The plan to which I committed here, in an entry to this blog, on

**17**September

**2011**--

http://feddialectics-miguel.blogspot.com/2011/09/goedelian-dialectic.html

-- appended at the bottom of this blog entry, has now been fulfilled [although not entirely inside this blog]!

Or, rather,

*fulfilled!*

__-__**HALF**A Part

**I**. of

**II**. of

**F**.

__.__

**E***.*

__D__**Vignette #4**, a text entitled

*"*has just been posted to the the www.dialectics.org web site --

**The Goedelian**"__Dialectic__of the Standard Arithmeticshttp://www.dialectics.org/dialectics/Welcome.html

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

Link to Vignette #4

Part

**II**. of

**II**. is expected shortly.

As Karl Seldon himself -- the co-founder of

**F**.

__.__

**E***. -- has recently written, this text is the most comprehensive "how-to manual" yet published by*

__D__**F**.

__.__

**E***. on dialectical-mathematical modeling using the Seldonian*

__D__**N**

**dialectical arithmetic/algebra.**

__Q__This text explains in copious detail how to construct systematically-ordered dialectical, categorial progression presentations of general subject matter, which can be captured in four-symbol-element mnemonic expressions of the form --

__a__^(2^s)-- [which take

**~**

**8**symbol-elements here, where we lack super-scripts, and super-scripts of super-scripts], exemplified by the expression --

__N__^(2^6)-- which, through the repeated application of three axiomatic arithmetical rules, generates a

**64**-category explication of the "

__atural" Numbers, "__

**N**__hole" Numbers, "__

**W****Integers**", "

**Rational**" Numbers, "

__eal" Numbers, "__

**R**__omplex" Numbers, and the first 'pre-vestige' of the "__

**C****amilton Quaternions", and --**

__H__

__A__^(2^4)-- which generates a

**16**-category explication of the Value-Form Chapters of Marx's three-volume <<

*>>,*

**magnum opus**__:__

**Capital***, including the Marxian categories of the Elementary Commodity Value-Form, the Expanded Commodity Value-Form, the General Commodity Value-Form, the Money Value-Form, the Money as Measure-of- Value Function category, the Money as Medium-of-Circulation Function category, the Money as Means-of-Payment Function category, the Capital Value-Form category, the Commodity-Capital category, the Money-Capital category, the Money-Capital-Mediated-and-Commodity-Capital-Mediated [Self-]Circulation of the Total Social Capital category, and the 'Self-Collision of the Capital-Value-System and the Transition Beyond It' category.*

**A Critique of Political Economy**In particular, the first part of this new text provides a comprehensive background exposition whose coverages includes the following topics [note: an <<

**>> [from ancient Greek] is a scientific**

*aporia**or*

**impasse***-- an apparently unsolvable problem] --*

**quandary****The****‘‘‘****Driver****’’’****for the Progression of the****“****Standard Arithmetics****”**:**The Solution of****“****Unsolvable****”****Equations**.

**The****‘****Dyadic Seldon Function****’****Equational****‘****Meta****-****Model****’****for the Progression of the****“****Standard Arithmetics****”**.

:**Dialectic****The Journey of the Progression of the**«».*Aporia*

**‘****The Gödelian Dialectic****’,****or****, “****The Incompletability of Mathematics****”**.

**The****Inherent****,****Ineluctable****‘****Self****-****Problematicity****’****of****‘****Ideo****-****Formations****’ [****as of****‘****Physio****-****Formations****’]**.

**The Pedagogical Strategy Guiding System Order Choices for Our Presentational****‘****Meta****-****Model****’**.

**The****‘****Evolute****-****ness****’****and****‘****Cumulativity****’****of Dialectical****,****Ontological****-****Categorial Progressions**.

**Rules of Computation of the Dialectic Rules****-****System**:**F****.**__E__**.**__D__**.’s***First*__Dialectical__Arithmetic**,****N**.__Q__

*Encyclopedia Dialectica***’***s***‘****Organonic Algebraic Method****’****for the Solution of Dialectical****-****Algebraic Equations**.

