__Full Title__: Part

**06**of

**29**--

#
__The Dialectica Manifesto__

__The Dialectica Manifesto__

##
__Dialectical Ideography__ and

__Dialectical Ideography__

##
the Mission
of F.__E__.__D__.

__E__

__D__

Dear Readers,

I am, together with

**F**.**.**__E__**. Secretary-General Hermes de Nemores, and**__D__**F**.**.**__E__**. Public Liaison Officer Aoristos Dyosphainthos, organizing to develop a new, expanded edition of the**__D__**F**.**.**__E__**. introductory documents, for publication in book form, under a new title --**__D__

__The__

__Dialectica__**:**

__Manifesto__

__Dialectical__

*Ideography*

*and the Mission of***Foundation**

**[**

__Encyclopedia____Dialectica__**F**.

**.**

__E__**.]**

__D__
-- and under the authorship of the entire

**Foundation**collective.
Below is the

**installment of a***sixth***29**-part presentation of this introductory material, which the**F**.**.**__E__**. General Council has authorized for serialization via this blog over the coming months, as we develop the material.**__D__
I plan to inter-mix these installments with other
blog-entries, including the planned additional

**F**.**.**__E__**. Vignettes, other**__D__**F**.**.**__E__**. news, my own blog-essays, etc.**__D__
Links to the earlier versions of these introductory
documents are given below.

Unlike the typical blog-entry, this series will attempt to
deliver an introduction to the

**Foundation**worldview as a**, in a***totality***, making explicit many of***connected account***.***the interconnexions among the parts*
Enjoy!!!

Regards,

Miguel

Part

**06**of**29**--#
__The Dialectica Manifesto__

__The Dialectica Manifesto__

##
__Dialectical ____Ideography__ and

__Dialectical__

__Ideography__

##
the Mission
of F.__E__.__D__.

__E__

__D__

*The Gödelian*

__Dialectic__

## and

##
F.__E__.__D__.’s* ‘*__Dialectic____al__* Meta**-**Axiomatics**’** Method of Presentation*

__E__

__D__

__Dialectic__

__al__

##
*Examples of* __Dialectic__, Example #0:

__Dialectic__

##
*The Gödelian *__Dialectic__.

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

*Platonic”*,

*‘*terms, and in

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

**-historical’**__a__*‘*terms, 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

**you can construct a proposition — in fact a proposition***formal system**of the***— which is certainly true if the system is***arithmetic of integers**free from***but***contradiction***[the foregoing summarizes Gödel’s “First Incompleteness Theorem” —***cannot be proved in the given system***F**.**.**__E__**.].”**__D__
“Now if
the system under consideration (call it

**) is***S**based on***, it turns out that**__the____theory____of____types__*exactly**the***not contained in***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**), Oxford University Press (NY: 1995), page 46;**bold**,*italics*,__underline__, and color emphasis added by**F**.**.**__E__**.].**__D__Again:

“If we
imagine that the system

**[a formal, logical, propositional-/predicate-calculus system inclusive of***Z**“***Natural**”*Numbers’*Arithmetic,**the full system of the positive and negative**__not__**, and zero [which is both, [or neither] positive [n]or negative], standardly also denoted by***Integers***Z**—**F**.**.**__E__**.] is successively enlarged by the introduction of variables for classes of numbers, classes of classes of numbers, and so forth, together with the corresponding**__D__**, we obtain a**__comprehension____axioms__**(continuable into the transfinite)**__sequence__**that satisfy the assumptions mentioned above, and it turns out that**__of____formal____systems__**(ω-consistency)***the consistency***.”***of any of these*__systems__is provable in__all____subsequent____systems__
“Also,

**[Gödel’s “First Incompleteness Theorem” —***the*__undecidable____propositions__constructed for the proof of Theorem 1**F**.**.**__E__**.]**__D__**; however,**__become____decidable__by the__adjunction__of__higher____types__and the__corresponding____axioms__*in the higher systems we can construct***by the***other undecidable propositions***.”***same procedure*
“...To
be sure, all the propositions thus constructed are

__expressible____in__**(hence are***Z**number**-*); they are, however,**theoretic propositions**__not____decidable____in__**, but***Z***...”**__only__in__higher____systems__
[Kurt Gödel;

__On Completeness and Consistency__(1931a), J. van Heijenoort, editor,__Frege and Gödel____:__, Harvard University Press (Cambridge: 1970), page 108;**Two Fundamental Texts in Mathematical Logic****bold**,*italic*,__underline__, and color emphasis and [square-brackets-enclosed commentary] added by**F**.**.**__E__**.].**__D__**»**

__aufheben__*-*progression of

*‘‘‘*, i.e., the advancing

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

**historically by him.**

__a__Gödel, as a ‘Parmenidean’, and a professed “mathematical Platonist” [in the sense of the

*earlier*rather than of the

*later*Plato; see below], didn’t intend this

*‘*— this

**-**__meta__**system**’*[*

__cumulative____diachronic____progression__of*axioms*-]

*— to serve as a*

__systems__*or*

__temporal__*model of the stages of human mathematical understanding, 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*

__pseudo__*-knowledges*] to which human thinking is susceptible within each such epoch.

But

*wish to explore its efficacy as such.*__we____do__*Note how, as Gödel narrates above, each successor system «*.

**aufheben**»-contains its immediate predecessor system, and, indeed,**of its predecessor systems; how each higher**__all____logical____type__«**aufheben**»-contains**predecessor**__all____logical____types__**Can Gödel’s theory of this**

*¿***,**

__cumulative__*‘‘‘evolute’’’*, «

**»**

*aufheben**progression of axioms-systems*, which we term

*‘*

*The Gödelian*__Dialectic__*’*, or

*‘*

**[**

*The Gödelian**Idea-Systems’ Ideo*-]

*Metadynamic**’*, provide at least an

*idealized*[i.e., a

*distorted*] image of this humanity’s

**, of the actual**

*actual history**psychohistorical struggle, process,*and

*progress*of mathematical aspects of the self-development of a humanity’s collective cognitive capabilities, hence of its

**and**

*knowledges*

*ideologies?*
We shall see.

Each of Gödel’s “undecidable”

*propositions of arithmetic*that plague each ‘‘‘epoch’’’ of this formal axiomatic expansion are propositions each asserting the

**of a different, specific**

__unsolvability__*“*

__diophantine__*”*[referencing Diophantus’

__«__; see more on this below]

**»***Arithmetica***.**

__unsolvable____equation__I.e., each “Gödel formula”, or “Gödel sentence”, which formally asserts the

*self-incompleteness-or-self-inconsistency*of its axioms-system,

*“*

*deformalizes**”*to one asserting the

**of a specific, algebraic**

__un__solvability*“*

__diophantine____equation__*”*:

“... The
Gödel sentence

**φ...**asserts its own*undeducibility*from the postulates... .”
“

where

*Deformalizing***φ...**we see that under the standard interpretation it expresses a fact of the form [for every**n**-ary list of number-components of the “diophantine” algebraic equation’s unknown,**x**, such that each number-component is a member of the set of ‘‘‘diophantine numbers’’’, or “**Natural**”**Numbers,**in use —**F**.**.**__E__**.]**__D__*...***ƒ**,*x ≠ gx...*where

**ƒ**and**are***g***n**-ary polynomials... .”
“An
equation

**ƒ**, where*x = gx***ƒ**and**are two such polynomials, is called***g**diophantine*[*see below for further information regarding Diophantus of Alexandria*—**F**.**.**__E__**.] ... .”**__D__
“By a
solution of the equation we mean an

