Theory

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Ross A. Finlayson

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May 10, 2021, 2:22:02 AM5/10/21
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I guess why the classical theory arises from deductive principles
which are unfounded, usually is via induction that "what would
be founded results inductively result that no theory is a theory".

Here I am mostly having adopted a very usual in fact old-fashioned
"all the canon and what philosophy has what reason arrives at logic",
for the usual post-Goedelian, that these days all the paradox of logic
is given to after Goedel, that there is no paradox that theories are true.

I mostly studied logic for foundations.

Here it is that mostly I assign the classical and modern and then
for direction: have it go through itself again, so it's classical
and modern again: instead of for example letting it out to
the computational. I.e., where "the computational" is
"the easy part" instead of "that it's paradoxical has it easier".

I.e., usual if large regular fragments of ZF for "set-like objects"
here what make for the unipotential or under concerns,
here with sets and partitions in models of real numbers with
my modern definitions of real numbers, for foundations
these are put back into the foundations, these as primitives.

So, I am mostly perfectly correct of course though that
it is a striking damnation - that just because it works out
in theory that there's no paradox to have a perfectly good
grasp of all theory that such hubris is easily damned -
it's perfect in theory what all natural theory has an
explanation as I've usually writ - that all perfect
theory naturally arises itself - why not it wouldn't.

Along for being "correct", if perfect - there's of course
both the outline and detail, foundations of course,
I mostly study the foundations and write in it.

Julio Di Egidio

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May 10, 2021, 2:50:22 AM5/10/21
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On 10/05/2021 08:21, Ross A. Finlayson wrote:
>
> I guess why the classical theory arises from deductive principles
> which are unfounded, usually is via induction that "what would
> be founded results inductively result that no theory is a theory".

Primitives aren't "unfounded". Induction indeed is basic to our
rationality, quite more fundamental than any logical method, and it is
the opposite of a give up: no-theory is just not-a-theory.

> Here I am mostly having adopted a very usual in fact old-fashioned
> "all the canon and what philosophy has what reason arrives at logic",
> for the usual post-Goedelian, that these days all the paradox of logic
> there is no paradox that theories are true.

Truth is relative to interpretation. Perfection in the Void? That goes
without saying... You are not even old-fashioned: your theoretics
remains plain vacuous and, in that, fundamentally counter-factual.

But I think I had said that already...

Have a nice day, Prometheus.

Julio

Peter

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May 10, 2021, 10:58:27 AM5/10/21
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Ross A. Finlayson wrote:
>
> I guess why the classical theory arises from deductive principles

The classical theory of _what_?
--
Just as 'beautiful' points the way for aesthetics and 'good' for ethics,
so do words like 'true' for logic. All sciences have truth as their
goal; but logic is also concerned with it in a quite different way:
logic has much the same relation to truth as physics has to weight or
heat. Frege in 'Thoughts' (Der Gedanke)

Mostowski Collapse

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May 10, 2021, 2:17:09 PM5/10/21
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The classical theory of intuitionistic logic?
Or maybe the intuitionistic translation of classical logic?

So that the intuitionistic translation provides
a classical theory of its own intuitionism.

LoL

Down the Rabbit Hole
https://www.youtube.com/watch?v=pHte24GGHD4

Julio Di Egidio

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May 10, 2021, 2:40:05 PM5/10/21
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A resident cunt reinjects some Frege, a retarded spammer joins in with some more bestiality.

As usual, way to go, both of you.

*Plonk*

Julio

Mostowski Collapse

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May 10, 2021, 5:34:23 PM5/10/21
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Nothing Frege, only:

Gödel–Gentzen translation
If T is a set of axioms and φ is a formula, then T proves φ using
classical logic if and only if T^N proves φ^N using intuitionistic logic.
https://en.wikipedia.org/wiki/Double-negation_translation

Mostowski Collapse

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May 10, 2021, 5:38:15 PM5/10/21
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Ha Ha Culio doesn't know the semantics of "--".
Whats below "--" is the footer, usually an address

banner in the old times, but Peter uses it for
some, varying, citation. Last it was something with

Specker, now its from Frege. LoL

Ross A. Finlayson

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May 16, 2021, 2:25:20 AM5/16/21
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The classical theory usually for logic and geometry,
for example there is mostly Aristotle and Euclid.

Atomism - not far behind.

