Unsolvable Algebraic Equations of the XQ Dialectical Arithmetics.

Unsolvable Algebraic Equations of the Seldonian _XQ_ Arithmetics for Dialectics.

Dear Readers,

Caveat.  This blog-entry -- unlike many others, typical within this blog -- is not about interpretations, or 'meta-model meta-equation' solutions, of the generic 'meta-numerals' that inhere in the _XQ_ algebras, wherein X represents one of the standard arithmetics, and wherein the '[pre-]subscriptization' of X indicates the subsumption of that standard arithmetic by one of the non-standard, dialectical arithmetical Q_ axioms-systems, and the subsumption of the set or space of numerals, X, by the set or space of 'meta-numerals' , {q_X} = _XQ.

On the contrary, this blog-entry is about the "uninterpreted", or, more accurately, is about the 'minimally-interpreted',  'meta-numbers' , and their arithmetics and algebras, themselves, i.e., in the generic forms of these 'meta-numbers', arithmetics, and algebras -- about the new mathematical  'ideo-ontology' that undergirds our 'meta-model meta-equations' & their interpretations/solutions.


As in the first text module pasted-in below, we have noted, in this blog, before, the expectation, related to Gödel’s epochal “incompleteness theorem”, that any axioms-system for a standard arithmetic that encompasses at least the so-called “Natural” numbers arithmetic will permit the formation, within its rules of syntax, of (one or more) algebraic equation(s), thus “well-formed”, which are unsolvable within that arithmetic, and one of which will the “deformalization” of the “Gödel formula”, G, for that arithmetic.

G is a formula for a proposition that asserts of itself that it cannot be proved from the axioms of that axioms-system for this standard arithmetic.  Hence, if that proposition can be shown, non-deductively, to be true, then that axioms-system is “formally incomplete”, and the algebra of this standard arithmetic is capable of forming at least one syntactically “legal” equation, whose solution(s) do(es) not exist within this standard arithmetic, and which is an equation that is tied to G.

If, on the contrary, G, in fact, can be formally deduced from those axioms, then that axioms-system for this standard arithmetic is “formally inconsistent”, i.e., is ‘“formally self-contradictory’’’ -- i.e., a false proposition can be deduced from its axioms.

¿How much can we rightfully expect this expectation to hold also for the axioms-systems of those non-standard systems of arithmetic that are the Seldonian ‘arithmetics for dialectics’?    

¿Must the Seldonian ‘arithmetics for dialectics’ also ‘‘‘contain’’’ algebraic equations, tied to their own versions of G, that are “well-formed” but unsolvable within them, out of formal-logical necessity?

Let us grapple, first, with a easier question:  ¿Do the Seldonian ‘arithmetics for dialectics’ also ‘‘‘contain’’’ algebraic equations that are “well-formed” within them, but that are also unsolvable within them?

Well, let’s start by looking at the first explicitly dialectical arithmetic in the Seldonian progression of dialectical arithmetics, the Natural-numbers, N, subsuming dialectical arithmetic, _NQ_.

The _NQ_ system of arithmetic encompasses, through its product rule axiom, axiom §9, an additions-only excerpt from Natural-numbers arithmetic, if only within its subscripts level.

And, indeed, the following equation of _NQ_’s algebra is, within the _NQ_ axioms-system, well-formed but unsolvable:  for every n in N, x = q_x is not in _NQ, if --

x * q_n   =   q_x * q_n  =  q_n.

And, as also with the ‘Gödelian Dialectic’ of the standard systems of arithmetic, for its systematic progression ['---)'] of number-spaces --

N ---) W ---) Z ---) Q ---) R ---) C ---)... .

-- the very next of the non-standard, dialectical arithmetics, in one of the two directions of the Seldonian progression -- the direction which follows the ‘Gödelian progression’ of the standard systems of arithmetic, just given, in terms of the next standard arithmetic that is subsumed by the next version of the Q_ non-standard arithmetics.