**n**-tuple**of natural numbers such that***α***ƒ**

*α*

*= g*

*α**.*

**.”**

*..*
“So

**φ...**asserts the unsolvability of the...equation**ƒ**, and the proof of [Gödel’s “First Incompleteness Theorem” —*x = gx***F**.**.**__E__**.]**__D__*produces... a particular diophantine equation that is really unsolvable, but whose unsolvability cannot be deduced from the postulates*...”
[Moshé Machover,

**, Cambridge University Press (Cambridge: 1996), pp. 268-269;**__Set Theory, Logic, and their Limitations__*emphasis***].***added*Per the standard modern definition, a “diophantine equation” is an equation whose

*[e.g.,*

__parameters__*coefficients*] and whose

*are restricted to the “natural” numbers [“positive integers”].*

__solutions__Each “Gödel sentence”-encoded equation truly is

**solvable**

__un__**the given axioms-system.**

__within__However, the proposition that it is so, also cannot be deductively proven

**that axioms-system — but it can be so proven within the**

__within__**axioms-system, the given axioms-system’s immediate successor-system — the**

__next__*, expanded axioms-system being created by means of the «*

__latter__**» [**

__aufheben__

__self__*-*]

__internalization__*’*of the ‘‘‘vanguard’’’, ‘‘‘meristemal’’’, highest [in

__logical__*] set idea-objects of the universe of discourse of that*

__type__*axioms-system.*

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

**»**

__aufheben__*‘*[

__self__*-*]

__internalizations__*’*.

If the

*“logical individuals”*or

*‘arithmetical idea-objects’*“existing”[constructed] per the

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

__comprehension____axioms__”*classes of*“natural” numbers, ... , all the way up to

*classes of classes of*... of “natural” 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 “natural” numbers, i.e., ‘class-objects’ of still higher

__classes____of____classes____of.__*“logical type”*.

Each successive higher class-inclusion of previous ‘class-objects’ can model [including via

*adjunction*of its corresponding

*“*, defining the

__comprehension____axioms__”*‘computative behavior’*of these new entities] a

*of arithmetical*

__new____kind__*‘idea-object’*;

*indeed,*

*.*

__a____new__,__higher____kind____of____number__Thereby, this

*qualitative expansion of each predecessor axioms-system*, in the formation of its successor axioms-system, this adjunction of the additional,

*“*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 axioms-system.

**-**__number____ontology__’**:**

__CONJECTURE__**Specifically**,

*the*__diophantine____algebraic____equation__that was

*as such*__un__solvable*within the*__predecessor____axioms__*-*,

__system__may itself become__solvable__

*albeit in a*__non__*-*,

**diophantine sense**

*within the next*[*as well as within*

*all subsequent*] successor axioms-systems in this*cumulative sequence of axioms**-*,

**systems**

*precisely by means of these***,**

__next__new kinds of numbers

*which will*

__not__

*be**‘*,

**diophantine numbers**’**.**

*i***.,**

*e*

**“**

__not__**natural**”

**; not in the number-set**

*numbers***N**.

We can see a

__kindred__*‘*__un__solvability*-*-- i.e., an**to**-__solvability__**’**__dialectic__*‘*__un__solvability*-*-- at work in the following examples, presented in a systematic order, rather than in the [psycho]historical order in which they arose for, and were solved by, our ancestors.**turning into**-**its**-**opposite**’__dialectic__
The systematic order
is that of the “filling in” of the so-called “

**R**eal” Number-Line, prior to the «**»***aufheben*__dialectical__**/***negation***/***elevation***of that Number-***conservation***in the irruption of the new ‘ideo-ontology’ of the “**__Line__**maginary” numbers, and of the “**__i__**C**omplex__Plane__*”*.**tep parameter, the “Whole number”**

__s__**s**-- which describes the progression of the axioms-systems of the arithmetics founded upon these ‘ideo-ontologically distinct’ number spaces, and which is written in terms of the Seldonian first dialectical arithmetic, viz.:

We shall not explicate the ‘dialectical meta-equation meta-model’ above in terms of its detailed workings at this juncture, letting that wait until readiness for this explication has been further cultivated in the course of this presentation.

For those who wish to leap ahead, a thorough narration of this ‘meta-model’ is available via the links below.

http://www.dialectics.org/dialectics/Vignettes_files/v.4.5,Part_0.,Prefatories,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,14FEB2013.pdf

http://www.dialectics.org/dialectics/Vignettes_files/v.4.4,Part_I.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf

http://www.dialectics.org/dialectics/Vignettes_files/v.4.4,Part_II.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf

**I**.

**. The equation [**

__The Paradox of__*Gainless*Addition**2**

**+**

**x**

**=**

**2**or

**x**

**=**

**2**

**-**

**2**] asserts a

**:**

*paradox***How can the**

*¿*

__add__*ition*of a number,

**x**, produce a result, a sum, that is

**that ‘known’ number, here**

__not__bigger than**2**, to which that “unknown” number,

**x**, is added

*?*

Given the

**N**«

**» of number,**

*genos***always means increase. It**

*addition**never means*.

**increase**__no__This equation is

__not__*solvable*within the set and within the system of arithmetic of the

*cardinal, or, sometimes,*

*“*

*N**atural”, numbers*,

**N**

**=**

**{**

**1**,

**2**,

**3**,

**. . . }**.

However, this equation

**solvable -- specifically by the**

__is__*‘*

__non__*-*

**diophantine number**’**0**-- within the

*‘*

*ideo**-*

**ontologically**’**of the “W**

*expanded system**hole numbers*”,

**W**

**=**

**{**

**0**,

**1**,

**2**,

**3**,

**. . .**

**}**.

The “number”

**0**belongs to a new set of numbers -- to a

*new***of number -- which we denote by**

*kind***a**, which stands for the set of the ‘‘‘

**ught’’’ numbers.**

__a__[

*¡**Adjunction of this*

*zero**concept may seem trivial to*].

**, yet it entailed a great and protracted conceptual travail for our ancient Mediterranean ancestors, and, with respect to issues surrounding**__us__**division by zero**, and the related issues of the**singularities**of, especially, the**linear integrodifferential equations, remains fraught with unresolved problems, “even” among we moderns today**__non__**!****II**.

__The Paradox of__

__'____Subtractive____'__**. The equation [**

__Addition__**2**

**+**

**x**

**=**

**1**] asserts a

**:**

*paradox***How can the**

*¿***of a number,**

*addition***x**, produce a result, a sum, that is

**that ‘known’ number, here**

*less than***2**, to which that “unknown” number,

**x**, is added

*?*
Within the

**W**«**» of number,***genos***always means a change that increases, or, at minimum, that results in no change at all, but***addition***.***it never means a*__decrease__
Our second equation
thus finds

**number among the so-called “**__no__**W***holes*” to solve it/satisfy it.
However, that
equation

**find a solution among the**__does__*“integers”*or*‘‘‘integral’’’*numbers, the expanded numbers-set**Z****≡****{****. . .**,**-****3**,**-****2**,**-****1**,**±****0**,**+****1**,**+****2**,**+****3**,**. . .****}**, which is a**, that is,**__qual__itatively*‘*__ideo__*-*, new-**ontologically**__expanded____kinds__*-of-numbers-*, meaning-of-“number”-expanded, or__expanded__*‘***-ing’-of-“number”-***meme***--**__expanded__*semantically**-*of “Number”,**-**__expanded__-- universe**of**-**discourse***vis-à-vis*the preceding «**», the***genos***W***universe**-*.**of**-**discourse**
The equation is