Here I have made modern continuum mechanics -
now under three instead of one "definition of continuous",
then with regards to signal recomposition versus measure.

It is an axiomless logic also as I have described.
(The geometry.)

I am finding deriving Euclidean geometry what is classical,
this way with my modern theories.

(Which is one theory.)

Mostowski Collapse

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May 16, 2021, 3:55:54 AM5/16/21
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What if Aristotle and Euclid were just pseudonyms,
like Satoshi Nakamoto? For Aristotele there are
biographic details that he went to Evia when he was old.

Peter

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May 16, 2021, 1:31:41 PM5/16/21
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Ross A. Finlayson wrote:
> On Monday, May 10, 2021 at 7:58:27 AM UTC-7, Peter wrote:
>> Ross A. Finlayson wrote:
>>>
>>> I guess why the classical theory arises from deductive principles
>> The classical theory of _what_?
>>> [...]
>
> The classical theory usually for logic and geometry,
> for example there is mostly Aristotle and Euclid.

You know, don't you, that Aristotle's logic was very limited? In
particular, it is quite useless for formalizing mathematics.

>
> Atomism - not far behind.
>
> Here I have made modern continuum mechanics -
> now under three instead of one "definition of continuous",
> then with regards to signal recomposition versus measure.
>
> It is an axiomless logic also as I have described.
> (The geometry.)
>
> I am finding deriving Euclidean geometry what is classical,
> this way with my modern theories.
>
> (Which is one theory.)
>


WM

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May 17, 2021, 10:13:31 AM5/17/21
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Peter schrieb am Sonntag, 16. Mai 2021 um 19:31:41 UTC+2:
> Ross A. Finlayson wrote:

> > The classical theory usually for logic and geometry,
> > for example there is mostly Aristotle and Euclid.
> You know, don't you, that Aristotle's logic was very limited? In
> particular, it is quite useless for formalizing mathematics.

That's why formalizing mathematics is quite useless?

Regards, WM

Mostowski Collapse

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May 18, 2021, 7:13:20 AM5/18/21
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WM the greatest moron on earth. No that is not
what is said. But Aristoteles term logic is weaker
than FOL or HOL, etc.. Since it features only monadic

predicates. Its basically a subset of monadic logic
i.e. first order logic with unary predicates only:

https://de.wikipedia.org/wiki/Begriffslogik

https://en.wikipedia.org/wiki/Term_logic

But Aristoteles and the Greeks did not only found
term logic, this is also a fallacy.

Euclid is a nice example. His background logic is not
only based on monadic logic. Many predicates for the
euclidean plane have a greater arity than only one.

Mostowski Collapse

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May 18, 2021, 7:16:18 AM5/18/21
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For some geometry formalization approaches:
https://geocoq.github.io/GeoCoq/

Peter

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May 18, 2021, 8:44:40 AM5/18/21
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You are shockingly illogical - the uselessness of Aristole's logic does
not imply (or even hint at) that formalizing mathematics is quite useless.

>
> Regards, WM

Mostowski Collapse

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May 18, 2021, 9:52:40 AM5/18/21
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To attribute uselessness to Aristole's logic is also
an exageration. After all you can reason for example:

all prime numbers are natural numbers
7 is a prime number
---------------------------------------------------------------------
7 is a natural number

Which is difficult in WM logic. What if there are dark numbers,
that bite you in the ass while taking a shit.

Mostowski Collapse

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May 19, 2021, 8:27:48 AM5/19/21
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Woa! sci.logic has a strong community, nobody reacting
on "All known evidence for dark numbers collected" by WM.

Thats like HODL-ing. LoL

Gus Gassmann

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May 19, 2021, 9:20:56 AM5/19/21
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On Wednesday, 19 May 2021 at 09:27:48 UTC-3, Mostowski Collapse wrote:
> Woa! sci.logic has a strong community, nobody reacting
> on "All known evidence for dark numbers collected" by WM.

Nobody has even read it. Let's hope it stays that way.

Mostowski Collapse

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May 19, 2021, 2:09:11 PM5/19/21
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The HODL broke already. :-( :-(
Some newbee?

Ross A. Finlayson

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May 20, 2021, 4:01:07 AM5/20/21
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Fast, forward to Frege, ....

ARistoteles I suppose then to Frege,
Frege;s quite profound.

