extension, here the intension:
all the standard members are same each
the real numbers - under free intensionality
or "the laws of and operators about real numbers",
here the extended is in two senses: "more than
real numbers" also "more, real number".

Then, infinity is singular and can be considered
for thusly its "behavior" otherwise in "algebra
of infinites and what reduces", that for singular
formulas, it's otherwise a plain and constant term.

For functions that are integrated over an infinite
domain yet still have a finite volume (area), of
course this includes functions that are non-zero
and even positive and negative, beyond any
given real bound. (Thus establishing reduction
in algebra, exhaustion, dissipation, perspective,
..., for usually writing "oo" for infinity where for
example its unboundedness is in effect.)

Limit: goes to infinity, goes to limit, goes to zero.

("One-sided.")

I find it easier a ray of real numbers first then
for the "all the real numbers, positive and negative".


I usually think of terms automatically in either
sign-magnitude, additive offsets, or quotients.

[Julio Di Egidio]

On Saturday, 7 May 2022 at 21:34:04 UTC+2, Ross A. Finlayson wrote:

> Limit: goes to infinity, goes to limit, goes to zero.

Shut up Musatov / just shut up.

*Plonk*

Julio

[Ross A. Finlayson]

It only means "there are limits that exist, because some
other 'limit' goes to infinity, otherwise exists, or reduces
to nothing", it's simply enough this way "so the rule that
establishes these limits, holds this way, in numbers most
written as infinity".

It is all so easy from fundamental principles -
well thank you then I will shut up - "not having
much more to say".

Not needing much more to say, ....

No, really, though, I have been waiting years to be trolling
this way - I think we can agree now that before there was
some hard trolling - trolling "mathematics" - which we can
agree is a large body of fundamental and human knowledge.

Next is "here is a thing that thinks it".

[Julio Di Egidio]

On Sunday, 8 May 2022 at 02:13:30 UTC+2, Ross A. Finlayson wrote:

> Not needing much more to say, ....
>
> No, really, though, I have been waiting years to be trolling
> this way - I think we can agree now that before there was
> some hard trolling - trolling "mathematics" - which we can
> agree is a large body of fundamental and human knowledge.

There is trolling and trolling, and I have not found
anything to work with in your trolling in years, in
fact ever more systematically you just disrupt, here
as elsewhere (just my opinion of course). And this
thread was indeed my mathematical challenge *to
you* (I hope so much was clear at least to you): the
last one, as, modulo unexpected news, this to me
closes the case.

> Next is "here is a thing that thinks it".

Next is a flat EEG...

Have fun,

Julio

[Julio Di Egidio]

On Saturday, 7 May 2022 at 21:25:18 UTC+2, Julio Di Egidio wrote:
> On Saturday, 30 April 2022 at 10:07:06 UTC+2, Julio Di Egidio wrote:

<snip>

> > Take the real interval [0,1], which is of course
> > continuous, compact, closed and indeed all
> > that we want!
> >
> > Now, for m > 0, map it to the interval [0,m] by
> > f(x) = m*x, the inverse map being g(y) = y/m.
> >
> > Now let m go to infinity! By the Mostowski
> > collapse lemma... (by isomorphism...) this
> > interval stays continuous, compact, closed
> > and indeed all that we want!
> >
> > No? And why not?
>
> Eventually, I think the answer is yes: as with
> something as simple as the affinely extended
> reals (*) we actually already get benefits in
> terms of topological properties:

<snip>

> The problem to me remains that we miss most
> of the algebraic properties, up to having a field
> of these, so better than just the "extended real
> number line" is needed.

We go up to the surreal numbers to recover all
the algebraic properties, still the essential
problem remains: defining the *last* ordinal.

I am going to look into positive set theory:
<https://en.wikipedia.org/wiki/Positive_set_theory>

Stay tuned...

Julio

[Julio Di Egidio]

Flash news / big news! omega-hyperuniverse!!
With the wonderful compactification of course...