The “number” *solved*/*“satisfied”*by the*‘*__non__*-***diophantine number**’**-****1**is an**E**[lement] of**m**which is**C**[ontained] in**Z**.**-**

**1**belongs to a new set of numbers -- to a

**of number -- which we denote by**

*new kind***m**, which stands for the set of the ‘‘‘

**inus’’’ numbers.**

__m__**III**.

__The Paradox of__

__'____Decreasive____'__**. Next, the equation [**

__Multiplication__**2x**

**=**

**1**] also asserts a

*‘‘‘*

*paradox**’’’*:

**How can the**

*¿**multiplication*of

**number, namely that of the “multiplicand”, denoted here by the algebraic “variable” or “unknown”,**

__any__**x**, by another, known, number, the “multiplier”, here

**2**, produce a “product” which is

**that “multiplier”, e.g., in this case, the “product”**

*less than***1**

*?*

Multiplication, within the

**Z**«

**» of number, always produces a “product” which is either increased**

*genos**in*relative to the “multiplicand” “factor”, or leaves the multiplicand unchanged, or turns it into zero.

__absolute__valueBut

**Z**multiplication can never turn a (

**+**)

**2**into a (

**+**)

**1**.

Such an equation is

**within the system of arithmetic of the “integers”; of the**

__not__solvable**Z**.

This equation

__is__**, however, via**

*solvable**‘ideo-ontological expansion’*to encompass the

__qual__itatively different*system*of arithmetic of the “

__Q__*uotient numbers”*, the

*“*, the

**-numbers”**__ratio__*“*nal” numbers, or

**-**__ratio__*“*, denoted by

**fractions**”**Q**, i.e., by an expansion that encompasses yet a new kind of

*‘*[per the modern sense of the term “diophantine”], the

**-**__non__**diophantine number**’*‘split*[i.e., the

__a__-tom’*‘cut*],

**cuttable’**__un__*‘*fragment’, or

**-**__monad__*“*ional value”,

__fract__**+**

**1/2**, part of --

**Q**

**=**

**{....**

**-**

**4/2...**

**-**

**3/2...**

**-**

**2/2...**

**-**

**1/2...**

**±**

**0/2...**

**+**

**1/2**

**...**

**+**

**2/2...**

**+**

**3/2...**

**+**

**4/2....}**.

**+**

**1/2**belongs to a new set of numbers -- to a

**of number -- which we denote by**

*new kind***f**, which stands for the set of the ‘‘‘

**ractional’’’ numbers.**

__f__**IV**.

__The Paradox of the__

__Odd Number__

__Unknown That Must Also Be an__*Even Number*__,__

**. The [algebraically]**

__or That Must Be__*NEITHER*an*Even Number**NOR*an*Odd Number***equation [**

__nonlinear__**x**

^{2}**=**

**2**] asserts a

*‘‘‘*too.

**paradox**’’’This equation requires

**x**to be a

**of number which is**

*kind**‘both*

**and**__odd__**at the same**__even__*time’*[see the classic «

**» proof of the**

*reductio ad absurdum**“*-ratio-nality”, or

__ir__*‘*ratio-ness’ [in terms of ratios

**-**__non__

*of***], of the**

*integers***square**-root of

**2**,

**\2/**

**2**.

This equation is

__not__

**solvable***‘‘‘*[

__ratio__*n*]

*ally’’’*.

This equation

__is__*via ‘ideo-ontic’ expansion to the so-called “*

**solvable****R**

*eal”*

*numbers*, this time by

**distinct**

__two__*‘*, due to the

**-diophantine numbers’**__non__**, “**

__nonlinear__**2**nd degree” character of this “unsolvable” equation, rather than by just

__one__*‘*, as were the preceding,

**-diophantine number’**__non__**[algebraically]**

**, degree**

*linear***1**“unsolvable” equations / ‘‘‘paradoxes’’’.

The

**solutions are the so-called “irrational” values**

*two*

**-****\2/**

**2**

**and**

**+**

**\2/**

**2**

**, both elements of the set --**

**R**

**=****{.....**

**-**

**p**

**i....**

**-**

**3....**

**-**

**e....**

**-**

**2....**

**-****\2/**

**2**

**....**

**-**

**1....**

**±**

**0....**

**+**

**1....**

**+****\2/**

**2**

**....**

**+**

**2....**

**+**

**e....**

**+**

**3....**

**+**

**p**

**i.....}**.

The “numbers”

**-****\2/**

**2**

**and**

**+**

**\2/**

**2**

**belong to a new set of numbers -- to a**

**of number -- which we denote by**

*new kind***d**, which stands for the set of the ‘

**iagonal numbers’ [i.e., for the set of the so-called “irrational numbers”].**

__d__**V**.

__The Paradox of__*Additive Inverse*=*Multiplicative Inverse*__'__

__Identicality____'__. Finally -- but “finally” only for the limited purposes of this selected spectrum of examples of “unsolvable diophantine equations” -- the algebraically

**equation --**

__nonlinear__**x**

^{2}**+**

**1**

**=**

**0**

-- asserts a

*‘‘‘*as well.

**paradox**’’’This equation implies that

**-**

**x**

**=**

**+**

**1/x**, requiring a kind of number whose

*additive inverse*,

**-**

**x**, equals its multiplicative inverse --

**1/x**, or

**x**

^{-}**-- whereas, among so-called “**

^{1}**R**eal” numbers --

**-**

**2**

**~=**

**+**

**1/2**,

**-**

**3**

**~=**

**+**

**1/3**,

**-**

**pi**

**~=**

****

**+**

**1/**

**pi**

-- etc., etc.

**On what basis do we hold that the number**

*¿***x**described by the equation [

**x**

^{2}**+**

**1**

**=**

**0**] must be such that its

**is always equal to its**

*additive inverse*

*multiplicative inverse?*
On the basis that this characteristic of

**x**is implicit in that**x**-defining equation itself.
To see that this is so, let us apply a little algebra to
this equation, [

**x**^{2}**+****1****=****0**], always performing exactly the same operation on both the Left Hand Side [LHS] and the Right Hand Side [RHS] of this equation, so that the relationship of equality is always maintained between those two sides, even as their form and content changes as a result of the operations performed on them.
First, let us subtract

**1**from both sides --**x**^{2}**+****1****-****1****=****0****-****1**-- yielding --**x**^{2}**+****0****=****-****1**.
Next, let us divide both sides of the resulting equation, [

**x**^{2}**=****-****1**], by**x**.
This action requires the assumption that

That is, [

**x ~=****0**, to preclude dividing by zero, but we already know this to be true, because the equation [**0**^{2}**+****1****=****0**] is a**assertion.***false*That is, [

**0**^{2}**+****1 ~=****0**], or, more specifically, [**0**^{2}**+****1****=****1**] is the**propositional***true***of that***negation***'posit-tion' / assertion.***false*
So, we obtain thereby --

**x**

^{2}**/**

**x**

**=**

**-**

**1**

**/**

**x**

-- which is equivalent to --

**+**

**x**

**=**

**-**

**1/x**

-- then, multiplying both sides of that equation by

**-****1**, we obtain that which we had asserted to be implicit in the equation [**x**^{2}**+****1****=****0**], namely, the equation between the**and the***additive inverse***of***multiplicative inverse***x**--**-**

**x**

**=**

**+**

**1/x**.

The equation [

**x**

^{2}**+**

**1**

**=**

**0**] is

**, or “satisfiable”, within any of the foregoing «**

__not__solvable**» of number, or of arithmetics, all the way up through the «**

*gene***» of the so-called “**

*gene***R**eal” numbers.