\

Ross A. Finlayson

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May 21, 2021, 10:23:53 PM5/21/21
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That's fair. Fast forward to Frege, it's mostly rules for derivations of what
result as (direct) implications quite throughout modus ponens and modus tollens
of course, square of opposition and such, according to a brief formalization
then after the ordinariness of transfinite sets, the extra-ordinary.

(What makes for axiomatics as syllogistic.)


About physics, it's not so far-fetched what it's a "theory of truth",
some truth of what results a logical mathematical universe -
for what as a system of mathematics it could eventually be
theoretical besides of course the usual empirical most for which
in matters of their relevant interpretation of physics suffices.

That "the theory" as logic is affirmatory and all in what results
then for identifity, tautology, ..., formula, that it plainly is a
"theory of truth" helps for interpretations like "there are no
paradoxes of logic only prototypes of improper derivation",
eg that in our ordinary "modern" world then (say, since Goedel),
it's a regular fragment of a usual ordinary universe like ZF set theory's.


Julio Di Egidio

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May 26, 2021, 9:16:16 AM5/26/21
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On Saturday, 22 May 2021 at 04:23:53 UTC+2, Ross A. Finlayson wrote:
> On Sunday, May 16, 2021 at 10:31:41 AM UTC-7, Peter wrote:

> > logic has much the same relation to truth as physics has to weight or
> > heat. Frege in 'Thoughts' (Der Gedanke)
> That's fair. Fast forward to Frege, it's mostly rules for derivations of what

Never will you fucking retarded and compromised cunts just do the right thing ever.

Stupid fucking white man, you deserve your own slavery and extinction...

Fuck you.

Julio

Ross A. Finlayson

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Jun 1, 2021, 12:52:36 AM6/1/21
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That is missing the point - I don't care yet still put the next in line.

Let us remember that one post where in any number of tirades
for which Julio can blame me - let's make it clear that anything
less than contemptuous rejection would be fair to such an affront
as some usual "definition of theory".

Because of course in definition of theory rests all what's affrontable.

So it's fair that he totally rejects the context or here swears at it.

Or, reading this, I put myself here.

That of course anyone else can do for themself also, or for example,
knowing my opinion and authority on such objective matters, reading
the slate as it were - later at least I can stumble over this and swear
I swore over it.

I hold understanding in foundations in very high opinion (and regard).
The semiotics of these things is mostly important, so I simply hold
this in usual high regard.

Julio, I thank you the constancy - it well reflects your continued opinion
and both can rely and not rely that when it's hitting you're spitting.

(Or, this theory hits.)


Julio Di Egidio

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Jun 1, 2021, 10:05:49 AM6/1/21
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On Tuesday, 1 June 2021 at 06:52:36 UTC+2, Ross A. Finlayson wrote:
> On Wednesday, May 26, 2021 at 6:16:16 AM UTC-7, ju...@diegidio.name wrote:

> Julio, I thank you the constancy - it well reflects your continued opinion
> when it's hitting you're spitting.

And of course never ever even an attempt to actually engage with the substance of my criticism. Fuck off, you delusional and always self-apologetic piece of spamming shit: *you* and the hordes of retarded cunts like you are the reason why this planet is a fucking ungodly shithole for everybody. So yes, I spit on you, eventually and to be charitable... you fucking vile cunts.

*Plonk*

Julio

Julio Di Egidio

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Jun 1, 2021, 10:44:16 AM6/1/21
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For the chronicle, meanwhile Google Groups has banned comp.lang.python, "identified as containing spam, malware or other malicious content", where comp.lang.python is in fact one of the few Usenet groups that has not yet been just hijacked by the spammers...

You are even more fucking pathetic than you are a crime against all life and intelligence.

Just fuck off.

*Plonk*

Julio

Mostowski Collapse

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Jun 1, 2021, 5:58:45 PM6/1/21
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Only because google has banned a group, doesn't mean the
group does not exist anymore. Just use a normal usenet
reader and some free news provider, and you will see, latest post:

Newsgroups: comp.lang.python
Subject: Async requests library with NTLM auth support?
Date: Tue, 01 Jun 2021 22:25:02 +0200

Hi,
I need to make thousands of requests that require ntlm authentication so I
was hoping to do them asynchronously. With synchronous requests I use
requests/requests_ntlm. Asyncio and httpx [1] look promising but don't
seem to support ntlm. Any tips?
Cheers!
Albert-Jan
[1] https://www.python-httpx.org/

Julio Di Egidio

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Jun 1, 2021, 6:21:51 PM6/1/21
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On Tuesday, 1 June 2021 at 23:58:45 UTC+2, Mostowski Collapse wrote:

> Only because google has banned a group, doesn't mean the
> group does not exist anymore. Just use a normal usenet

Of course you cannot even guess what I was complaining about, you top-posting imbecile and other spammer for decades.

