__Full Title__: ‘

**The Gödelian Dialectic**’,

**or**, “

**The Incompletability of Mathematics**”.

Dear Readers,

The following is an excerpt on the unconscious Marxian-dialectical essence of what the mathematical Platonist, Kurt Goedel, called "The Incompletability of Mathematics", or "The Inexhaustibility of Mathematics".

It is excerpted from

**F**.

__.__

**E***. Vignette*

**D****#4**--

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

http://www.dialectics.org/dialectics/Vignettes_files/v.4,Part_I_of_II,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_10OCT2012_1.pdf

**"**Kurt Gödel, arguably the contributor of the greatest leaps forward in the science of formal logic since classical antiquity, described an ‘axiomatic dialectic’ of mathematics, albeit in “[‘early’-]

*Platonic”*,

*‘‘‘*and

**-psychological’’’**__a__*‘‘‘*terms, hence in

**-historical’’’**__a__*‘*terms as well, as follows --

**-**__a____psychohistorical__’**“**It can be shown that

**— whether it is based on**

*any formal system whatsoever***or not, if only it is free from contradiction —**

__the____theory____of____types__**.**

*must necessarily be deficient in its methods of proof*Or to be more exact: For any formal system you can construct a proposition — in fact a proposition

**— which is certainly true if the system is**

*of the arithmetic of integers***but**

*free from contradiction***[**

*cannot be proved in the given system***F**.

**.**

__E__**.: the foregoing summarizes Gödel's "First Incompleteness Theorem"].**

__D__Now if the system under consideration (call it

**) is**

*S***, it turns out that**

*based on*__the____theory____of____types__

*exactly***not contained in**

*the next higher type***is necessary to prove this**

*S***proposition, i.e., this proposition**

*arithmetic***if you add to the system**

*becomes a provable theorem*

*the next higher type***.**

*and the axioms concerning it***”**

[Kurt Gödel, “The Present Situation of the Foundations of Mathematics (*1933o)”, in S. Feferman,

*et. al.*, editors,

**:**

__Kurt Gödel__**(**

*Collected Works**Volume*III:

*Unpublished Essays and Lectures*),

*ibid*., page

**46**;

**bold**,

*italics*,

__underline__, and color emphasis added by

**F**.

**.**

__E__**.].**

__D__Again --

**“**If we imagine that the system

**Z**[

**F**.

**.**

__E__**.: a formal, logical, propositional-/predicate-calculus system inclusive of**

__D__*“*

**Natural**”*Numbers’*Arithmetic,

**the full system of the positive and negative**

__not__**, and zero ... standardly also denoted by**

*Integers***Z**] is successively enlarged by the introduction of variables for classes of numbers, classes of classes of numbers, and so forth, together with the corresponding

**, we obtain a**

__comprehension____axioms__**...**

__sequence__**that satisfy the assumptions mentioned above, and it turns out that**

__of____formal____systems__**...**

*the consistency***.**

*of any of these*__systems__is provable in__all____subsequent____systems__Also,

*the*__undecidable____propositions__constructed for the proof of Theorem**[**

*1***F**.

**.**

__E__**.: Gödel’s “First Incompleteness Theorem”]**

__D__**; however,**

__become____decidable__by the__adjunction__of__higher____types__and the__corresponding____axioms__**by the**

*in the higher systems we can construct other undecidable propositions***. ...**

*same procedure*To be sure, all the propositions thus constructed are

__expressible____in__**Z**(hence are

*number**-*); they are, however,

**theoretic propositions**

__not____decidable____in__**Z**, but

**...**

__only__in__higher____systems__**”**

[Kurt Gödel;

__On Completeness and Consistency__(1931a), J. van Heijenoort, editor,

**:**

__Frege and Gödel__**, Harvard University Press [Cambridge:**

*Two Fundamental Texts in Mathematical Logic***1970**], page

**108**,

**bold**,

*italic*,

__underline__, and color emphasis [and square-brackets-enclosed commentary] added by

**F**.

**.**

__E__**.].**

__D__The cumulative «

**»-progression of**

__aufheben__*‘‘‘*, i.e., the advancing

**conservative extensions**’’’*‘ideo-cumulum’*of axiomatic systems which Gödel describes above was viewed

**historically by him.**

__a__Gödel was an avowed ‘‘‘neo-Parmenidean’’’, and a professed “mathematical Platonist” [in the sense of the

*earlier*rather than of the

*later*Plato].