In this case, it is the _WQ_ dialectical arithmetic, that renders solvable, within itself, that equation.  That equation is solvable by setting q_x = q₀, and  by applying the ‘double-conservation «aufheben» evolute product rule, axiom §9 [see second module below] --

q_x is in _WQ, for every w in W, if -- q_x * q_w  =   q_w; 

q₀ * q_w  =  q_w + q_w+0  =  q_w + q_w  =  q_w, so q₀ * q_w   =   q_w, including --

q₀ * q₀  =  q₀ + q₀₊₀  =  q₀ + q₀  =  q₀, so q₀ * q₀   =   q₀  [thus, by its subsuming W, the Boolean pattern rises up again within the generally 'contra-Boolean' Q_, in the form of the exceptional value, q₀].

q₀ is not in _NQ  &  q₀ is in _WQ.

However, also as with the “standard systems of arithmetic”, this next arithmetic, _WQ_, has its own unsolvable equation(s) -- in this example, an equation not even expressible -- not “well-formed” -- within _NQ_, because q₀ is not an element of the ‘meta-number space’ _NQ --

for every w in W, q_x is not in _WQ, if --

x * q_w   =   q_x * q_w  =  q₀.

In this case, it is the _ZQ_ dialectical arithmetic, that renders solvable, within itself, the equation immediately above.  That equation is solvable by setting q_x = q_-w, and by applying a new version of axiom §9, e.g., the ‘meta-heterosis convolute product rule’, as follows --

q_x is in _ZQ, for every z in Z, if -- q_x * q_z  =   q₀; 

q_-w * q_+w  =  q_{+w + -w}  =  q_w-w   =  q₀, so --

q_-w * q_+w  =   q₀, and, in general, q_-z * q_+z   =   q_+z * q_-z   =   q₀, for every z in Z.

q_-w is not in _WQ  &  q_-w is in _ZQ.

However, also, and, again, as with the “standard systems of arithmetic”, this next arithmetic, _ZQ_, has its own unsolvable equation(s) -- in this example, an equation involving the operation of ‘ “purely”-qualitative, categorial division’ --

q_x is not in _ZQ, if, for all a, b, and c in Z, such that a ≠ b ≠ c ≠ a --

x    =   q_x   =   q_a /( q_b + q_c ).

The operation of ‘ “purely”-qualitative, categorial division’, '_ /_', is not defined for the _NQ_ arithmetic, so that the equation immediately above would not be “well-formed” within it.

The operation of ‘ “purely”-qualitative, categorial self-division’ can be defined within the _WQ_ dialectical arithmetic, such that, for every w in W, q_w /q_w    =    q_w  *  q_-w   =   q₀.

Unrestricted binary ‘qualitative division’ is defined for the _ZQ_ dialectical arithmetic, e.g., via the ‘meta-heterosis convolute product rule’ version of axiom §9,  such that --

q_a /q_b  =  q_a *  q_-b  =  q_{a + -b}  =  q_a-b.

However, even in _ZQ_, and in _QQ_, and in _RQ_, the ‘qualitative division’ of a category by a cumulum of categories --

q_a/( q_b + q_c )

-- is not a value that is contained in any of their ‘meta-number spaces’.

One has to progress in the other direction of the Seldonian double-progression [see bottom-most two modules, below] of systems of arithmetics for dialectics, to find an arithmetic in which a value of a form similar to that of --

q_a /( q_b + q_c ) 

-- is contained, namely, in the form b /(a₁ + a₂).

That arithmetic, one that accomodates "purely" qualitative, 'qualo-fractal' '''fractions''' of that form, is the 24th arithmetic in that Seldonian progression of dialectical arithmetics --

q_BA [---) q₂₄ 

-- and that 24th system of arithmetic is capable of interpretation to represent two-level, e.g., «genos»-over-«species», ‘ideo-taxonomic’, trans-Platonian «arithmoi eidetikoi» dialectics.


For more regarding these issues, see the F.E.D. Primer by F.E.D. Secretary-General Hermes de Nemores --

www.dialectics.org

http://www.dialectics.org/dialectics/Primer.html

http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

-- pages A-15 through A-17. 

Regards,

Miguel