## Sunday, 9 March 2008

### The unified theory of all we can do with it (seriously)

I have posted the following message to the Mathforum yesterday at 4:43 AM (GMT+00). Starting from around 6:00 AM today, the Mathforum has become less and less responsive, until at 8:00 AM it has finally stopped working.

I am reposting the message here (untouched), because at the moment it doesn't appear in any other publicly available sci.math repository.

I have also attached an exchange with Mr Aiya-Oba, who was so kind to support in dis-closing the proof.

WARNING: MIND YOU!!
This is no fucking around, handle with care and take at your own risk.

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Julio Di Egidio
The unified theory of all we can do with it (seriously)
Posted: Mar 8, 2008 4:34 AM

[I wish to just warn you I am not a professional in any of the fields I am going to touch, I am rather some kind of abstract logician plus a professional programmer, so please mind the step(s). What I'm after is for (dis-)proofs to the following chain of assertions. As to the "discussion", that might very well start later, and I do mean it. OTOH, questions are always welcome. Again, please mind the step(s), there are none.]

Subject: The unified theory of all we can do with it (seriously).

Freely mentioning Russell, Cantor, Goedel, Turing, Complexity, Walster, Golden, and Di Egidio.

Below, "dis-x" stands for "x and only x", where the connotations are thought to be even more interesting.

>> Walster shows[*] that a number is a set, and that the empty set is a number.

Walster extends interval arithmetic with the empty interval and intervals with one or both infinite end-points. He first defines operations on the empty interval; from there, he closes arithmetic to any operator and function combinations on the entire domain. In his system, the empty interval is dominant to any other, including the entire interval, i.e. {}.rel.X = {} and X.rel.{} = {}, for all X in IR*, for all interval relation. (More from Walster later.)

(1) We now can say: a number dis-is a set, and the empty set is dominant to any number, dis-including itself.

>> Russell asks what is the set of all sets not having themselves as elements.

(2) We now can say: the set of all sets not having themselves as elements dis-is the empty set, i.e. the paradoxical set of paradoxical sets not having any elements, dis-including themselves.

>> Cantor wanders how the diagonal argument leads to entities outside the domain.

(3) We now can say: the diagonal argument leads to the all-encompassing void of the empty set, i.e. the paradoxical number of paradoxical numbers (or, more simply, the "without" (outside) of the domain, as seen from "within" (inside) the domain).

>> Goedel proves that self-referential entities must exist, yet undecidably.

(4) We now can say: the purely self-referential set dis-is the empty set, by foundation; in fact, a progression up the chain of provability systems can be seen from "I can't be proved", to "I am not true", up to "You fail", where this last sentence dis-is the empty set and expresses the limit ad infinitum of goedelization.

Incidentally, "You fail" might express more than the abstract sentence ~Bf ("consis"; can read "not-believe-that"), where belief is the foundational meaning behind logical negation and falsehood. I cannot say (for lack of specific knowledge) what this strictly entails on the "incompleteness arguments in a general setting" (Smullyan, 1992), but the introduction of the empty set as a full fledged and foundational entity, and the closedness of algebraic systems it brings, seem to suggest a profound impact. For instance, it seems quite evident that belief should itself be founded on "us", the subjects, and aren't we, with respect to the system, just its very external domain? In a sense, ultimately, "we" are the empty set and the self-referential dis-proof of all proofs.

Indeed, OTOH, I must note that, as far as "real" systems are concerned (i.e., in any form of "engineering"), the introduction of the empty set is in itself enough to formally found our very daily "practices", where we are used to discard apparently incorrect answers (i.e., dis-answers within our accepted domain), and where we, ultimately, improve throw failure (i.e., breaking out of the boundaries of the accepted). This is in the lights of what straight follows.

>> Turing, in shades of Hilbert's tenth problem, restates the question in terms of the "halting problem".

(5) We now can say: All machines indeed stop, sooner or later; in the worst case, it is "us" (see my preceding note) stopping them, and, in any meaningful sense, the machines "we" stop have failed, and dis-belong to the void of the empty set, which happens to be the void "we" are.

In simpler terms, the set of failing machines is the set of machines that are "not machines" at all, with respect to the bounds "we" impose on the accepted domain.