He didn't intend this

*‘*— this

**-**__meta__**system**’*[*

__cumulative____micro__-__diachronic____progression__of*axioms-*]

*— to serve as a*

__systems__*or*

__temporal__*model of the stages of human mathematical cognition, as reflective of the stages of the self-development of humanity's collective cognitive powers as a whole; of the*

__psychohistorical__**to which each such epoch of those powers renders access, and of the**

*knowledges**“historically-specific”*

**[or**

*ideologies**] to which human thinking is susceptible within each such epoch.*

__pseudo__-knowledgesWe of

**F**.

**.**

__E__**., however, do wish to explore its efficacy as such.**

__D__Note how, as Gödel narrates this axioms-system

**-progression above, each successor system «**

__s__**»-contains its immediate predecessor system, and, indeed,**

*aufheben***of its predecessor systems; how each higher**

__all____logical__

__type__«

**»-contains**

*aufheben***predecessor**

__all____logical__

__types__.

Can Gödel's theory of this

**, ‘evolute’, «**

__cumulative__**» progression of axioms-systems, which we term**

*aufheben**‘*, or

**’**__The____Gödelian____Dialectic__*‘*[Idea-Systems'

__The____Gödelian__

*Ideo**-*]

__Metadynamic__*’*, provide at least an idealized [i.e., a distorted] image of

**actual history**, of the actual psychohistorical struggle, process, and progress of mathematical aspects of the self-development of a humanity's universal labor [Marx]; of its collective cognitive capabilities, hence of its knowledges and ideologies?

We say “yes!”.

[To clarify our terminology here: we mean, by an ‘

*ideo**-*, a pattern that characterizes the ‘int

**dynamic**’**-systemic’ process of, e.g., deductively proving ever-more theorems within a single axioms-system, in the context of a diachronically presented ‘**

__ra____meta__-system[atic]’ progression of axioms-system

**.**

__s__We mean, on the contrary, by the neologistical term

*‘*, the pattern characterizing the ‘int

**ideo**-**-**__meta__**dynamic**’**-systemic’,**

__er__

__non__*-*deductive-logic-al process of moving from inside each given predecessor axioms-system, to outside it, to inside its successor axioms-system].

Each of Gödel's "undecidable" propositions of arithmetic that plague each ‘‘‘epoch’’’ of this formal axiomatic expansion is a proposition asserting,

**-- ‘‘‘**

*of itself**’’’ -- that it is a formula which cannot be deductively demonstrated from the axioms of its axioms-system of arithmetic.*

**self**-**reflexively**That is, each ‘‘‘Gödel formula’’’ states that

*“*-- not a provable statement of the axioms-system of arithmetic inside which it immanently arises, as a well-formed formula within that system.

**I am not a theorem of my axioms**-**system**”If such a proposition is true -- if the proposition actually

**be deductively demonstrated from the axioms of the axioms-system in which it arises -- then things are**

*can*__not__**from an immanent, formal-logical point-of-view; then the system in which that proposition arises immanently is logically**

*very bad**“*-- i.e., is incapable of deductively demonstrating all of the true propositions which arise as well-formed formulas within the rules of that system, because its Gödel proposition exhibits that there is

**incomplete**”

__at__**this one**

__least____true__proposition within it that it cannot deductively prove.

If such a proposition is false -- if the proposition actually

**be deductively demonstrated from the axioms of the axioms-system in which it arises immanently -- then matters are**

__can__**from an immanent formal-logical point-of-view; then the axioms-system of arithmetic in which this proposition arises immanently is even**

*even worse***;**

*contradictory**‘‘‘*, i.e., is capable of deductively demonstrating at least one

**self**-**’’’**__in__consistent**proposition, as well as the [**

*false***] negation of that proposition, and thus of deductively “demonstrating” two [or more] mutually**

*true*

*formally***propositions, because its Gödel proposition exhibits that there is at least this one**

*contradictory***proposition within it that it can, erroneously, deductively “prove”.**

*false*That is, the axioms-system can derive,

**, a proposition that says of itself “I am not a theorem of this axioms-system.”, and which, thus proven as a theorem, is thereby shown to be a**

*as a theorem***proposition.**

*false*That is the “meta-mathematical” content of a ‘‘‘Gödel Formula’’’, its meaning at the level of the “meta-language” which talks about axioms-systems, provability from axioms, etc.