**From****‘‘‘****Formal Subsumption****’’’****to****‘‘‘****Real Subsumption****’’’,****and the****‘****Culminant****’****/Epitome of the Latter**.

Enjoy!!!

Regards,

Miguel

P. S. I am appending my earlier entry on this topic below, FYI --

"

**of the Axioms-**

*The Goedelian*__Dialectic__**of the "Standard Arithmetics" -- a System**

*Systems***-Progression, 'Presentational Diachronic Meta-System', or**

__s__*"*,

**-**__Meta__**System**-**atic**"__:__

*Dialectic*Dear Readers,

The

**F**.

**.**

__E__**.**

__D__*"*theorizes, per

**"**__Dialectical__Theory of Everything**F**.

**.**

__E__**., not just the totality of the "external-to-human-mind", <<**

__D__**>>, "phys-io", "phys-ic-o", or "phys-ic-al" reality that humans experience, but also the totality of the "internal-to-mind" realm of**

*physis**"*, of

**memes**"*"*object

**idea**-

*s**"*, or of <<

**>> that humans create and experience -- of '''the Human Phenome'''.**

*eide*The latter, historically later realm includes human

**in general, including also "**

*languages***", "artificial", "ideogram-ic"**

__formal__**, such as**

*languages***of arithmetic, represented by their**

*systems*

*axioms**-*systems, their <<

**>> of**

*arithmoi***[their assemblages of axiom-units], i.e., of special**

*axioms***stating their**

*sentences***.**

*rules*Per

**F**.

**.**

__E__**., the "historical material" of human history, includes not only what humans experience as**

__D__*"*ternal-to-mind", physical material, but also the

__ex__*collective, inter-subjective "*historical material" of this latter realm -- humanly experienced as

__-__**psycho***"*ternal-to-human-minds" -- of language, of

**in***"*, of

**memes**"*objects, of <<*

**idea**-*>> -- of the "human*

**eide***.*

**"**__phe__nome**F**.

**.**

__E__**. holds that this latter realm has its own special**

__D__*"*of self-development, tied to the historical "self-meta-evolution" of the human social individual in terms of the

**laws**"*"*[

**two****]**

__qual__itatively*[Marx,*

**different sides**"**] of the human-social individual, namely, the "side" of the human**

*Grundrisse**"*

**social**

__relations__*, and the "side" of the human*

**of production**"*"*

**social**__forces__

**of***.*

**production**"**F**.

**.**

__E__**. also holds that these latter, later special**

__D__*"*of human-historical self-development constitute the "subject-

**laws**"*, the*

**"**__matter__*'*, of the "meta-science" of

**psychohistorical**'__material__*"*, or of

**psychohistory**"*"*.

**psychohistorical**"__dialectics__Now, per

**F**.

**.**

__E__**., a major discovery in the history of this**

__D__*"*[not explicitly presented as such by its discoveror] occurred

**-**__psychohistorical____material__**ism**"*circa*

**1933**, in the pit of the last Global Great Depression, a discovery achieved by the logician, mathematician, and

*de facto*physicist, Kurt Goedel.

Below, I paraphrase [''']

**F**.

**.**

__E__**.'s account of this, his discovery, with interspersed commentaries, and other textual variations, of my own.**

__D__'''You may never have heard of Kurt Goedel, a man who died in the mid

**1970**s.

However, the "news" about his extraordinary life story, and about his extraordinary accomplishments, is slowly leaking out to a wider world.

He is arguably the greatest logician since Aristotle, and the greatest mathematical logician of all time, to-date.

He was a close bud of Einstein's during their later years, when they were both contemporaneously honored by appointments to the prestigious Institute for Advanced Study, in Princeton, NJ.

Goedel even discovered a special solution to Einstein's Equations of General Relativity -- the first system of integro-differential equations to be capable of modeling the cosmos as a whole, at least from the point of view of the universal gravity field -- and a notoriously difficult-to-solve, or even, supposedly, "unsolvable in general", system of ten "simultaneous"

**partial differential equations.**

__NONLINEAR__Goedel's celebrated special solution even predicted the possibility of a certain kind of

*time***!**

*travel*But the "physicality" of this solution is still in doubt, and it was but a backwater to the core of Goedel's scientific contributions.