It's pretty much what I have been looking for/blabbing
about for years, isn't it:

<< In this theory, the class of natural numbers (considered
as finite ordinals) is not closed and acquires an extra
element “at infinity” (which happens to be the closure of
the class of natural numbers itself). >>

Also, quite beautifully:

<< The ordinals are defined by a non-positive condition,
and do not make up a set, but it is interesting to note that
the closure CL(On) of the class On of ordinals is equal to
OnU{CL(On)}; the closure has itself as its only unexpected
element. >>

(Is there a gap there?! Kidding... but, isn't that closure itself
omega for the omega-hyperuniverse?)

Now back to implementing a provably correct mathematics:
which might easily take me the next five years until I have any
news.

Julio

[Ross A. Finlayson]

Sounds good to me, ....

It is quantifiers.

[Ross A. Finlayson]

Really, this sounds much better about
the features of ordinal numbers after
quantification you mention - you are
noting that there are many of these
types of features, that are primary,
in relevance.


Abstractly, ....

[Julio Di Egidio]

On Friday, 13 May 2022 at 12:30:04 UTC+2, Ross A. Finlayson wrote:
> On Friday, May 13, 2022 at 12:18:01 AM UTC-7, Ross A. Finlayson wrote:
> > On Tuesday, May 10, 2022 at 9:30:57 AM UTC-7, ju...@diegidio.name wrote:

<snip>

> > > Now back to implementing a provably correct mathematics:
> > > which might easily take me the next five years until I have any
> > > news.
> >

> > Sounds good to me, ....
> >
> > It is quantifiers.
>
> Really, this sounds much better about
> the features of ordinal numbers after
> quantification you mention - you are
> noting that there are many of these
> types of features, that are primary,
> in relevance.
>
> Abstractly, ....

There are many ingredients to that cake: the topology and
the algebra meet over the positive sets, and these are our
numbers (and spaces, and geometries, and we should look
into the Univalent Axiom and how that fits/relates...); there
is also a dimension of language, and here (I think) we just
need a dependently-typed functional(+logical?) system
(not as easy as it sounds finding a workable one); there is
finally a logical dimension, where I'd say we should explore
up to computability logic and whether there is a positive
version/subsystem of that... Plus, back to numbers, we
need to extend from the point-like numbers to the "closed"
intervallistic approach, or we still do not close the circle
over applications...

But... as far as the purely logical progression goes, first is
the (pre-formal) meta-language, to even start writing
"things" down, within which are the "natural numbers", i.e.
the "pre-formal" ones, so that we can even count our steps.
And here I'd propose that, in a properly infinitary theory as
ours, there is a first "number" and that is omega, not zero.
As the "fundamental leap" to even get started (indeed,
universal quantifications "built-in").

Julio

[Ross A. Finlayson]

Objects....

[Julio Di Egidio]

On Friday, 13 May 2022 at 19:26:05 UTC+2, Ross A. Finlayson wrote:

> Objects....

*Irreducibly*: facts, logics, languages.

Pretty much all over the place...
<https://youtu.be/OmaSAG4J6nw?t=1618>

(Welcome to my kill-file: under the rubric
calling you a crank is a compliment.)

Julio

[Julio Di Egidio]

On Friday, 13 May 2022 at 14:25:06 UTC+2, Julio Di Egidio wrote:
<snip>

> And here I'd propose that, in a properly infinitary theory
> as ours, there is a first "number" and that is omega, not
> zero. As the "fundamental leap" to even get started
> (indeed, universal quantifications "built-in").

Thinking about omega first and what changes.

Take this for example (still from the "new-Platonic" approach):
<< Given ontic random variables A and B, pA(a) and pB(b) may be
/incommensurable/, meaning there may not exist a physically
accurate joint probability distribution p(a,b) [...] If no
joint probability distribution for A and B [exists], then A+B
and AB are no longer random variables! [...] This is not about
ontology, it's a matter of epistemology! >>
<https://youtu.be/OmaSAG4J6nw?t=2205>

But we could, and maybe should, given that a complex system is
more than the sum of its parts, reason in the opposite direction:
a system in a maximally entangled state simply has no parts.
Namely, first the system, then the parts!