The equation [

**x**

^{2}**+**

**1**

**=**

**0**]

__is__*‘*-diophantinely solvable’, via expansion to the “

__non__**omplex” numbers, which we denote by [given that**

__C__**r**

**=****+**

**1.0...**, and that

**\2/**

**-**

**1**denotes the

**square**-

**root**of

*negative***R**] --

*eal unity***C**

**=****{**

**Rr**+

**R**·

**\2/**

**-**

**1**

**}**

**=****{**

**R**+

**Ri**

**}**

-- by, again, due to its

**2**nd degree

*algebraic***,**

*nonlinearity*

*two**‘*diophantine’ numbers, in the case of this equation, by

**-**__non__**so-called “pure**

*two***maginary” numbers --**

__i__**x**

**=**

**+**

**\2/**

**-**

**1**

**=**

**0**

**r**

**+**

**1**

**\2/**

**-**

**1**

**=**

**0**

**r**

**+**

**1**

**i**

**=**

**+**

**i**

-- and --

**x**

**=**

**-****\2/**

**-**

**1**

**=**

**0**

**r**

**-**

**1**

**\2/**

**-**

**1**

**=**

**0**

**r**

**-**

**1**

**i**

**=**

**-****i**.

The “numbers”

**-****i**and

**+**

**i**belong to a new set of numbers -- to a

**of number -- which we denote by**

*new kind***i**, which stands for the set / totality of the so-called “

**maginary numbers”.**

__i__*Note how each successor*«

**»**

*genos**, or universe, of number ‘‘‘«*

*aufheben**»-*

**’’’ all of its predecessor universes of number -- i.e., is a ‘‘‘**__contains__**conservative extention**’’’ of all of its predecessor ‘‘‘universes-of-discourse’’’ about number and about arithmetic -- while also**determinately**/negating__changing__**&**

*.*

**into an ‘ideo-ontologically’ richer level, all of its predecessors**__elevating__Such a [

*‘*

__Qual__o*-*or

**Peanic**’*‘*

*Meta**-*] «

**Peanic**’**»**

*aufheben**‘*

*consecuum**’*of number-

*«*

*gene**» -- of number ‘*--

**ideo**-**ontology**’*evinces part of the essence of what we mean by a*

*‘*

__dialectic__*’*; by a

*‘*

__dialectical____process__*’*, i.e., by an ‘[

*ideo-*]

__meta__*-*,

**dynamical**

__meta__*-*[

*ideo-*]

**[**

*system***]**

*at***,**

*ic***and also**

**[**

*ideo-*]

__meta__*-*[

**evolutionary***self-*]

**[**

*progression of**axioms-*]

*systems**’*, [self-]launched from an original,

*«*

*arché**»*[

*ideo-*]

**, in this case, from the axioms-system of “**

*system***N**atural Numbers” arithmetic.

**But how might this**

*¿**potentially-*infinite, constructed progression of

*«*

*gene**»*and of

*‘*

__species__*’*of number,

**, map to**

*required for*__equational____solvability__*“sets of sets of ... of sets*”

*?*One way that sets of higher

*“*can model [

**logical type**”**'**

**(---)**'] higher, later

**[**

*kinds of*

__non__*-*]

**diophantine****is as**

*numbers***of earlier**

*ordered pairs***/ earlier**

*kinds of numbers***.**

*kinds of sets*
We have already noted, above, that

**can be modeled via certain kinds of***ordered pairs***.***sets**“*, for example, can be modeled as ordered pairs of “

**Integers**”**W**

*hole numbers*

*”*, i.e., as sets of logical type

**2**— if we take the “

**W**

*hole numbers*

*”*to be our ‘

**base**objects’ — with the

**Z**s, the

*“*

*integers**”*, being defined, via their

*“*, as

*comprehension axioms*”*, i.e., as*

__differences__*, viz., as --*

__subtractions__**{**

**{**

**1**

**}**,

**{**

**1**,

**0**

**}**

**}**

**=****<**

**1**,

**0**

**>**

**(---)**

**1**

**-**

**0**

**=**

**+**

**1**

**~=**

**-**

**1**

**=**

**0**

**-**

**1**

**(---)****<**

**0**,

**1**

**>**

**=**

**{**

**{**

**0**

**}**,

**{**

**1**,

**0**

**}**

**}**.

**can, in turn, be modeled as**

*Rational numbers***of**

*ordered pairs**“*

*integers**”*, defined, via their

*“comprehension*, as

*axioms”**, i.e., as*

__quotients__*, viz., as --*

__divisions__**<**

**+**

**1**,

**+**

**2**

**>**

**(---)**

**(**

**+**

**1)**

**/**

**(**

**+**

**2)**

**=**

**+**

**(1/2)**

**~=**

**+**

**(2/1)**

**=**

**(**

**+**

**2)**

**/**

**(**

**+**

**1)**

**(---)**

**<****+**

**2**,

**+**

**1**

**>**.

**W**

*hole numbers*

*”*, or to ‘sets-of-sets of sets-of-sets’ of “

**W**

*hole numbers*

*”*, that is, to ‘sets-of-sets of “

**W**

*hole numbers*

*”*

__squared__*’*, meaning that these sets-of-sets

**these very sets-of-sets themselves, per a '**

*operate upon**multiplicand-ingestion*' set-product rule, so --

**+**

**(1/2)**

**(---)**

**<**

**+**

**1**,

**+**

**2**

**>**

**(---)**

**{**

**{**

**+**

**1**

**}**,

**{**

**+**

**1**,

**+**

**2**

**}**

**}**

**<**

**<**

**1**,

**0**

**>**,

**<**

**2**,

**0**

**>**

**>**

**(---)**

**{**

**{**

**<**

**1**,

**0**

**>**

**}**,

**{**

**<**

**1**,

**0**

**>**,

**<**

**2**,

**0**

**>**

**}**

**}**

-- which translates to --

**{**

**{ {**

**{**

**1**

**}**,

**{**

**1**,

**0**

**}**

**}**,

*}***{ {**

**{**

**1**

**}**,

**{**

**1**,

**0**

**}**

**}**,

**{**

**{**

**2**

**}**,

**{**

**2**,

**0**

**}**

**} }**

**}**

-- which is a class-object of

__logical____type__**4**, i.e., of logical type

**2**, or

^{2}*“*

*two squared**”*,

*w.r.t*. the

**W**

*hole numbers*

*”*taken to be the

*‘*

*base**-*objects’.

Further on in this axioms-systems progression,

**of “**

*ordered pairs***R**

*eal numbers*” may, in turn, model the space of the “

__C__*omplex numbers”*,

**C**,

*viz.*--

**<**

**+**

**1.000**,

**+**

**2.000**

**>**

**(---)****1r**

**+**

**2i**

**~=**

**2r**

**+**

**1i**

**(---)****<**

**+**

**2.000**,

**+**

**1.000**

**>**

-- such that

**C**can be modeled as the

**-dimensional space of a**

*two***of direction-denoting, as well as magnitude-denoting, ‘‘‘directed line segment’’’, or ‘‘‘**

*special kind***2**-dimensional vector’’’.