From the times of Plato, Aristotle, and Euclid, circa

**364**B.C.E., right up until the dissemination of Goedel's celebrated

**, circa**

*Incompleteness Proof***1933**C.E. -- in the pit of the last Global Great Depression -- the consensus of academic philosophers, mathematicians, and scientists held that determining a set of -- supposedly "self-evident" -- fundamental sentences or propositions, called "postulates" and/or "axioms", and generating theorems from those premises by strict deductive logic, was the only secure way to true mathematical and scientific knowledge.

In

**1930**, Goedel proved, in his Ph.D. dissertation, his

*"*-- that the basic axiomatic deductive system

**"**__Completeness__Theorem**-- called the "first order predicate calculus" -- was "complete" in the "semantic" sense -- in the sense of "meaning-content": essentially that the axioms of deductive logic were capable of supporting the deduction of every single sentence that is true or valid per our "intuitive" grasp of deductive logic.**

*of deductive logic itself*Circa

**1933**, the world was learning of Goedel's second major, world-historical accomplishment -- his two

*"*.

**"**__In__completeness TheoremsGoedel's

**prove that, within any deductive logical system able to deduce as little as "**

__In__completeness Theorems**N**atural" Arithmetic -- the arithmetic of the "counting numbers",

**{1, 2, 3, ... }**-- or more, there must be a "well-formed" sentence of that system which can be shown, 'meta-deductively', to be, in fact, true, but which cannot be "decided" -- neither proved true, nor proved false -- by strict logical deductions from the axioms/postulates of that logico-arithmetical system,

**that system is "logically useless", i.e., is "**

__unless__**consistent" -- meaning**

__in__

**propositionally****-**

__self__**-- able to deduce**

*contradictory***a proposition and its formal**

__both__**.**

*negation*Goedel exploited, in proving his

**, a close analogue of the ancient paradox of Epimenides, which can be rendered, in essence, by the following sentence:**

__In__completeness Theorems*"*

*This sentence is false**"*.

If you think through that Epimenides sentence, you will see that its truth-value appears to oscillate endlessly, back-and-forth, from True to False to True to False again, and again, and again, in your mind

**. . .**.

Goedel's "undecidable" sentence essentially

*says of itself***that it is**

__not__*false*, but that it is

**of the logico-mathematical deductive system inside which it arises, i.e., that it is**

*not a theorem***from the axioms/postulates of that logic/arithmetic system.**

__not__logically deducibleIf such a sentence is true, then its logic/arithmetic system is

**-- unable to deduce all of its own, internally true propositions.**

__in__completeIf such a sentence is false, then, even worse, its logic/arithmetic system is

**-- able to deduce a**

__in__consistent*false*proposition.

So, Goedel thus proved that the method of rigorous logical deduction from axioms/postulates could never grasp "the whole truth" -- not even the "partial whole" of the truths contained within the logical system carved out by any given, limited set of axioms / postulates.

But Goedel's

**finding should**

__In__completeness**be taken -- as it typically is -- to be a**

__not__**truth about any sufficiently-rich system of deductive logic/arithmetic.**

__static__As Goedel himself was repeatedly at pains to point out, his

**findings locate the driver of a [potentially] infinite progression of**

__in__completeness*ever richer and more descriptively powerful logico-mathematical-scientific systems/languages, each successor such system «*

**cumulatively****»-containing, but also «**

*aufheben***»-advancing beyond, and thus «**

*aufheben***»-superseding, its immediate predecessor system.**

*aufheben*The process of this

__in__completeness*-*

*to**-*

*greater**-*

*partial**-*

**of each such successor system/language of mathematics, as a**

*completeness movement**"*

*conservative extension**"*of its immediate predecessor mathematical system, thus