A parade of retarded and brainwashed cunts, that's what you are, indeed too vile to just do the right thing ever. Fucking pathetic.

*Plonk*

Julio

Mostowski Collapse

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Jun 1, 2021, 6:44:44 PM6/1/21
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You're not micro penis, are you?

Ross A. Finlayson

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Jun 5, 2021, 12:54:06 AM6/5/21
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Regular axiomatics as numerical resources themselves,
mostly leaves out for no theory of paradoxes, what all
usual concerns make for measure besides quantity.

I've left it myself all accommodative all "tower of rain"
and "metal numbers", consistent, complete, constant,
concrete. (Current.)

Newberry

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Jun 5, 2021, 5:57:07 PM6/5/21
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Mostowski Collapse wrote:
> WM the greatest moron on earth. No that is not
> what is said. But Aristoteles term logic is weaker
> than FOL or HOL, etc.. Since it features only monadic
>
> predicates. Its basically a subset of monadic logic
> i.e. first order logic with unary predicates only:

It does not deal with empty subjects, i.e. it does not deal with vacuous
sentences. This kind of logic has been formalized by Strawson. It can be
extended to polyadic logic.

Mostowski Collapse

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Jun 6, 2021, 5:42:14 AM6/6/21
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The fascinating thing about monadic logic, it is
a decidable fragment of FOL. So if all your predicates
are unary, a finite theory of such axioms, becomes

decidable. Thats kind of amazing, that Aristoteles in
his exposition chose a decidable logic. Maybe its the
most used logic in common sense, because it is

decidable? A folk corollary is as follows:

Folk Corollary from Monadic Logic Decidability:
- An undecidable FOL theory must have at least have
one predicate which is not unary.

Lets make a reality check, give an example, which
is not meant as a proof, only an illustration, take
FOL+ZFC, ZFC has indeed a predicate which

isn't unary and ZFC is undecidable.
The membership relation ∈ is binary!!

Mostowski Collapse

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Jun 6, 2021, 5:48:53 AM6/6/21
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But I dont know whether the antique Greeks could relate to
notions such as decidable and undecidable. They had
curiosities such as incommensurability leads to infinite

Euclidean algorithm. But they didn't have explicitly halting
problem? Since they didn't have universal turing machines
explicitly? Not a single item here is antique Greeks?

https://en.wikipedia.org/wiki/List_of_undecidable_problems

Ross A. Finlayson

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Jun 6, 2021, 11:59:33 AM6/6/21
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I just keep tipping up the slate of paradox/no-paradox and
infinity/infinity-infinity, must be foundations.

I luxuried study in the types and so when there is inspection
and reflection on the mathematical objects, there are all the
paradoxes of the mathematics and logic which results in objects
what so confirm for example for geometry all their truth besides
how it's possibly comprehensible, if at all.

I.e., the paradoxes like Zeno's, or the paradox of motion,
are simply laws either way with respect to what must be time.
Still, motion in geometry is for the usual _objective_, i.e. that
geometry the theory allows theory is objective. (If not "my theory",
subjective, "the theory sublime to any other theory including mine",
objective.)

So it's a luxury when "for my theory what is actually sublime to
this formalism where paradox means stop not build a bridge to
the other side of complete objective formality", it's what foundations
is relevant or not for the applied, that these total questions in theory
are just mindless inference.

Then for countability/uncountability or "infinity / no, infinity",
I already had a clock hypothesis besides a lattice hypothesis,
so uncountability in finite domains is largely about combinatorics,
besides what the continuous is sublime to it. For "no paradox"
then is laws both ways, all real numbers.


Going from Cartesian what are the usual inner products about
that going from binary, 0-1, to any more values in the valency of
the logic from the multi-valued or multi-valent not being the
bi-valent Cartesian case, I define my functions regularly as
Cartesian about the Cantorian.

For example, where for function theory is defined for the
actual case in continuum mechanics about Descartes' lattice
in space and Cantor's lattice in space, each the same space
in ordinals, it's a usual result after primary function theory.
(In foundations.)