... and so on, to the axioms-systems for the arithmetics of the

**H**amilton Quaternionic [

**H**] hypercomplex numbers, the Cayley/Graves

**ctonion [**

__O__**O**] hypercomplex numbers, the Clifford hypernumbers [

**K**], the

**rassmann hypernumbers [**

__G__**G**], the

**S**edenions [

**S**], etc., etc. ... .

Thus, e.g., the

*“*

*rational**”*

**are seen as “analogues” -- as**

*numbers**‘*

*meta**-*

**fractal**

*similants**’*-- of the

*“*

*integers**”*; the

*“*

*integers**”*as

*‘*

*meta**-*

**fractal**

*similants**’*of the

*“*

*whole numbers**”*.

Even though these successive numbers-systems are of

*different*

**, differing in**

*kind***, their**

__qual__ity**‘idea-objects’, or numbers — their**

*base*

*universes**-*— may be constructed and

**of**-**discourse****,**

*presented***, in and as**

*systematically**different*

**s**teps in a progressive-cumulative ‘[

*self**-*]iteration’ of one and the

*same*«

**»**

*aufheben***of**

*operation**‘*

__self__*-*, of

**’**__internalization__*‘*

*self**-*, of

**incorporation**’*‘*

*self**-*, of

**subsumption**’*‘*

*self**-*/

**combination**’*‘*

*self**-*, of

**combinatorics**’**, i.e., of**

*sets***.**

*ordered pairs*This

*number**-system*, modeled by the

__s__**progression**

*self**-*iteration of the

*‘*or

**ordered pairs of**’*‘*operations, constructs a “logical” or ‘‘‘idea-object’’’ version of what we term an «

**sets**’__of__**»,**

*aufheben**‘*

*meta**-*[

**fractal***ideo-*]

*cumulum**’.*

This cumulum is a ‘[meta-]

__fractal__*ideo-cumulum’*because it constructs and presents a structure which is

*self**-*at successive “scales”.

**similar**This cumulum is a

*‘*

**[-fractal]**

__meta__*ideo-cumulum’*because its “scales” or ‘‘‘levels’’’ are not purely quantitative, as they supposedly are for “mere”

**, but are ‘quanto-**

*fractals*

__qual__itative*’*, or ‘quanto-

__ontological__*’*.

That is, such a

*‘*

*cumulum**’*[self-]

*and/or [self]-*

__constructs__*as a*

__presents__**chronic progression of ‘[**

__dia__*ideo-*]

__meta__*-*«

**»’ -- of**

*arithmoi***of**

*sets***-- and**

*ordered pairs**as a synchronic, nested “tree” of an «*

__persists__**»-«**

*arché***» -- e.g., the**

*arithmos***of “**

*set***W**

*hole numbers” as*«

**base****», or**

*monads*

*base**“*-- pooled with a graduated scale of ‘[

**elements**”*ideo-*]

**-«**

__super__**»’ -- of**

*arithmoi***of higher, more-inclusive**

*sets***-- each one made up out of**

*ordered pairs***of ‘[**

*ordered pairs**ideo-*]

__sub__*-*«

**»’-as-**

*arithmoi**‘*

__meta__*-*«

**»’ -- e.g., the**

*monads***of**

*set**“*

*integers**”*made up out of

**which are**

*elements*

*ordered**-*of “

**pair****sets****W**

*hole numbers”*, the

**of**

*set**“*

*rational**”*made up out of

**numbers****which are**

*elements*

*ordered**-*of

**pair****sets***“*

*integers**”*, ... , the

**of “**

*set***C**

*omplex” numbers*made up out of

**which are**

*elements*

*ordered**-*of “

**pair****sets****R**

*eal” numbers*, etc. -- and also such that each constituent ‘[

*ideo-*]«

**»’ of**

*arithmos**number-*«

**» -- e.g.,**

*monads***N**,

**W**,

**Z**,

**Q**,

**R**,

**C**,

**H**, ... -- is, in turn, ‘‘‘populated’’’ via a different

**of ‘[**

*kind**ideo-*]«

**»’; in this example, by**

*monad**different*

*kinds**of*[via

__number__*different*

**[y][ie]**

__unit__**, viz.**

__s__**--**

**1**, versus

**(**

**0...01**

**)**, versus

**(**

**+**

**0...01**

**)**, versus

**(**

**+**

**0...01/**

**+**

**0...01**

**)**, versus

**(**

**+**

**0...01**

**.0...**

**)**, versus

**(**

**+**

**0...01**

**.0...**

**)**

**r**

**.**

*vs***(**

**+**

**0...01**

**.0...**

**)**

**i**

-- etc.]

*.*

This may be seen in that the later

*‘*involve adjunctions of new [

**similants**’*idea-*]

**, of**

*ontology**new, higher*

**of sets; new**__logical____types__

*kinds**of*

*sets**; new*of

**kinds**

**ordered pairs**; new

*kinds**of*

**numbers**,**, ‘**__qual__itatively*ideo-*, and thus, and

**ontologically**’ different from all of their earlier ‘**similants**’,**just**__not__**different therefrom,**__quant__itatively**«**__because__**aufheben**»-‘‘‘**’’’ all of their earlier ‘**__containing__**similants**’ in a**nested**fashion**, also**

__thereby__*‘*

__meta__*-*

**fractally**’*‘*

*scale**-*i.e.,

**escalated**’,

*«*

*aufheben**»-‘‘‘*, with respect to all of their earlier

**’’’**__elevated__*‘*.

**similants**’We also find, in the known history of nature to date,

**,**

__physical__*‘*’

**ternal-objective**__ex__*‘*

*meta**-*; a cosmological

**fractal**’**structures***‘*

**-**__physio__

*cumulum**’*, of

*‘*

**-**__physio__

*meta**-*«

**»’, each one**

*arithmoi**made up out of multitudes of ‘*

**physio**-

__sub__*-*«

**»’ -- made of different**

*arithmoi***of**

*kinds**‘*«

**-**__physio__**»’:**

*monads*
·

**. . .**;
·

**molecules**as*‘***-***meta***atoms**’, each one made up out of a heterogeneous multiplicity of**atoms**;
·

**atomic nuclei**as*‘*__meta__*-***sub**-**atomic**“particles” ’, each one made up out of a heterogeneous multiplicity of**nuclear sub**-**atomic**“particles”;
·

**nuclear sub**-**atomic**“particles” as*‘*__meta__*-***pre**-**nuclear**-“particles” ’, each one made up out of a heterogeneous multiplicity of**pre**-**nuclear**-“particles”;
·

**. . .**.On the basis of the foregoing, we therefore hold that —

**(1)**the ‘human-minds-

**ternal’, ‘inter-subjective’,**

__in__*‘*,

**-**__idea__**-ive’**__object__*mathematical-progress-driving, conceptual process*of

*‘*

*The Gödelian*__Dialectic__*’*, i.e., of

*‘*[as modeled, using the ‘‘‘algebra’’’ of

**-**__ideo__**onto**-**dynamasis**’**, via**

__dialectical____ideography__

*generalized*

*self**-*,

**multiplication***‘*

*quadraticity**’*, or

*‘*

*ideo**-*— to appropriate Diophantus’s term for

**onto**-**’**__dynamis__*‘‘‘*

*squaring**’’’*, i.e., for the

**, or**

*second**“*

*quadratic**”*, “degree”, or “power”, of a variable], and;

**(2)**the ‘human-minds-

**ternal’, “objective”, “natural” process driving the**

__ex__

*self**-*of ‘physical Nature’, or «

**development****» — both of ‘‘‘pre-human Nature’’’, diachronically, and also of ‘‘‘extra-human Nature’’’, synchronically, as well as of ‘‘‘human Nature’’’ — i.e.,**

*physis**‘*

__physio__*-*, share a similar, «

**onto**-**dynam**’__as__is**»**

*aufheben**/ ‘*

*meta**-*— i.e., a generally single, singular

**fractal**’**logic**

__dialectical__*— or*

__logic__*pattern of ‘followership’*, which we term

*‘*

*meta**-*.