Incidentally, for all this to be (believed) true, we must accept that classical mathematics (pardon my lack of "sharpness" on that, but, strictly speaking, we could go back to Aristotle), from its very logical foundations onward, is rooted on a fallacy. I wish to stress here that correcting that fallacy doesn't mean throwing to the bin all of the great accomplishments so far, it rather means we could enlarge our perspectives beyond what is today believed to be out of our reach.

As an anticipation, Walster talks about an "exception free system", which is a way to show that this "new" system (in the sense just given) must be simpler, not more complex than our current systems. This "new" system must still show some kind of instrinsic limit, though, and that is what I am (at a naive level) dis-expressing within the very last sentence in this document. However, let's finish the tour first.

>> Within Complexity, NP problems are said to be "intractable", and it is yet an open question whether or not P = NP.

Going back to Walster: "The use of interval methods provides computational proofs of existence and location of global optima. Computer software implementations use outwardly-rounded interval (cset) arithmetic to guarantee that even rounding errors and bounded in the computations. The results are mathematically rigorous." For example, the Kepler conjecture has been proved after 300 years by means of computers and outwardly-rounded cset arithmetic.

(6) We now can say: NP is too in the tractable domain, though NP is not P in that they represent the two opposite approaches to problem solving, the latter finding the correct solutions, the first discarding the incorrect ones.

Incidentally, if NP happened to be intractable, in real life we would forever be stuck in undecidedness, which is apparently not the case (and here I mean, apart from any heuristics: human beings don't need heuristics in every-day life, though every-day life is full of undecidable questions, rooted into the very structure of natural language).

>> Di Egidio asks what then is a "number" (or a "set") for the sake?

(7) We now can say (extensionally): a number (or a set) dis-is all we can do with-in and with-out it, i.e. the closed number system exhausts the whole domain of tractability, by tautological-within-self-referential foundation.

>> Golden shows[**] a family of number systems called "polysign numbers", having a natural number of signs.

(8) We now can say: Walster shows the "natural" closure of real numbers (avoiding undefined numbers and bounding computational errors); Golden shows the "natural" interplay between natural and real numbers (avoiding the asymmetries introduced with the imaginary numbers[***]); finally, Di Egidio shows (yet to be dis-proved), the general "natural" meaning of "number", as rooted into the concept of a subject which is the empty set (avoiding undefined reasoning).

>> Di Egidio screams, what's the outcomes then?

(9) We now can say: the outcome is the unified theory of all we can do with it (seriously).

>> Di Egidio cries, come on, you're saying "seriously"!?

(10) The outcomes are still to be inspected and worked out, but... even if only half of what is stated here has some reasonable foundation, then its incidence on everybody's lives could be dramatic, and to the better(!!). For instance, we have here the foundation for a final convergence of natural and logical languages, and, if you cannot imagine how that could only and only only change our lives for the better, then "You" fail.

If you managed to get to this point, I'll be looking forward to your knowledgeable dis-proofs.

(As you may guess, I do really need your feed-back.)

Thank you very much,

Julio

--------------------------
Julio Di Egidio
Analyst/Programmer
http://julio.diegidio.name

[*] More on Walster's Closed Sets in this thread (it's the sci.math group): http://mathforum.org/kb/message.jspa?messageID=6126425&tstart=0

[**] More on Golden's Polysigned Numbers on his web site: http://www.bandtechnology.com/PolySigned/index.html

[***] The asymmetries are not completely avoided, but I guess Mr Golden might not have heard of closed number systems, yet.

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Anthony A. Aiya-Oba
Prime Two Exclusiveness Conjecture
Posted: Mar 8, 2008 11:33 AM
http://mathforum.org/kb/message.jspa?messageID=6128731

Other than two, there exists no prime, whose sum of its factors equals prime. -Aiya-Oba (Poet/Philosopher).

Thus, P/1 + P/P = 3, is solely, and solely, P = 2.
Q E D

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Julio Di Egidio
Re: Prime Two Exclusiveness Conjecture
Posted: Mar 8, 2008 12:30 PM
http://mathforum.org/kb/message.jspa?messageID=6128733

2 dis-is Q E D

-LV

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