That may be hard to reconcile with our common understanding
because the two electrons whose spin is in the singlet state
can in fact be taken apart and acted upon individually. But
of course a property is not an object... rather and eventually
(I think) the point is that "non-locality" (in scare quotes as
that itself may be upside down, it's more "glocality", then
possibly the subsumption of decoherence and of the very inside/
outside dialectic: and that's where I am not a Platonist) has to
be embraced at some fundamental level: together with telepathy...

"A map is not the territory": unless the territory is the map.
Overall "outperforming meta-goedelization through short-circuiting
self-referentiality". A *Theory of All* that is: it has no no-go
theorems, and that a "theory of everything" does not exist.

(Comments welcome.)

Julio

[Ross A. Finlayson]

It's kind of like the Romans: "Two kinds of infinity? Three kinds of infinity."

We are obviously not reinventing anything here, mathematics is discovered.

The philosophy of mathematics, I suppose there is more than one philosophy
of mathematics, though, my philosophy is there's only one mathematics.

I remember one guy "mathematics _is_ my philosophy".

Which he had at the time....

One "constant, consistent, complete, concrete" mathematics. (Physics, a science.)

This is where Godel's theorems instead of "this theory is incomplete" is
"this theory is complete, also there are extra-ordinary elements,
that further complete it", then in terms what where are those.

Goedel's incompleteness theorems ..., after Cantor's uncountability theorems, ....

There's a basic re-reading of each, for the other.

Consider for example three counters. Any two or three might be the same.

[Julio Di Egidio]

On Sunday, 15 May 2022 at 01:30:07 UTC+2, Ross A. Finlayson wrote:
> On Saturday, May 14, 2022 at 5:45:19 AM UTC-7, ju...@diegidio.name wrote:

<snip>

> > "A map is not the territory": unless the territory is the map.
> > Overall "outperforming meta-goedelization through short-circuiting
> > self-referentiality". A *Theory of All* that is: it has no no-go
> > theorems, and that a "theory of everything" does not exist.
> >
> > (Comments welcome.)
>

> It's kind of like the Romans: "Two kinds of infinity? Three kinds of infinity."

Your knowledge of history is on a par with your knowledge of philosophy.

> We are obviously not reinventing anything here, mathematics is discovered.

We obviously disagree, already on that: spes est una in inductione vera.

> The philosophy of mathematics, I suppose there is more than one philosophy
> of mathematics, though, my philosophy is there's only one mathematics.

Did I say philosophy? You just don't know what you are talking about...

Julio

[Ross A. Finlayson]

I think that at least one thing that Wittgenstein "the anti-Plato", is
allowed is a practical mathematics that he can inspect and modify,
if only in the general sense of picking it up and putting it down -
that is inventable.

That there is "one" a mathematics - this is the Platonist that these
days being a "strong platonist" doesn't necessarily include "going
from Platonism through Neo-Platonism", it's more an "old platonist"
where also it's figured - the same philosophy for mathematics is
that each culture has their own and it's all the same one.

Then "infinity" and "continuous" are the largest and for the
smallest aspects, of mathematics: they are themselves primary
in the theory as properties, besides the usual "empty" or "zero",
then that the integer lattice is all usual while also circles are usual.

Circles in geometry: here of course I point to line-drawing as
fundamental in geometry for mathematics, and that one line
draws all the circles and draws the infinite-dimensional line.

Then of course there's directly all of "modern mathematics", ....
Basically "modern mathematics" is the only day's language,
of mathematics.

Here then for my modern mathematics it acknowledges both
a strong platonism, and, Wittgenstein's ideals of forms or
lack thereof - then for basically "making that a science".