So, the slate for uncountability is a raft of results, what for
the usual conscientious formalist admit of course all sound
reason and that most outstanding questions, in mathematics,
often these days for example reflect questions about infinity
and continuum mechanics.

Thanks, I had enough time here to study foundations for its
own sake as much as usual for example matters what take
less time - that when people have issues or notice usual total
grand facts about things that I also found so profound and
of course for example confirming larger logic must be correct,
I mostly of course wrote immediately whatever possibly I could
use in a sentence that could possibly advance my "cause".

(What is to have written a foundations.)




Mostowski Collapse

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Jun 6, 2021, 3:00:49 PM6/6/21
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Paradoxes are more Martin Gardner math entertainment.
The real work starts after Paradoxes. Ask Dan, he started
with Barber Paradox, but he will soon do some real work.

LoL

Mostowski Collapse

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Jun 6, 2021, 3:02:36 PM6/6/21
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His distraction is feeding a troll. Whats your distraction Ross?

Ross A. Finlayson

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Jun 7, 2021, 10:57:19 AM6/7/21
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It's an idealism.

Continuity and the mathematical foundation of the continuous,
more than less connects geometry and arithmetic with continuity.

Of course then studying the canon to understand what is the
relayed wisdom of continuity, it takes a while to understand
Eudoxus/Dedekind/Cauchy, for a usual standard curriculum in
real analysis and its foundations, and with respect to set theory
and indiscernibles, then about that "classical infinitesimals" of
the sort like fluxions or the differential, needed a more direct
placement in the center of things as a prototypical line segment
with respect to constant motion and continuity.

About the continuous and discrete, then finding for example all
these properties of the "natural/unit equivalency function" with
respect to real-functions and here not-a-real-function-but-a-function-
nonetheless, and about how ordinary set theory's functions are
Cartesian while for example this one isn't, makes that then: in
such a study, there arise plentiful what otherwise would be paradoxes,
that are instead resolved via distinction of form, here with at least
three models of all real numbers each as different sets, it's found that
the resulting more replete character of the reals is better defined and
that a "foundation" for continuity doesn't lack them.

Also "ex falso nihilum" not "ex falso quodlibet".

Then, besides the unique and surprising probabilitistic and differential
character(s) of this central function, after that it is a spiral space-filling
curve this line-drawing as what makes for an origin anywhere, for
singularity theory for the origin besides the asymptote, then also there
is the factorial/exponential identity after what these days is called Ramsey
theory, to help explain and indeed provide a foundation for, the quasi-
invariant in modern measure theory. (And from having written a measure
theory and fundamental theorems for calculus for these line reals after
field reals, that of course formally they must satisfy embodying the
analytical character of the real numbers and all what results in the
hypercomplex numbers.)

So, it's an idealism with a fortuity of having found the formal development
in such a state: perceived lacking these things mentioned above, and having
basically penchant, interest, and time.




Julio Di Egidio

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Jun 7, 2021, 12:03:07 PM6/7/21
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On Monday, 7 June 2021 at 16:57:19 UTC+2, Ross A. Finlayson wrote:

> Also "ex falso nihilum" not "ex falso quodlibet".

"Ex falso quodlibet", that is mathematics and how derivation works: "ex contradictio nihilum", that is logic and how that works.

Write on your wall: "logic is not mathematics and I just have no clue what logic is". Or try and Read Strawson, Introduction to Logical Theory, if you are actually interested in logic.

Julio

Daniel Pehoushek

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Jun 7, 2021, 2:56:13 PM6/7/21
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intellectual strolling with bob nay quite trolling for "fish bites"

my program bob reasons one million times faster than the author with many fewer errors of detail