But the real secret of the meaning of such ‘‘‘Gödel Formulae’’’, seldom mentioned in the standard accounts of Gödel’s “First Incompleteness Theorem”, is their “deformalized” content, their “mathematical”, i.e. arithmetical and algebraic content; their meaning in the “object language”, which talks about numbers, algebraic [“diophantine”] equations, and their solvability, etc.

At that deeper, more concrete level, a ‘‘‘Gödel Formula’’’ is a proposition asserting the

**of a specific**

*solvability*__un__*“*

__diophant__**ine**” [

**F**.

**.**

__E__**.: referring to the circa**

__D__**250**C.E. proto-ideographic-algebraic work by

**us of Alexandria, the «**

__Diophant__**»] algebraic**

__Arithmetica__**.**

__equation__I.e., each “Gödel formula”, or “Gödel sentence”, which, at the “meta-mathematical” level, asserts the

__either__*-*self-incompleteness-

__or__*-*self-inconsistency of its axioms-system, “deformalizes” to one which asserts the

**--**

*solvability*__un__**that system of arithmetic, and given the**

__within__*limitations*of the

*kinds*of

*numbers*that are at that axioms-system’s disposal -- of a specific, algebraic, “

**” --**

__diophantine____equation__**“**... The Gödel sentence

**φ**

**...**asserts its own

*undeducibility*from the postulates....

Deformalizing

**φ**... we see that under the standard interpretation it expresses a fact of the form [

**F**.

**.**

__E__**.: for every**

__D__**n**-ary list of number-components of

**such that each number-component is a member of the set of ‘diophantine numbers’, e.g., of integers] ...**

*x***ƒ**... , where

*x*~=*gx***ƒ**and

**are**

*g***n**-ary polynomials [

**F**.

**.**

__E__**.: I.e., are**

__D__**n**-ary

*polynomial*

**, whereas**

__functions__**ƒ**and

*x***denote their**

*gx**function-*]....

__values__An equation

**ƒ**

*x***=**

**, where**

*gx***ƒ**and

**are two such polynomials, is called**

*g**diophantine*....

By a solution of the equation we mean an

**n**-tuple

**of natural numbers such that**

*α***ƒ**

*α***=**

**... .**

*gα*So

**φ**

**...**asserts the unsolvability of the...equation

**ƒ**

*x***=**

**, and the proof of [**

*gx***F**.

**.**

__E__**.: Gödel’s “First Incompleteness Theorem”]**

__D__*produces... a particular diophantine equation that is really unsolvable, but whose unsolvability cannot be deduced from the postulates*...

**”**

[Moshé Machover,

__Set Theory____,__, Cambridge University Press [Cambridge:

**,***Logic**and their Limitations***1996**], pages

**268**-

**269**,

*emphasis*

*and*square-brackets-enclosed commentary

**by**

*added***F**.

**.**

__E__**.].**

__D__Each of the [‘“locally”’]

*unsolvable*algebraic equations that we will evoke, in the core section of this essay, in our narration of the Dyadic Seldon Function ‘meta-model’ of the dialectic of the Standard Arithmetics, will be seen to formulate, and to ‘explicitize’, a

**for that concept of [counting] number which is implicit in the “[syn]thesis” arithmetical axioms-system term**

__paradox__

__X__**, of the ‘**

__#__

*aporial**antithesis-sum’*,

__X__

__#__**+**

__x__**.**

__#__Per the modern definition, a "diophantine equation" is an equation whose

*[e.g.,*

__parameters__*coefficients*] and whose

*are restricted to the “integers”, or, sometimes, to “rational” numbers [in the case of Diophantus himself, positive rational numbers only, excluding zero, etc.].*

__solutions__Each “Gödel sentence”-encoded equation truly is unsolvable

*the given axioms-system.*

__within__However, the proposition that it is so,

**be deductively proven**

*can*__not__*that axioms-system.*

__within__But that proposition

**be so proven within the**

__can____next__axioms-system, its immediate successor — the

*being created through the «*

__latter__**»**

__aufheben__*‘*

__self__*-*of the ‘‘‘vanguard’’’, ‘meristemal’, highest [in

**’**__internalization__*“logical*

*type”*] set idea-objects of the universe of discourse of the

*axioms-system, a ‘self-internalization’ which produces sets of the next higher*

__predecessor__*“logical type”*.

That proposition can also be so proven within all subsequent successor-systems, created by yet-further such «