**monadization**’*‘*of such

**’**__Dialectical__Models*‘*

__ideo__*-*are presented in

**onto**-**dynamasis**’**to this text [see: http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf ], and also in**

*Supplement A***F**.

**.**

__E__**. Vignette**

__D__**#4**, entitled

*“*

*The Gödelian*__Dialectic__of the Standard Arithmetics*”*.

[see: http://www.dialectics.org/dialectics/Vignettes_files/v.4.5,Part_0.,Prefatories,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,14FEB2013.pdf

http://www.dialectics.org/dialectics/Vignettes_files

/v.4.4,Part_I.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf

http://www.dialectics.org/dialectics/Vignettes_files/v.4.4,Part_II.,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,last_updated_29NOV2012.pdf ].

*‘*of such

**’**__Dialectical__Models*‘*

__physio__*-*are presented in

**onto**-**dynamis**’**to this text.**

*Supplement B*[see: http://www.dialectics.org/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf

__].__

A fuller exploration — and a

*‘‘‘*

__dialectical__

**model***’’’*— of the

*‘*

*Gödelian Ideo**-*, as actually observable in the [

**Meta**-**Evolution**’*psycho*]

*history of arithmetics*, is forthcoming in

__volume__**. of**

__II__**, entitled**

__Dialectical Ideography__

__The Meta__*].*

__-__**Evolution of Arithmetics**##
F.__E__.__D__.’s* ‘*__Dialectical__* Meta**-**Axiomatics**’** Method of Presentation*:
*A *__Bi__*-***Systematic Method of Presentation**.

Gathering together, and integrating/synthesizing
what we have reviewed so far regarding the Platonian «*arché*» of __dialectic__, and regarding the
subsequent historical development of __dialectic__, and
regarding the __dialectical__, *immanent
critique* of the *ideology* impairing modern mathematics, now calls for us to frame a
new perspective on mathematical systems and their optimal *mode of exposition*, as well as, in later *exposition*,
a new, more optimal *mode
of mathematical discovery*.

The new, rationally and pedagogically advantaged *mode of exposition* that we advocate has come to be called, in our internal
deliberations, *‘*__Dialectical__ Meta*-***Axiomatics**’.

*‘*__Dialectical__ Meta-**Axiomatics**’ names a new, *complex unity*, or *dialectical synthesis*, of the
traditions of axioms/postulates/primitives/definitions/lemmas/theorems *mode of exposition*, of which Euclid’s __Elements__
provide the «*arché*», with that of the, later, *‘‘‘**Systematic *__Dialectics__*’’’* and *‘**Meta**-***Systematic
Dialectics**’ *modes
of exposition*.

Plato’s proposal cited above, Hegel’s
achievement, in his «__Logik__»
and «__Encyklopädie der philosophischen
Wissenschaften__», and Marx’s achievement, in his __Capital__, provide the chief models of
the latter *modes* extant to-date.

**F**.__E__.__D__.
advocates *‘*__Dialectical__ Meta*-***Axiomatics**’ as the preferred *mode of exposition*
for __Dialectical__ Science, including
for the *‘**psych*[*e*]*ohistorically**’* expanded *Science of Mathematics*.

*‘*__Dialectical__ Meta-**Axiomatics**’ «*aufheben*»*-***conserves**, without apology, the full rigor of
formal-logical/mathematical-logical deductive proof, of «*verstand*» / ‘«*dianoetic*»’ reason, __within__ each Axioms-System of a *Gödel**-***Incompleteness**-**driven**,
*‘**Gödelian**-***ideo**-__dialectical__’ **progression** of Axioms-System__s__ [that __systematically__*-*__ordered__ system__s__-**progression** of Axioms-System__s__ constituting what
we term the *Gödelian*, __dia__chronic *‘*__Meta__*-***System**’ for those successive Axioms-System__s__].

But it also applies __dialectical__ reason in the __trans__*-***deductive** realm of the necessarily __non__*-*deductive derivation of the possible,
candidate Axioms, and to the __rational__ __justification__ of the
choice of Axioms from among those possibilities.

Moreover, it applies __dialectical__ logic also to the «*aufheben*» transitions __between__ predecessor /-successor pairs
of Axioms-System__s__, from a predecessor Axioms-System to its ‘‘‘conservative
extention’’’ in its successor Axioms-System.

Such transitions partly «*aufheben*»*-***conserve** the Axioms of the predecessor Axioms-system in the Axioms
of the successor Axioms-System, while also adding, via «*aufheben*»*-***transformation **/ *elevation*, the new comprehension Axioms, and the new ‘ideo-ontology’
that they implement, rendering “decidable”, within the successor Axioms-System,
the “undecidable” propositions of the predecessor Axioms-System, which was thus
Gödel-Incomplete with respect to [at least] those propositions.

E.g., such transitions form *new kinds of
numbers*, able to solve the diophantine
equations that are __un__solvable
within the number ‘ideo-ontology’ of the predecessor Axioms-System -- equations
the __un__solvability of
which is asserted by the “deformalized” *undecidable propositions* of that predecessor Axioms-System.

Such transitions form these *new*, *higher kinds of numbers* as *new kinds*/*higher logical**-***types** of *sets*, __qualitatively__,
‘ideo-ontologically __different__’ from the predecessor logical-types of *sets*, within the “power-set” «*aufheben*» *‘**self**-***internalization**’ / *‘**subsets-subsumption**’*, of those *sets* of the predecessor Axioms-System’s ‘ideo-ontology’, which
formed the farthest horizon of the number-concept extant *within* that predecessor Axioms-system and its implied ‘ideo-ontology’.

This «*aufheben*» process
renders the relative truth of the formerly undecidable propositions __provable__
via the new “comprehension Axioms” added to form the new-Axioms-component of
the successor Axioms-System, and renders the formerly __un__solvable diophantine equations __solvable__ within the successor
Axioms-System, using the *new kinds*/*higher
logical**-***types** of *sets*, defining the *new*, *higher kinds of
numbers* thus «*aufheben*»*-***created** within the successor Axioms-System.

*‘*__Dialectical__ Meta-**Axiomatics**’ drops the pretence
that each Axiom in the Axioms-set of an Axioms-System can be “self-evident”, and
uniquely-determined, with __no__
possible Axiom-alternatives.

By this pretense, the function of __dialectical__ reason, as the __non__*-deductive
*__derivation__ of
multiple candidates for a given key Axiom, and as the __justification__ of the choice of one Axiom from among those candidates,
has for so long been dogmatically denied [*¡*ever
since Plato, for the *anti**-*__dialectical__ traditions of academia, for which *the Occidental Dark Ages* have never yet ended*!*].