+1 is the founding brick of bobs reasoning structure
all control flow uses +1 throughout bob

some summary sentences are proofs of bobs corredness

corredness means what it should after being derived from correctness

bob searches all of boolean space using +1 tree traversal from all booleans zero through and to all booleans one

bob counts every good answer
leaving the good questions for later high reasoning with allqbfs
on sufficiently small core theory
where one inference takes ten cpu cycles to complete


my pspace solver bob does one trillion inferences per hour on five giga hertz cpu clock
source code (of one thousand lines is available by email)
of detailed c++ mathematical reasoning
for corredly pspace solving all small graph coloring formulas using qspace
graph coloring is an np complete area of colorful high reason
two favorites are
c3d5n180 and c4d9n150
what can be built using all solutions of those two sets alone
corredness of completely reasoned solutions is the major claim

monotone reason is high order common sense reasoning
where the monotone Q formula decides qbf validity completely

boolean formulas with only zero negations are called monotone
the truth of monotone sentences is decidable in linear time space
monotone reason is high order common sense reasoning
where the monotone Q formula decides qbf validity completely


i claim to have sat scores of 1530/1600 and gre scores of 2380/2400 from 1979 and 1983
i have worked for over thirty years studying sub atomic details of computerized reason
my program bob reasons one million times faster than the author with many fewer errors of detail

+1 is the founding brick of bobs reasoning structure
all control flow uses +1 throughout bob

some summary sentences are proofs of bobs corredness
jdp


just look at his ugly meaningless strings:
Also "ex falso nihilum" not "ex falso quodlibet".
"Ex falso quodlibet", that is mathematics and how derivation works: "ex contradictio nihilum", that is logic and how that works.
Write on your wall: "logic is not mathematics and I just have no clue what logic is". Or try and Read Strawson, Introduction to Logical Theory, if you are actually interested in logic.

on my wall
reasoning of all pspace
moderately well solved
by bob with less puntaytion
the abcdefghijklmnopqrstuvw yz0 momday alphabet
now that we begin with a good alphabet
how clearly are the azioms of bobs reason
they are also my azioms but but but
bob uses them one million times faster than i

any "fish bites" replies shall be treated with the utmost resped i may muster
pot is becoming legal in all fifty states
for best results stifle coughs with good pot

now to see if my toilet is unplugged so i may shower

soap on ocean correlationary with snows on mountains global weather management agendas

Ross A. Finlayson

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Jun 9, 2021, 10:40:56 AM6/9/21
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The point is that "material implication" with its false antecedent
is not a proper derivation rule, in terms of a usual stroke or strike
in the validity of tableau that result formulas.

Then with regards to the unit line segment as a prototypical
countable continuous domain, or ran(EF), it's as among matters
of mathematical fact what conscientious formalists should find.

Reading Sarton's "Appreciation of Ancient and Medieval Science
During the Renaissance", he helps distinguish the value of philology.

Julio Di Egidio

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Jun 9, 2021, 11:38:54 AM6/9/21
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On Wednesday, 9 June 2021 at 16:40:56 UTC+2, Ross A. Finlayson wrote:
> On Monday, June 7, 2021 at 9:03:07 AM UTC-7, ju...@diegidio.name wrote:
> > On Monday, 7 June 2021 at 16:57:19 UTC+2, Ross A. Finlayson wrote:
> >
> > > Also "ex falso nihilum" not "ex falso quodlibet".
> > "Ex falso quodlibet", that is mathematics and how derivation works:
> > "ex contradictio nihilum", that is logic and how that works.
> >
> > Write on your wall: "logic is not mathematics and I just have no clue
> > what logic is". Or try and Read Strawson, Introduction to Logical Theory,
> > if you are actually interested in logic.
>
> The point is that "material implication" with its false antecedent
> is not a proper derivation rule, in terms of a usual stroke or strike
> in the validity of tableau that result formulas.

But material implication is not a derivation rule, material implication is a binary operator, indeed with truth table. So no, you are conflating things, material implication and provability are just not the same thing: and then neither of them, and not even theoremhood, i.e. mathematical truth, are yet logic proper.

> Then with regards to the unit line segment as a prototypical
> countable continuous domain, or ran(EF), it's as among matters
> of mathematical fact what conscientious formalists should find.

All right: show me your formalisation of infinity.

> Reading Sarton's "Appreciation of Ancient and Medieval Science
> During the Renaissance", he helps distinguish the value of philology.

Sure, in the negative. Your point being?

Julio

Ross A. Finlayson

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Jun 11, 2021, 10:36:09 AM6/11/21
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1/0 = infinity

(Wait for it....)

Infinity the usual lemniscate or oo as it's written, of course there's a
theory of limits about fixed-point that infinity is written as among
quantities, here instead it's almost easier to define infinitesimals
before defining naturals, what result that all the infinitesimals together
is one just like all the naturals together (one for each) is infinity.