**»**

__aufheben__*‘*

__self__*-*.

**’**__internalizations__*“Logical type”*works like this.

If we say that the “universal set”, or “universal class”, containing all of the “logical individuals”, e.g., all of the individual numbers, that constitute a given “universe of discourse [of arithmetic]”, e.g.,

**{0, 1}**, for the “universe of discourse” of the arithmetic of Boolean algebra, is of

*“logical type”*one, then the set of all of its subsets,

**{**

**{0}, {1}, {0, 1}, { _ }**

**}**,

with ‘

**{ _ }**’ denoting the “empty set”, is of

*“logical type”*two, and the set of subsets of that set of subsets --

**{**

**{{0}}, {{1}}, {{0, 1}}, {{ _ }}, { {0}, {1}, {0, 1}, { _ } }, {{0}, {1}}, {{0}, {0, 1}}, {{0}, { _ }},**

**... }**

-- is of

*“logical type”*three.

That is, we are, in effect, counting the ‘“depth”’ of the braces of a set, including counting the main, outer, braces, to assess the

*“*of that set, i.e., to measure the ‘‘‘depth’’’ of ‘sets-

**logical type**”

__as__*-*content of the set in question.

**’**__elements__If the

*“logical individuals”*, or

*‘arithmetical idea-objects’*, “existing” per the

*“*of a given axioms-system, are limited to “

__comprehension____axioms__”**N**atural” Numbers,

*classes*of “

**N**atural” Numbers,..., all the way up to

*classes of classes of*... of “

**N**atural” Numbers, e.g., to ‘class-objects’ up to a given

*“logical type”*, then the next system will

*expand those ‘‘‘existential’’’ limits by one step, to include also*

__cumulatively__*classes of classes of*.. of “

__classes__.**N**atural” Numbers, i.e., ‘class-objects’ of next-higher

*“logical type”*.

Starting from the “universal class”, each second and higher class-inclusion of previous ‘class-objects’ can model [including via

*adjunction*of those object’s corresponding

*“*, defining the

__comprehension____axioms__”*‘computative behavior’*of these new entities] -- e.g., via the new, higher logical type-level of those special kinds of sets called “ordered pairs”, that arise, for the first time, in the second step of universal class self-inclusion -- a

*of arithmetical*

__new____kind__*‘idea-object’*;

*indeed,*

__a__*.*

__new__,__higher____kind____of____number__Thereby, this

*qualitative expansion of each*, in the formation of its

**predecessor axioms**-**system**

*successor axioms**-*, together with the adjunction of the additional,

**system***“*to the previous, predecessor axioms, corresponds to a

__comprehension____axioms__”*expansion of the*

__qualitative__*‘idea-ontology’*-- of the

*‘arithmetical ontology’*, i.e., of the

*‘*-- of that

**-**__number____ontology__’

*predecessor axioms**-*, thereby transforming it into its

**system**

*successor axioms**-*.

**system**__Hypothesis__: Specifically,

*the*

__diophantine____equation__that*was*as such

__unsolvable__*within the*‘“Natural”’ Numbers

**-**__predecessor____axioms__**itself becomes**__system__**, albeit in a**__solvable__**-diophantine sense, i.e., by a**__non____-diophantine type of number -- a number beyond the__**non***-- within the*[

**next***as well as within all subsequent*]

*successor axioms**-*in this

**system**__s__*cumulative sequence of axioms-systems, precisely by means of these next higher kinds / logical types of “ordered pair” sets, and by means of the*

**new**which they ground, which will__kinds__of numbers__not__*be ‘diophantine numbers’, e.g.,*‘“Natural”’ Numbers.

**not**The result is a progression of qualitatively-distinct, ‘ideo-ontologically’ distinct, ‘number-space distinct’, axiom

*-systems of arithmetic.*

__s__Each successor arithmetical axioms-system “contains” the kinds of numbers, and at least some of the axioms, of all of its predecessor arithmetical axioms-systems.

