**of Axioms-**

*The Goedelian*__Dialectic__**of Arithmetic -- a**

*Systems**System*

__s__*-Progression*, or

*"*

**-**__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 as such, but also the totality of the "internal-to-mind" realm of**

*physis**"*, of

**memes**"*"*object

**idea**-

*s**"*, or of <<

**>> that humans create and experience.**

*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***, i.e., of those very special**

*axioms***that state their**

*sentences***.**

*rules*Per

**F**.

**.**

__E__**., the**

__D__*"*of human history, includes not only what humans experience as

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

__ex__*"*of this latter realm -- humanly experienced as

**-**__psycho__**historical material**"*"*ternal-to-human-minds" -- of

__in__**, of**

*language**"*, of

**memes**"

*idea**-*objects, of <<

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

*eide*

__phe__nome*"*.

**F**.

**.**

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

__D__*"*of self-development, tied to the

**laws**"

*historical**"*of the human social individual in terms of the

**self**-**-**__meta__**evolution**"*"*[

**two****]**

__qual__itatively

*different sides**"*[Marx,

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

__Grundrisse__*"*

**social**

__relations__

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

*"*

**social**__forces__

*of*

*production**"*.

**F**.

**.**

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

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

**laws**"*"*, 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 materialism**"*circa*

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

Below, I paraphrase [''']

**F**.

**.**

__E__**.'s account of this 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, at the 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"

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

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

In

Circa

Goedel's

Goedel exploited, in proving his

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,

Goedel's "undecidable" sentence essentially

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

If such a sentence is false, then its logic/arithmetic system is

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, if that logical system encompasses as little as the "

But Goedel's

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

**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 philosophers, mathematicians, and scientists held that determining a set of 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 within that system, 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,

**. . .**.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, although it is a well-formed sentence of that deductive system, it is***not a theorem***from the axioms/postulates of that logic/arithmetic system,**__not__logically deducible**is its**__nor__**, its propositional***contrary***,***negation***from the axioms/postulates of that logic/arithmetic system.***logically deducible*If 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 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, if that logical system encompasses as little as the "

**N**atural" Numbers arithmetic, or more.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 [**__in__completeness**] infinite progression of***potentially***ever richer and more descriptively powerful logico-mathematical-scientific systems/languages.***cumulatively*In

The process of this

That is, Goedel's

Goedel thus discovered, via his

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

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

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

**F**.**.**__E__**.'s terms [not Goedel's terms], each successor such system «**__D__**»-contains, but also «***aufheben***»-advances beyond, and thus «***aufheben***»-supersedes, its immediate predecessor system.***aufheben*The process of this

__in__completeness*-*of each such successor system/language of mathematics, as a**to**-**greater**-**partial**-**completeness movement***"*of its immediate predecessor mathematical system, thus**conservative extension**"**, 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__**, per***process***F**.**.**__E__**.**__D__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 -- what Goedel called***aufheben**the**"***inexhaustibility**"**.***of mathematics*Goedel thus discovered, via his

**,**__In__completeness Theorems*'*, a**The****Gödelian****'**__Dialectic__**that is logically, deductively**__dialectic__*immanent***mathematics!***in*Goedel himself formulates this in terms of philosopher Bertrand Russell's "theory of

*logical types**"*. First, let's outline Russell's*"*theory.**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 saw 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

Here is the formulation of what

"...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

"Now if the system under consideration (call it

Again:

"If we imagine that the system

"Also, the

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

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

"... The

"Deformalizing

[

where

"An [in]equation

**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 what

**F**.**.**__E__**. [not Goedel] terms**__D__*'*, in Goedel's own words --**The Goedelian**'__Dialectic__"...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**.**.**__E__**.] is**__D__**by the introduction of variables for classes of numbers,***successively enlarged***, and so forth, together with the corresponding comprehension axioms, we obtain**__classes____of____classes__of numbers**(continuable into the transfinite [***a sequence**per those who accept assertions about 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**--***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 "*— M.D.],**"**__un__solvable**equations**that I plan to present in a subsequent posting to this thread, on "**The Goedelian**"__Dialectic__of the Standard Arithmeticswhere

**f**and**g**are**n**-ary polynomial[*function*]s. ...""An [in]equation

**[for**__every__**x****from***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***--**

**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**.

**.**

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

__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 asserting that a certain algebraic equation cannot be solved [**

*true***the given logical system of arithmetic] -- not only becomes a decidable, provable proposition within the next higher 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 type**Goedel's Incompleteness Theory is thus the theoretical prediction and explication

*of the**--*

**in****F**.

**.**

__E__**.'s**

__D__

*terms**--*«

**psycho-historical**--__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 blog entry here on**

*equations***.**

*The 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**

__kinds__**'''**

*?*In a subsequent entry to this blog, I plan to present my reconstruction of the

**F**.

**.**

__E__**. equational**

__D__

__dialectical__*"*, of

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

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

__Q__*"*, and via a

**First**"__Dialectical__Arithmetic**, modeling the processes, and the contents, of**

*Dyadic Seldon Function**this "*, i.e., for --

**Goedelian****"**__Dialectic__

__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**,

**. . .**.

Regards,

Miguel

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