*‘*__Dialectical__ Meta-**Axiomatics**’ admits that
axiomatic ‘alternativity’ veritably abounds, and that Axiom-choice needs to be *justified *__dialectically__,
i.e., *‘**self**-refl*__e__xively’ / *‘**self**-refl*__u__xively’, that is, in light
of each candidate Axiom’s consequences within the context of the «*arithmos*» of Axioms it is candidate to join, and in light of the
purposes for which the Axioms-System it is candidate to join is designed.

Examples of such ‘alternativity’ -- of the “independence”
or Gödel-undecidability of key Axioms with respect to the rest of the Axioms of
a given Axioms-System -- include the choice of the Euclidean ‘fifth Axiom’, the
Parallels Postulate, versus one of its possible contraries, for the Axioms-System
of Euclidean Geometry versus for those of the several __Non__*-*Euclidean Geometries, and
the choice of the Generalized Continuum Hypothesis, versus one of its possible
contraries, and/or of the Axiom Of Choice, versus one of its possible
contraries, for the Axioms-Systems of *Totality Theories* [“Set
Theories”].

*‘*__Dialectical__ Meta-**Axiomatics**’ also rejects any
pretence that first-order-logic Axioms-Systems have but one possible, “categori__c__al”, unique “model”, or “interpretation”,
an old dogma that has been refuted both by the Löwenheim-Skolem Theorem, and by
the first order co-applicability of the Gödel Completeness Theorem and the
Gödel __In__completeness
Theorem.

*‘*__Dialectical__ Meta-**Axiomatics**’ is also based on a
grasp of the *‘**intra**-***duality**’ / *‘**intra**-***multiality**’ of the ‘interpretabilities’
of a first order Axioms-System as marking a potential «*arché*» for a *‘*__meta__*-*__system__atic __dialectical__’, *‘*__categorial__*-***progression**’, *‘**Axioms-System*__s__-**progression**’ exposition, and for a __dialectical__*-***algebraic
modeling**, of alternative,
“non-standard” models of that first order Axioms-System.

*‘‘‘*__Dia__chronically’’’, __between__ each predecessor/successor
pair of Axioms-System__s__, the *‘*__Dialectical__ Meta*-***Axiomatics**’ __methodology__ practices an expository, pedagogical
discipline, using an *heuristic*, __intuition__*-***involving**, *“**intensional**” ***derivation** of the «*aufheben*» *progression* of Axioms-System__s__:
of the ‘Axioms-__Meta__*-*System’.

*‘‘‘*__Syn__chronically’’’, __within__ each, successive
Axioms-System in that ‘Axioms-__Meta__*-*System’,
*‘*__Dialectical__ Meta*-***Axiomatics**’ justifies the *theorems* implied by
that Axioms-System’s ‘Axioms-collective’, definitions, primitives, and rules of
inference via *rigorous
deductive logic*.

*Theorems* are __also__ justified, and
explained *conceptually* and *intuitively* [«*begrifflichkeit*»], without apology.

Indeed, the main text, in a work of *‘*__Dialectical__ Meta*-***Axiomatics**’, is the *intuitive/conceptual
exposition*, but with a __parallel__ stream of formal-logical,
algorithmic/mechanical deductive proof [which may often compel a human mind to
assent to a proposition *without
comprehension*] relegated to a subordinate narrative -- e.g., to End-Notes,
or to [an] Appendi[x][ces] -- as a necessary verification check on the
conceptual/intuitive narrative’s flow/progression of claims/assertions.

The two parallel texts should thus each contain
‘“pointers”’ -- i.e., cross-references and bridging commentary; ‘‘‘__transversals__’’’
and asides -- linking from the deductive proofs to the intensional-heuristic / intuitive
narrative, and from the intensional-heuristic / intuitive narrative to the
deductive proofs.

We can therefore ‘visualize’ the
‘content-structure’ of an exposition constructed in accord with the tenets of *‘*__Dialectical__ Meta*-***Axiomatics**’ as follows --

The present work, overall, remains an *‘*__intuitively__*-*__ordered__ narrative’.

We plan for volume **2** of our treatise, __A
Dialectical Theory
of Everything__, to contain instantiations of *‘*__Dialectical__ Meta*-***Axiomatics**’ method of exposition.

*‘*__Dialectical__ __Meta__*-*__Axiomatics__’ «__aufheben__»*-***conserves** the full logical rigor of deductive proof-based «__dianoesis__», without apology.

But *‘*__dialectical__ __Meta__*-*__Axiomatics__’ also exceeds that «__dianoesis__» in rigor by virtue of its unified recognition of:

**(i)** the __non__-self-evidence
*of appropriate and optimal ***axioms **generally;
the exercise of choice and skillful design required in their development
and selection, and the abounding *‘*__alternativity__*’* which that activity
confronts;

**(ii) **the *axioms**-***dependence** or *assumptions**-***relativity**
of** all ****formal proofs**, hence of all *formal “*__truths__”;

**(iii)** the *logical*
*‘equi-coherence’* *of* __non__*-*__standard__ __models__
of *“first order logic”* axioms-[__sub__]systems with respect to
the *standard models* with which those __non__*-*standard models are associated, and with
reference to which they are defined as *“*__non__-standard”;

**(iv)** the *formal*
__independence__, or *Gödel-*__un__decidability,
of key axioms of *“higher-than-first-order-logic”* axioms-systems with
respect to the rest of the axioms, hence the *logical* *‘equi-coherence’*
of *alternative* axioms-systems, built on contraries of those key axioms,
and especially;

**(v)** *‘*__The__ __Gödelian__ __Dialectic__*’*; the *‘‘‘**psychohistorical**-*__dialectical__’’’, «__aufheben__»/*evolute**-***cumulative progression** of *de facto* Axioms-system within the *social*
and ‘socio-cognitive’ [*‘‘‘**psychohistorical**’’’*], *‘**Phenomic**’ ***progression** of a human species.

That is, the *controversial**,* __dialogical__, __dialectical__ __process__ of discovery, exploration, comparative evaluation, and
rational selection of assumptions [of *premises*, *postulates*, *axioms*,
*definitions*, *primitives*, and *rules of inference*] is not a
final, once-for-all, ‘finishable’, *synchronic* human activity.

Not all possible alternative and/or *incremental*
axioms are known, __or__ __even__ __knowable__, *for humanity*, at any given moment in
human history -- i.e., at any given stage in the self-development of the
complex unity of the *‘‘‘**Human Phenome**’’’*/*“**Human Genome**”*.

This *‘meta-axiomatic’* __dialectic__ process is, on the contrary, an ever-renewed, ongoing, and
cumulative process, a *diachronic* activity of expansion of the axiomatic and *‘**ideo**-***ontological**’ foundations that are
accessible to humanity -- i.e., that have newly become new parts of *‘‘‘**The Human Phenome**’’’*.

It produces an «__aufheben__»*-***progressive** [*‘*__Qual__o*-***Peanic**’
or* ‘**Meta**-***Peanic**’ ] *historic*[*al*]* *__sequence__ __of__ __systems__
of logic and mathematics.

That [*‘*__Qual__o*-***Peanic**’
or* ‘**Meta**-***Peanic**’ ] «__aufheben__»*-***progression** reflects the *immanent emergence*, within human society, of psycho-cultural ‘readiness’ for
each next new epoch of *cognitive*, *axiomatic*, and *‘**ideo**-***ontological**’ *expansion*, borne in the interconnexion between:

**(****1**)
*“**technical**”* or *‘**technique**-***al**’, *“**technological**-***ontological**” *self**-***expansion** of the activities/practices of a __generally__ __acceleratedly__*-*__expanding__ *human**-societal self-reproduction*;
i.e., of *“*__human__ __species__ social*-***reproductive **__praxis__” [“growth of the
social forces of production”], and;

**(****2**) maturation in the prevailing level of ‘exo-somatically’
acquired, ‘trans-genomically’ transmitted cognitive and affective development within
the typical “social individual”, hence of the *global human culture* and *“**memes pool**”* [or *‘‘‘**Phenome**’’’*].