**, but also**

__negates__**, and also**

__conserves__**, into a**

__elevates__**,**

__higher__

__more__**, system, its immediate predecessor system, thus meeting the definition of an «**

__inclusive__**»**

*aufheben***, that is, of a**

*process*

__dialectical__**.**

*process*That is, Goedel's

**logically predict a potentially infinite**

*Incompleteness Theorems***«**

*cumulative***»-progression of number sets / number systems / number spaces, and of axiomatic systems of arithmetic for those numbers and their spaces -- what Goedel called**

*aufheben*

*the**"*

**inexhaustibility**"**.**

*of mathematics*Goedel thus discovered, via his

**,**

__In__completeness Theorems*'*, logically

**The****Gödelian**__'__**Dialectic**

**immanent***mathematics!*

**in**Goedel himself formulates this in terms of philosopher Bertrand Russell's "theory of

*.*

**logical types**"Let's say that a set that "contains" only the names of

**-set idea-objects, "**

__non__*Ur*-objects", like, say, that of the Earth and that of the Sun and that of the Moon, is of Russellian

*"*

**logical type**"**0**, since it "contains" no braces.

Then a set which "contains", as its "deepest" object/member, the

__set__**{ Sun, Moon }**itself, i.e. --

**{**

**Earth, Sun, Moon, { Sun, Moon } }**

-- denotes, in "extensional" form, the "intension", or

**, that both the Sun and the Moon share in common, and is thus of**

*quality*

**logical type****1**[ since it "contains"

**1**pair of braces -- "contains"

**1**level of

**{...}**-parenthetical enclosure-depth

**or**

*more***than the minimal,**

*deeper***0**-depth, outer-only parenthetical enclosure of set content, which defines

**logical type****0**, as we described above].

A set containing, as its "deepest" object/member, the

*set**-*

**containing**

__set__**{Earth, { Moon, Sun } }**, e.g. --

**{**

**Earth, Sun, Moon, { Sun, Moon },**

**{ Earth, { Moon, Sun } }**

**}**--

-- denotes the

**that the Earth shares in common with the common quality shared by the Sun and the Moon, and is of**

*quality*

**logical type****2**[since it "contains" braces to depth

**2**], and so on.

Here is the formulation of

*'*

**The**

*Goedelian*__Dialectic__*'*in Goedel's own words --

"...For any formal system you can construct a proposition — in fact a proposition of the arithmetic of integers — which is certainly true if the system is free from contradiction but which

**[**

*cannot be proved**or*

**—**

__dis__proved**F.**]

*.*__E__*.*__D__**" [**

__in__the given system*the foregoing summarizes Goedel's*

*"*—

**First**"__In__completeness Theorem**F.**].

*.*__E__*.*__D__"Now if the system under consideration (call it

**) is based on**

*S**, it turns out that exactly*

**the theory of types****[**

*the next*__higher__**—**

*logical***F.**]

*.*__E__*.*__D__**not contained in**

__type__**is necessary to prove this arithmetic proposition, i.e., this proposition**

*S***a**

*becomes***theorem if you**

*provable***the**

__add__to the system**[**

*next*__higher__**— M.D.]**

*logical*

__type__**." [Kurt Goedel, "The Present Situation of the Foundations of Mathematics (***

*and the axioms concerning it***1933o**)", in S. Feferman, et al., editors,

**:**

__Kurt Goedel__**[**

__Collected Works__**:**

__Volume III__**], Oxford Univ. Press [NY:**

*Unpublished Essays and Lectures***1995**], p.

**46**,

__emphases__

*added*

*by***F.**].

*.*__E__*.*__D__Again:

"If we imagine that the system

**[**

*Z**a formal, logical, propositional-/predicate-calculus system inclusive of "*

**N**

*atural" Numbers' Arithmetic*,

__NOT__*the full system of the positive and negative Integers, and zero*[

*zero being both*[,

*or neither*]

*positive*

*and*[

*nor*]

*negative*—

**F.**],

*.*__E__*.*__D__*standardly also denoted by*

**Z**—

**F.**] is

*.*__E__*.*__D__**by the introduction of variables for classes of numbers [M.D.: thus of logical type**