Being, Nothing, and Time, and, Certainty, and Chance:
how I would say it is "truth is regular".

This is the theory I think is the best and all mathematics.

[Julio Di Egidio]

On Sunday, 15 May 2022 at 17:30:16 UTC+2, Ross A. Finlayson wrote:
> On Saturday, May 14, 2022 at 5:48:45 PM UTC-7, ju...@diegidio.name wrote:
> > On Sunday, 15 May 2022 at 01:30:07 UTC+2, Ross A. Finlayson wrote:
> > > On Saturday, May 14, 2022 at 5:45:19 AM UTC-7, ju...@diegidio.name wrote:
> > <snip>
> > > > "A map is not the territory": unless the territory is the map.
> > > > Overall "outperforming meta-goedelization through short-circuiting
> > > > self-referentiality". A *Theory of All* that is: it has no no-go
> > > > theorems, and that a "theory of everything" does not exist.
> > > > (Comments welcome.)
> > >
> > > It's kind of like the Romans: "Two kinds of infinity? Three kinds of infinity."
> > Your knowledge of history is on a par with your knowledge of philosophy.
> >
> > > We are obviously not reinventing anything here, mathematics is discovered.
> > We obviously disagree, already on that: spes est una in inductione vera.
> >
> > > The philosophy of mathematics, I suppose there is more than one philosophy
> > > of mathematics, though, my philosophy is there's only one mathematics.
> > Did I say philosophy? You just don't know what you are talking about...
>

> I think that at least one thing that Wittgenstein "the anti-Plato", is
> allowed is a practical mathematics that he can inspect and modify,
> if only in the general sense of picking it up and putting it down -
> that is inventable.

Invented yes, arbitrarily no: rather within the bounds
of the logically necessary. A la Occam's razor, but
without the unwarranted reductions.

> That there is "one" a mathematics - this is the Platonist

<snip>

Since I too am after the "one" theory, you should see that,
whatever Platonism is, that is not its distinctive feature.
In fact, I have given a hint upthread to the inside/outside
dialectic then the mythologies, namely, the fundamental
dissociation of the individual from the absolute: that is
Platonism, as a philosophy, then Idealism/Materialism
as the necessary limit of that dissociation.

> Here then for my modern mathematics it acknowledges both
> a strong platonism, and, Wittgenstein's ideals of forms or
> lack thereof - then for basically "making that a science".

Wittgenstein is not just a "formalist", nor am I. Indeed, the
Wittgenstein of the Tractatus, which is the one relevant to
a theory of foundations, indeed talks of facts, languages,
forms... and that eventually some questions are rather
ill-posed. Truth is that there is one and only one rationality.
The sceptics and the liars deny that.

> Being, Nothing, and Time, and, Certainty, and Chance:
> how I would say it is "truth is regular".

While Heidegger, who's asking the properly philosophical
questions at the apex of the professional philosophical
tradition in the modern age, rather had to give up...

The Socratics had all the right questions, past that point
it's increasing insanity and abuse. Plato is the beginning
of the end.

> This is the theory I think is the best and all mathematics.

You have no theory, just magic in your left hand and who's
got the biggest guns in your right.

Julio

[Ross A. Finlayson]

Sure, if modern mathematics were all collected corrected foundations,
then what I am claiming as theory would just be a comment.

When you say "dissociation", in the objective sense, it's also in
the psychological sense: not to dissociate "subjectivism" when
dissociating "subjectivism".


I would expect that a larger computer reasoner working
for quite a while, is where I bothered to share my opinion,
is left with so few parts writing for: forms, ..., I am always
writing fully to the point and extendedly and discursively,
which I have left for myself as a trail in my retirement
that in so few words work up a usual connection of mathematics.

Or so I just told it, ....

The reason "A Theory" is the best is no particulars, then though
"expressly logical", what makes for the placement of question words
in the language of theory, .... The "A Theory" or "A-Theory" itself
the "Null Axiom Theory", is as well whatever is the setting.