It also contains a new kind of number, “absorbing” and “converting” into itself -- into its kind -- all of the previous-systems’ kinds of number that it “contains”, and, with this new kind of number, solving a kind of algebraic equation that was unsolvable in its immediate predecessor arithmetical axioms-system, and proving a kind of proposition that was “undecidable” -- neither provable nor dis-provable -- in its immediate predecessor arithmetical axioms-system.

But this successor arithmetical axioms-system also has its own, new kind of undecidable theorem, and its own, new kind of unsolvable algebraic equation, a ‘self-incompleteness’ which leads on to its own successor axioms-system of arithmetic.

We term such a progression a

*‘*

*Gödelian*

__Dialectic__*’*.

What we present herein, in the core section of this essay, is, precisely,

*‘*

*The*

*Gödelian*__Dialectic__of the Standard Arithmetics*’*, as encoded in a ‘dialectical meta-model’ of its systematically-ordered method-of-presentation, expressed via the ‘Dyadic Seldon Function Dialectical Equation’ introduced in the third sub-section of this background section.

Given the [

__potentially__*-*infinite character of this dialectical, «

**» system**

*aufheben**-progression of axioms-systems of arithmetic, driven by the provability-incompleteness and the equational-unsolvability that characterizes every possible axioms-system of arithmetic in this progression of axioms-systems of arithmetic, Gödel himself describes this progression as manifesting*

__s__*“*

*the incompletability or inexhaustibility of mathematics*

*”*,which, from our point of view, represents also the

*potential*

*of*

__interminability__*‘*[

*The***]**

*Gödelian*

__Dialectic__*’*--

**“**The metamathematical results I have in mind are all centered around, or, one may even say, are only different aspects of, one basic fact, which might be called

*. ...*

**t****he incompletability or inexhaustibility of mathematics**The phenomenon of

*, however, [is] always present in some form, no matter what standpoint is taken.*

**the inexhaustibility of mathematics**So I might as well explain it for the simplest and most natural standpoint, which takes mathematics as it is, without curtailing it by any criticism.

From this standpoint all of mathematics is reducible to abstract set theory. ...

So the problem at stake is that of axiomatizing set theory.

Now, if one attacks this problem, the result is quite different from what one would have expected.

Instead of ending up with a finite series of axioms, as in geometry, one is faced with [

**F**.

**.**

__E__**.: a**

__D__*] infinite series of axioms, which can be extended further and further, without any end being visible and, apparently, without any possibility of comprising all of these axioms in a finite rule producing them.*

__potentially__This comes about through the circumstance that, if one wants to avoid the paradoxes of set theory ... the concept of set must be axiomatized in a stepwise manner.

If, for example, we begin with the integers, that is, the finite sets of a special kind, we have at first the integers and the axioms referring to them (axioms of the first level), then the sets of sets of integers with their axioms (axioms of the second level), and so on for any finite iteration of the operation “set of”. ...”

[Kurt Gödel, “Some Basic Theorems on the Foundations of Mathematics and Their Implications (*1951)”, in S. Feferman,

*et. al.*, editors,

**:**

__Kurt Gödel__**(**

*Collected Works**Volume*III:

*Unpublished Essays and Lectures*),

*ibid*., pages

**305**-

**306**].

**Is this Gödelian ‘Dialectic of Arithmetics’ -- this “inexhaustibility” and “incompletability” of arithmetics -- still alive and in evidence today**

*¿*

*?***Are new, ‘ideo-**

*¿**ontologically*expanded’ --

*kinds-*of-numbers-expanded -- arithmetics still being discovered

*?***I.e, do we today**

*¿**to search for a Gödelian “next” arithmetic, with its new kinds of numbers: an arithmetic that has eluded us so far*

__need__

*?***Do we**

*¿**that next, new arithmetic, and its new kind of numbers, in order to solve equations that have remained “unsolvable” ever since their discovery, so far, and right up until to today*

__need__

*?***Do we need it to enable**

*¿**calculations*that we cannot perform even with today’s most advanced mathematics, hence to enable

*predictions*that we cannot, even with today’s most advanced

*‘ideo-technology’*, discern, hence to enable new ‘physio-technologies’, new technologies that are crucial to the very survival of the Terran human «