*successively enlarged***0**, if individuals numbers are taken to be the non-set

*"Ur-*objects"],

__classes____of____classes__**[M.D.: thus of logical type**

*of numbers***1**, if individuals numbers are taken to be the non-set

*"Ur-*objects"], and so forth, together with the corresponding comprehension axioms, we obtain

**(continuable into the transfinite [**

*a sequence**per those who accept assertions about "actually infinite" constructions that, because infinite, cannot actually be performed by mathematicians*—

**F.**] )

*.*__E__*.*__D__**formal**

*of***that satisfy the assumptions mentioned above, and it turns out that the consistency ... of any of these systems is provable in all**

*systems***systems."**

__subsequent__"Also, the

**constructed for the proof of Theorem**

__un__decidable propositions**1**[

*Goedel's "First Incompleteness Theorem"*—

**F.**]

*.*__E__*.*__D__**[**

*become decidable by the adjunction of*__higher__**— M.D.]**

*logical***; however, in**

__types__and the corresponding axioms**we**

*the*__higher____systems__

*can construct*

__other__

__un__decidable**by the same procedure."**

*propositions*"...To be sure, all the propositions thus constructed are expressible in

**(hence are number-theoretic propositions); they are, however,**

*Z*

__not__decidable in**, but**

*Z***..." [Kurt Goedel,**

*only in*__higher____systems__*On Completeness and Consistency*(

**1931a**), in J. van Heijenoort, editor,

**:**

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

*Two Fundamental Texts in Mathematical Logic***1970**], p.

**108**,

__emphases__**[square-brackets-enclosed commentary] added by**

*and***F.**].

*.*__E__*.*__D__Now, here's there clincher --

For each, successive logico-arithmetical system in the Goedelian progression of such systems, Goedel's "undecidable" meta-mathematical proposition, denoted

**G**-- neither provable nor disprovable from the premises of its logical system -- maps into the "object language" of that logical system of arithmetic+ inside which it arises, that is, it

*"*, to an

**deformalizes**"*"*

**"**__un__solvable**--**

*algebraic equation*"... The

**oedel sentence**

__G__**G**... asserts its own undeducibility from the postulates...."

"Deformalizing

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

*for every*

**n**

*-ary list of number-components of*

**x**

*such that each number-component is a member of the set of 'diophantine', or "*

**N**

*atural", Numbers in use*—

**F.**] ...

*.*__E__*.*__D__**[for**

*every*x in

*Z***--**M.D.

**]**

**fx**

**does not equal****gx**...

[

*the expression to the left is just a generalized, generic expression of the kind of simple "*

**"**__un__solvable**algebraic equations**that I plan to present in a subsequent posting to this thread*, on "*— M.D.],

**The Goedelian**"__Dialectic__of the Standard Arithmeticswhere

**f**and

**g**are

**n**-ary polynomial[

*function*]s. ..."

"An [in]equation

**[for**

__x from__*every*

*Z***--**M.D.

**]**

**fx**

**does not equal****gx**,

where

**f**and

**g**are two such polynomial[

*function*]s, is called diophantine [

*after*

**Diophantus of Alexandria**, a crucial figure in our story, as we shall see, who, circa**250**C.E., wrote the earliest known book on arithmetical proto-ideographic Algebra, entitled*"*

**«**

__The__**»" —**

__Arithmetiké__**F.**] ...."

*.*__E__*.*__D__"By a

**of the equation we mean an**

*solution***n**-tuple

**a**of natural numbers such that

**[**there is at least one

**a from**

*Z***such that --**M.D.