We describe *‘*__Dialectical__ __Meta__*-*__Axiomatics__’ as a ‘*A *__BI__*-***Systematic Method of
Presentation**’, because both the *‘*__Meta__*-*__System__atic
__Dialectical__’ *method of presentation*, and
the **Euclidean**/**Newtonian** *method
of presentation* -- recall the ordering of the theorems, etc., presented in
Euclid’s __Elements__
and in Newton’s __Principia__
-- can exhibit a rigorous, *determinate*, *systematic
ordering* of their contents.

Thus, both the *intuitive expository stream*, and the *deductive**-‘***dianoic**’**
expository stream** -- as depicted in
the graphical visualization above -- are systematically-ordered narrative
streams.

__E__

__D__

__Dialectical__

*A*__Bi__*-*.

**Systematic Method of Presentation***arché*

__dialectic__

__dialectic__

__dialectical__*immanent critique*

*ideology*

*mode of exposition*

*exposition*

*mode of mathematical discovery*

*mode of exposition*

__Dialectical__Meta**Axiomatics**’

**-**

__Dialectical__Meta**Axiomatics**’

*complex unity*

*dialectical synthesis*

*mode of exposition*

__Elements__*arché*

*Systematic*

__Dialectics__*Meta*

**Systematic Dialectics**’

*modes of exposition*

__Logik__

__Encyklopädie der philosophischen Wissenschaften__

__Capital__*modes*

__E__

__D__

__Dialectical__Meta**Axiomatics**’

*mode of exposition*

__Dialectical__Science*psych*

*e*

*ohistorically*

*Science of Mathematics*

**-**

__Dialectical__Meta**Axiomatics**’

*aufheben*

**conserves**

*verstand*

*dianoetic*

__within__*Gödel*

**Incompleteness**-

**driven**

*Gödelian*

**ideo**-

**’**

__dialectical__**progression**

__s__

__systematically__**-**

__ordered__system__s__**progression**

__s__*Gödelian*

__dia__chronic

__Meta__**System**’

__s__

__dialectical__reason

__trans__**deductive**

__non__

__rational____justification__

__dialectical__logic*aufheben*

*aufheben*

**conserve**

*aufheben*

**transformation**

*elevation*

*new kinds of numbers*

__un__

__un__*undecidable propositions*

*new*

*higher kinds of numbers*

*new kinds*

*higher logical*

**types**

*sets*

*sets*

*aufheben*

*self*

**internalization**’

*subsets-subsumption*

*sets*

*within*

*aufheben*

__un__

__solvable__*new kinds*

*higher logical*

**types**

*sets*

*new*

*higher kinds of numbers*

*aufheben*

**created**

**-**

__Dialectical__Meta**Axiomatics**’

__no__

__dialectical__reason

__non__

__derivation__

__justification__*¡*

*anti*

__dialectical__traditions*the Occidental Dark Ages*

*!*

**-**

__Dialectical__Meta**Axiomatics**’

*justified*

__dialectically__*self*

**xively’**

__e__*self*

**xively’**

__u__*arithmos*

__Non__*Totality Theories*

**-**

__Dialectical__Meta**Axiomatics**’

__c__

__In__**-**

__Dialectical__Meta**Axiomatics**’

*intra*

**duality**’

*intra*

**multiality**’

*arché*

__meta__

__system__atic**’**

__dialectical__

__categorial__**progression**’

**-**

__s__**progression**’

__dialectical__**algebraic modeling**

**’’’**

__Dia__chronically

__between__

__Dialectical__Meta**Axiomatics**’

*heuristic*

__intuition__**involving**

*intensional*

**derivation**

*aufheben*

*progression*

__Meta__**’’’**

__Syn__chronically

__within__

__Meta__

__Dialectical__Meta**Axiomatics**’

*theorems*

*rigorous deductive logic*

*Theorems*

__also__*conceptually*

*intuitively*

*begrifflichkeit*

__Dialectical__Meta**Axiomatics**’

*intuitive/conceptual exposition*

__parallel__*without comprehension*

__Dialectical__Meta**Axiomatics**’

__intuitively__**’**

__ordered__narrative

__A Dialectical Theory of Everything__

__Dialectical__Meta**Axiomatics**’

__Dialectical__

__Meta__**’**

__Axiomatics__

__aufheben__**conserves**

__dianoesis__

__dialectical__

__Meta__**’**

__Axiomatics__

__dianoesis__**(i)**the

__non__-self-evidence*of appropriate and optimal*generally; the exercise of choice and skillful design required in their development and selection, and the abounding

**axioms***‘*

__alternativity__*’*which that activity confronts;

**(ii)**the

*axioms**-*or

**dependence**

*assumptions**-*of

**relativity****all**

**formal proofs**, hence of all

*formal “*;

*”*__truths__**(iii)**the

*logical*

*‘equi-coherence’*

*of*

__non__**of**

*-*__standard____models__*“first order logic”*axioms-[

*]systems with respect to the*

__sub__*standard models*with which those

__non__*-*standard models are associated, and with reference to which they are defined as

*“*standard”;

**-**__non__**(iv)**the

*formal*

*, or*

__independence__*Gödel-*of key axioms of

**decidability,**__un__*“higher-than-first-order-logic”*axioms-systems with respect to the rest of the axioms, hence the

*logical*

*‘equi-coherence’*of

**axioms-systems, built on contraries of those key axioms, and especially;**

*alternative***(v)**

*‘*

__The____Gödelian____Dialectic__**; the**

*’**‘‘‘*

*psychohistorical**-*, «

**’’’**__dialectical__**»/**

__aufheben__

*evolute**-*of

**cumulative progression***de facto*Axioms-system within the

*social*and ‘socio-cognitive’ [

*‘‘‘*

*psychohistorical**’’’*],

*‘*

*Phenomic**’*of a human species.

**progression***controversial*

__dialogical__

__dialectical__

__process__*synchronic*

__or____even____knowable__*Human Phenome*

*Human Genome*

__dialectic__process*diachronic*

*ideo*

**ontological**’

*The Human Phenome*

__aufheben__**progressive**

__Qual__o**Peanic**’

*Meta*

**Peanic**’

*historic*

*al*

__sequence____of____systems__of logic and mathematics

__Qual__o**Peanic**’

*Meta*

**Peanic**’

__aufheben__**progression**

*immanent emergence*

*cognitive*

*axiomatic*

*ideo*

**ontological**’

*expansion*

**1**)

*technical*

*technique*

**al**

*technological*

**ontological**”

*self*

**expansion**

__generally__

__acceleratedly__

__expanding__

__human____species__social**reproductive**”

__praxis__**2**)

*global human culture*

*memes pool*

*Phenome*

__Dialectical__

__Meta__**’**

__Axiomatics__*A*

__BI__**Systematic Method of Presentation**

__Meta__

__System__atic**’**

__Dialectical__*method of presentation*

*method of presentation*

__Elements__

__Principia__*determinate*

*systematic ordering*

*intuitive expository stream*

*deductive*

**dianoic**’

**expository stream**

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