**», e.g., to mitigate external hazards -- ‘exolithic bombardments’, solar coronal mass ejections; solar and other-stellar mega-flares; magnetars; gamma ray bursts/hypernovae -- and to mitigate internal hazards, e.g., of a New/Final Ice Age, and of a New/Final Dark Age, due to dialectical, internal, immanent [self-]contradictions of the descendant phase of our present «**

*species***» of global [proto-]human civilization**

*species*

*?*

**I.e., do we need this new math. to grow the social forces of production to the next level**

*¿*

*?*Consider the greatest scandal of modern science, which, given its vast magnitude, is mainly mentioned as such only in whispers: the “unsolvability” of “most”

*linear integrodifferential equations.*

__non__This problem has fettered the advance of modern science since the inception of modern science, since the very discovery of such equations, over

**300**years ago, and it encompasses the “unsolvability” of the most important of such equations, those that constitute Terran humanity’s most advanced formulations of its “laws” of nature to date.

These include the “many-body” Newtonian ‘‘‘gravitic’’’ equations, the Einsteinian General-Relativistic ‘‘‘gravitic’’’ equations, the Navier-Stokes equations of

*‘“*

**id”’ dynamics [the dynamics of**

__rhe__*“*

**ological”, or flowing, matter, in the form of electrically nearly-neutralized liquids and gases, thus of weather, etc.], the Maxwell-Boltzmann-Vlasov equation [for electrically non-neutral, plasma**

__rhe__*‘‘‘*

**ids’’’], etc.**

__rhe__This

*is not some merely esoteric, ethereal, rarefied, merely conceptual failure, of concern only to specialists.*

__central____failure____of____modern____science__*It is a failure also in deeply practical terms*.

Were we to discover how to analytically solve the Navier-Stokes equations, we would probably, given the disproportionalities of cause/effect inherent in nonlinearity, be enabled to stop hurricanes in their tracks, by applying a presently-‘harnessable’ amount of energy to their “Achilles-heals”; and to nip tornados in the bud.

Were we to learn to analytically solve the Vlasov equation, we could design a global grid of zero pollution fusion power reactors, harnessing radioactivity-free advanced fuel regimes, emitting only electrons.

The irruption of human capability to analytically solve “most” nonlinear integrodifferential equations would represent an enormous leap forward in “universal labor” [Marx], that could quickly translate into an enormous upsurge in the level of development of

*the human-*

*social*

*forces of*[human society’s self-expanding self-re-]

*production*,

*the core -- the evil, craven,*

__if__*technodepreciation-terrified*, and

*technologically-educated-*middle-working-class-terrified ruling faction -- of the global ruling plutocracy could be prevailed upon,

*by popular insistence*, to desist from their strategy of global enfetterment and reversal of those forces [negative growth].

*Sans*that irruption, and that persuasion, Terran humanity is headed downwards, into a new, global, and, this time, likely

*final*Dark Age.We have mentioned that the «

**» of the “unsolvable equations” that motivate our [meta-]models’ dialectic movement from arithmetic to higher arithmetic will seem “muted” -- because we already know**

*aporia**they were eventually solved, and because we already know*

__that__*they were solved, and because we already know the new kinds of numbers which made those solutions possible.*

__how__Many scientists and mathematicians are wont to say, today -- and without proof

*!*-- that most nonlinear differential equations must remain

*‘‘‘*

*forever*unsolvable’’’.

The «

**» of the “unsolvability” of most nonlinear equations is**

*aporia*

__drastically__*alive*

*today*, but the perception of that ‘drasticity’ is muted

*by despair*, by

*the ideology of eternal unsolvability*.

Still, new “closed-form”, “exact”, “analytical” solutions to vast classes of ‘‘‘minimally nonlinear’’’ partial-differential “evolution equations”,

*including to*

*the*

__Non__*linear*[

*cubic*]

*Schrödinger*

*equation*, keep being discovered -- e.g., those called “nonlinear wave”, or “solitary wave” solutions; the “soliton” solutions, for waves that act like “particles”.

Many such “exact solutions” don’t even require

*new*“transcendental functions”.

*Old*“transcendental functions”, e.g., the hyperbolic-tangent & hyperbolic-secant functions, provide their “exact”, “analytical”, “closed-from” solutions.