Of course, these aren't "standard" real numbers: but there are only
and everywhere elements of the continuum in (0,1), so, when there
are infinity-many infinitesimals (and no less) then these "iota-values",
where iota-values are these standard infinitesimals, they sum to one.

It's a "scalar" infinity then in a sense, and of course back in the standard
arithmetic and algebra, infinity is usually enough left out, while it's still
reflected in the structure of a linear continuum's unit line segment. (1.0.)

lim n->d d->oo (n/d) = 1

This function EF from naturals 0, ... to segment [0,1] then results it's 1-1 and onto.

(Extent, density, completeness, measure, ... = countable continuous domain.)



False antecedents don't belong in truth tables.

Julio Di Egidio

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Jun 12, 2021, 7:55:20 PM6/12/21
to
On Friday, 11 June 2021 at 16:36:09 UTC+2, Ross A. Finlayson wrote:
> On Wednesday, June 9, 2021 at 8:38:54 AM UTC-7, ju...@diegidio.name
wrote:
> > On Wednesday, 9 June 2021 at 16:40:56 UTC+2, Ross A. Finlayson wrote:
<snipped>
> > > Then with regards to the unit line segment as a prototypical
> > > countable continuous domain, or ran(EF), it's as among matters
> > > of mathematical fact what conscientious formalists should find.
> >
> > All right: show me your formalisation of infinity.
>
> 1/0 = infinity

I find that's a good starting point in many ways:

oo := 0^(-1)

Hence, reciprocally: oo^(-1) = 0

It's a good starting point in the sense that, as seen with the surreal
numbers, standard infinite ordinals are the reciprocals of
infinitesimals, not of Zero! (*) Which is not surprising if one
considers that *all* standard infinite sets, not just the set of natural
numbers, are *potentially* infinite sets. The Infinity we are talking
about here is *absolute*, aka actual infinity, indeed the reciprocal of
Zero.
(*) <https://en.wikipedia.org/wiki/Surreal_number#Infinity>

Still, how do we define succ(oo), whence the arithmetic? Notice that,
if succ(oo) := oo, as it should be (...), we quickly run into problems,
e.g.:

oo = oo + 1 <==> 0 = 1
oo = 2 * oo <==> 1 = 2

We could restrict the algebra, we could require that one "absorbs" all
finite quantities into the Infinities before the usual algebraic rules
can be applied:

oo = oo + 1 <==> oo = oo
oo = 2 * oo <==> oo = oo

Would that work? I am not readily aware of the (co-)implications.

> Of course, these aren't "standard" real numbers

The surreal numbers certainly "subsume" the standard real numbers, but I
don't know how calculus works with the surreal numbers, not even when
restricted to the standard real numbers. Is the definition of limit
even the same? Certainly worth investigating...

OTOH, notice that there is no Infinity proper in the surreal numbers
either: in fact, division by Zero remains undefined. Which is another
way to see that the relevant game with Infinity happens at the level of
the natural numbers already: to be clear, contra your insistence on
continuity, divisibility, the unit interval, etc.

> lim n->d d->oo (n/d) = 1

That can indeed be derived, more basic is: the limit of every
monotonically increasing sequence of natural numbers *is* oo. Period.
The "point at infinity".

> False antecedents don't belong in truth tables.

Still a meaningless statement, truth tables don't have "antecedents".

The theory of all: that counts.

Julio

Julio Di Egidio

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Jun 12, 2021, 8:30:13 PM6/12/21
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On 13/06/2021 01:55, Julio Di Egidio wrote:
> On Friday, 11 June 2021 at 16:36:09 UTC+2, Ross A. Finlayson wrote:

>   oo = oo + 1  <==>  oo = oo
>   oo = 2 * oo  <==>  oo = oo
>
> Would that work?

For one thing, it does not seem to address the problem of what oo/oo
(equiv. oo*0, or 0/0) is. Maybe oo*0 = 0, which seems consistent with
how multiplication is defined inductively from (addition and,
eventually) successor.

Maybe... Where is a "conscientious formalist" when you need one?

Julio

Daniel Pehoushek

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Jun 16, 2021, 10:59:18 PM6/16/21
to
any comments about the novelty of
using the opposite diagonal from cantor
as the foundation for reason?
bob does one trillion logical inferences
per hour. see graphcolor.zip
daniel

Ross A. Finlayson

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Sep 30, 2022, 12:36:26 PMSep 30
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Heh.

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