**]**

**fa**

**=**

**ga**... ."

"So

**G**;... asserts the

**of the...equation**

__un__solvability**fx**

**=**

**gx**, and the proof of [

*Goedel's "First Incompleteness Theorem"*—

**F.**] produces... a particular diophantine equation that is

*.*__E__*.*__D__**[**

*really*__un__solvable

__un__*solvable*— M.D.],

**the**__within__**given**axiomatic**system**of arithmetic,**and**also within**all**of__its__**predecessor****systems**of arithmetic, but**within**__solvable__**all subsequent systems**of arithmetic, starting with the next/**successor system**of arithmetic, incorporating**the**, and the axioms governing that__next____higher____logical____type____higher__**, which corresponds to**__logical____type__**the next**, as we shall see__higher____kind__of__number__**..." [Moshé Machover,**

*but whose*__un__solvability cannot be deduced from the postulates**, Cambridge University Press [Cambridge:**

__Set Theory, Logic, and their Limitations__**1996**], pp.

**268-269**,

**[square-brackets-enclosed commentary] added by**

__emphases__and**F.**].

*.*__E__*.*__D__That is, the undecidable proposition into which Goedel's undecidable "I am not a theorem of this logical system of arithmetic" proposition

*"*-- a

**deformalizes**"**proposition, but one that is "undecidable" -- one that cannot be either proven or disproven within the axioms-system of arithmetic in which it arises -- a true proposition asserting that a certain algebraic equation cannot be solved [**

*true***the given logical system of arithmetic] -- not only becomes a decidable, deductively-provable proposition within the next higher[-in-logical-type] logical system of arithmetic,**

*within*

*but the**"*.

**"**__un__solvable**equation of that proposition, and of that predecessor system, becomes a**__solvable__equation in that same successor system, as well as in all of__its__successor systems, using the__new____kind____of____number__that results from the successor system's incorporation of sets of the next higher logical typeGoedel's Incompleteness Theorem is thus the theoretical prediction and explication

*of the**--*

**in our terms**--**psychohistorical**

*dialectical**--*«

**or****»**

*aufheben**--*, of

__progression__of the expansion of arithmetic**-- i.e., of the expansion of**

*the expansion of the*__kinds__of number*'*

**number****-**

*idea*

__ontology__*'*-- and of the

**of**

*conversion**"*

**"**__un__solvable**into**

*equations*

__SOLVED__**, that we will see by way of examples in my planned subsequent post here on**

*equations**.*

**T****he Goedelian**__Dialectic__of the Standard ArithmeticsThe discovery of the zero

*"*of number", of the "negative"

**kind***"*of number", of the "fractional"

**kind***"*of number", of the "irrational"

**kind***"*of number", and of the "imaginary"

**kind***"*of number", etc., are all behind us in history -- the legacy of our ancestors' solutions to their ancestors' "unsolvable" problems.

**kind**But, here and now, what is the

**expansion of**

__next__*'*

**number****-**

__idea__

__ontology__*'*, beyond the

*of number officially known today, that is needed to solve the "unsolvable" equations, and the related theoretical and practical, technological and general societal problems, of our own time*

**kind****'''**

*?*In a subsequent post to this thread, I plan to present my reconstruction of the

**F**.

__.__

**E**__. equational__

**D**

**dialectica**__"meta-model", of__

**l***"*of the Standard Arithmetics' Axioms-Systems, expressed via the algebra of the

**The Goedelian****"**__Dialectic__**N**

__"First Dialectical Arithmetic", and via a Dyadic Seldon Function, modeling the processes, and the contents, of__

**Q***this "*, i.e., for [using spectral-ordinality textual color-coding:

**Goedelian****"**__Dialectic__**red**-

**orange**-

**yellow**-

**green**-

**blue**-

**indigo**-

**violet**] --

__N__--->__W__--->__Z__--->__Q__--->__R__--->__C__--->__H__---> . . .--wherein --

**connotes the axioms-system of the arithmetic of the "**

__N____atural Numbers",__

**N****N**;

**connotes the axioms-system of the arithmetic of the "**

__W____hole Numbers",__

**W****W**;

**connotes the axioms-system of the arithmetic of the "Integers",**

__Z__**Z**;

**connotes the axioms-system of the arithmetic of the "Rational Numbers",**

__Q__**Q**;

**connotes the axioms-system of the arithmetic of the "**

__R____eal Numbers",__

**R****R**;

**connotes the axioms-system of the arithmetic of the "**

__C____omplex Numbers",__

**C****C**, and;

**connotes the axioms-system of the arithmetic of the**

__H____amilton "Quaternions",__

**H****H**,

**. . .**"

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