Are nonlinear differential equations possibly a kind of Gödelian “diophantine equation”, such as could be the subject of a “Gödel Formula” for the

*de facto*axioms-set of today’s most advanced system of mathematics

**?**

Yuri Matiyasevich, on the way to his solution of Hilbert’s Tenth Problem, his proof that arbitrary diophantine equations are “semidecidable” as to their solvability [in terms of ‘diophantine numbers’] -- including the proof of the ‘diophantinicity’ of the prime numbers, and the derivation of diophantine algebraic [finite] polynomial equations with only integer coefficients, whose positive solution sets, the sets of all positive values that they yield for integer values of their variables, is exactly the set of all prime numbers -- found that, indeed,

**“**the problem of the existence of the solutions of a Diophantine equation in natural numbers can be reduced [

*sic*] to the problem of the existence of a solution of a system of polynomial

__differential__*equations*of first order.

**”**

[Yuri Matiyasevich,

**, MIT Press [Cambridge:**

__Hilbert’s Tenth Problem__**1994**], pages

**xix**-

**xx**,

**46**,

**54**-

**56**,

**85**,

**176**,

__emphasis__**].**

*added*But let us consider an “unsolvable” diophantine equation that is, apparently, a little less “esoteric”, and “much simpler”, than a nonlinear differential equation -- that is a little “closer to home”:

x

x

**=**

**z/0**,

for any

**z**in

**Z**[including for

**z**

**=**

**0**in

**Z**:

**x**

**=**

**0/0**].

That kind of diophantine equation is unsolvable among the

**N**, the

**W**, the

**Z**, the

**Q**, the

**R**, the C, the H -- is still, to this very day, unsolvable in any of the “Standard Arithmetics”.

This «

**», the quandary of**

*aporia**division by zero*, is still very much alive in our own times, and, again, its intensity is muted

*only by despair*, only by the

*hopelessness*that most of us are taught to feel about ever “fixing” this locus where our Standard Arithmetic

*breaks down*, where it

*fails to work*; by the official, consensus denial that there can exist any axioms-system of arithmetic in which zero division could

*“*

*make sense*

*”*; in which this kind of diophantine equation could be

*.*

__usefully__solvedActually, our nonlinear differential equation example is not so remote from our “more homey” zero division example after all: Zero division is the proximate cause of

**, of the “meaningless” or “undefined” values that plague, especially**

*the problem of*__singularity__

__and__*, the*

__essentially__*linear differential equations, which are so inherently prone to singularities, helping to thwart their solvability/“integrability” under their present-day arithmetical undergirding.*

__non__If you want to experience the “bite”, the intensity, the fierceness, the searing mental mood, of a real dialectical «

**» -- still alive and still burning --**

*aporia***muted, then simply let yourself**

__un__*let go of*

*your despair*about ever finding an arithmetic in whose context, e.g.,

**r/0**

**=**

**x**,

for

**r**in

**R**is rendered intelligible.

Simply imagine believing that a solution is “out there”, ready to be found, to that “diophantine equation”.

Simply imagine that finding that arithmetic will solve

**, and, even more, help to unlock the door to the secrets that will habilitate to a full breach of the**

*the singularity problem**‘‘‘*

*Nonlinearity Barrier**’’’*-- including its aspect of the ‘patterned-ness’ of the never-repeating -- a breakthrough to beyond the world-historical «

**» that presently blockades Terran humanity from access to the higher theories, and to the higher technologies, that reside beyond that**

*aporia***, and upon whose acquisition the matriculation of this humanity, from its looming**

*Barrier***‘Meta-Darwinian Planetary Selection Test**’, and from its “prehistory”, in Marx’s sense, so vitally depends.

The “solution” that

**r/0**

**=**

**oo**

**"**

**=****"**

**infinity****=**

**x**,

for

**r**in

**R**is known to incur devastating problems in physical models -- indeed, it leads to ‘infinity residuals’, to ‘infinite error’, to ‘infinitely wrong answers’.

There is almost equal despair, in the consensus view of today’s physics and applied mathematics communities, that the

**of the extant nonlinear partial differential equation models of the “laws” of nature can ever be “solved”.**

*singularities***Can the Gödelian Dialectic, pursued to a sufficiently advanced**

*¿*__s__tage, overcome this ‘‘‘incompleteness’’’ too

*?"*Regards,

Miguel

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