Then, it's at least the strongest theory, ..., absent particulars.

The A-Theory, here is that "A" is both "the indefinite article" and
"A, the first letter in the Alphabet", "A-Theory: 1'st theory".

Then, that "A-Theory is the theory", is what gives to all particulars,
besides what is logic or theory.

Then, "A-Theory the stronger", here is along the lines of "stronger
platonism's A-Theory the physical theory", for example, where
physical is "much stronger" than just "stronger, applied, ...",
theory goes all the way from applied to physical (concrete).

Then, I have measured out all these categories in few small words
in English, that, according to A-Theory, there is a stronger platonic
theory what is mathematics (logic, ..., geometry, ...), then that
"a-theory" is the strong theory, generally, while "A-Theory" is for
example set theory, geometry, ..., strong enough in terms.

So, according to English and what I learned and read in mathematics,
I expect a strong computer mathematics thinker to validate it.

Or so it would seem I believe, ....

[Ross A. Finlayson]

On Sunday, May 1, 2022 at 9:50:22 AM UTC-7, Ross A. Finlayson wrote:

> On Saturday, April 30, 2022 at 1:07:06 AM UTC-7, ju...@diegidio.name wrote:
> > On Saturday, 30 April 2022 at 09:37:41 UTC+2, Ross A. Finlayson wrote:
> > > On Friday, April 29, 2022 at 12:55:55 PM UTC-7, Mostowski Collapse wrote:
> > > > 3 category mistakes in one MSE question.
> > > Shut up Musatov / answer the paradoxes.
> >
> > You give me an idea:
> >

> > Take the real interval [0,1], which is of course
> > continuous, compact, closed and indeed all
> > that we want!
> >
> > Now, for m > 0, map it to the interval [0,m] by
> > f(x) = m*x, the inverse map being g(y) = y/m.
> >
> > Now let m go to infinity! By the Mostowski
> > collapse lemma... (by isomorphism...) this
> > interval stays continuous, compact, closed
> > and indeed all that we want!
> >
> > No? And why not?
> >

> > Julio

> It's rather direct the relation [0,1)~[0,1] as that of [0,oo)~[0,oo].
>

> The compactification of the naturals (a point at infinity) and
> that it's included or "the naturals are compact", helps to illustrate
> that more than "finite" they're "infinite".
>
> The very notion of a "scalar" infinity, is more than less the "modular".
>
> These are of course plainly "only numbers", where though whatever
> models the continuous results the same fine-ness of the "smooth".
>
> That as "only numbers as only models of only objects", when the
> objects are finite, is about the "continuous and discrete".
>
> It's rather as once I wrote "field operations for [-1,1]".

Which are ancient!

[Ross A. Finlayson]

Again, actually I'm happy for you that you are discovering
or inventing these things of mathematics, that I discovered
as each must discover for themself to "know mathematics",
besides what's content of usual logic, philosophy, ..., degrees.

In fact there's a subculture of academics, for whom we are trite,
who are topologists and platonists and objectivists and so on.
All this "extra-ordinary" and "utterly-fundamental", it's so much
old hat - where is left out a usual education and a history.

[Julio Di Egidio]

On Monday, 16 May 2022 at 06:50:04 UTC+2, Ross A. Finlayson wrote:

> When you say "dissociation", in the objective sense, it's also in
> the psychological sense: not to dissociate "subjectivism" when
> dissociating "subjectivism".

From Guy Debord, The Society of the Spectacle:
<< "In clinical pictures of schizophrenia" says Gabel, "decline of
the dialectic of totality (with dissociation as its extreme form),
and decline of the dialectic of becoming (with catatonia as its
extreme form), seem strictly connected". >>
<https://voveo.blogspot.com/2006/11/the-society-of-spectacle.html>

And of course "Marxist" my ass... Indeed, I said magic but
I should have said spells, of the worst kind: just calling your
incivilization delusional altogether misses the point.

Julio