__Full Title__:

**--**

*The Goedelian*__Dialectic__of the Standard Arithmetics**Rendered via Pictures**

Dear Readers,

The 'Dialectical Meta-Equation' that generates the 'Meta-Systematic Dialectical' Presentation of the Standard Arithmetics, depicted below, is defined, symbolic-element-

*by-*symbolic-element, as follows --

'

**, [**

*The Goedelian*

*Meta**-*]

*Systematic*__Dialectic__of the Standard Arithmetics*'*:

*--*

**First**__Tri__ad__Initial Overview Depiction__:

*'*

**The**

**Dialectic**

**of**

**the Standard Arithmetics***'*--

The

*denoted by*

**Axioms**-**System**

**N**

__#__--__tages__

**s**

**/**__teps of__

**s**

**presen***--*

__t__ation**s**:

__#__= 0

**N**

**#****s**:

__#__= 1

**N**

__#__~+~

__a__**#**__The "__

**=****N**atural" Numbers Arithmetic, plus their '

__ught' numbers axioms sub-system__

**a***, refuting any claim of the "*

**counter**-**example****N**atural" Numbers System to '''completeness''' regarding [Goedel: "diophantine"] algebraic equation-solving capability;

The

*denoted by*

**Axioms**-**System**

**W**

__#__--**s**:

__#__= 2

**W**

__#__~+~

__m__**#**__The "__

**=**__hole" Numbers Arithmetic, plus their '__

**W**__inus' numbers axioms sub-system__

**m**

*counter**-*

**, refuting any claim of the "**

*example*__hole" Numbers System to '''completeness''' regarding [Goedel: "diophantine"] algebraic equation-solving capability;__

**W**The

*denoted by*

**Axioms**-**System**

**Z**__--__

**#****s**:

__#__= 3

**Z**

__#__~+~

__f__**#**__The "Integers" Arithmetic, plus their '__

**=**__ractional' numbers axioms sub-system__

**f**

*counter**-*

**, refuting any claim of the "Integers" System to '''completeness''' regarding [Goedel: "diophantine"] algebraic equation-solving capability;**

*example*The

*denoted by*

**Axioms**-**System**

**Q**

__#__--**s**:

__#__= 4

**Q**

__#__~+~

__d__**#**__The "__

**=****Rational**" Numbers Arithmetic, plus their '

__iagonal' numbers [__

**d***'*

**-**__ir__**rational**' numbers] axioms sub-system

*counter**-*

**, refuting any claim of the "**

*example***Rational**" Numbers System to '''completeness''' regarding [Goedel: "diophantine"] algebraic equation-solving capability;

The

*denoted by*

**Axioms**-**System**

**R**__--__

**#****s**:

__#__= 5

**R**

__#__~+~

__i__**#**__The "__

**=**__eal" Numbers Arithmetic, plus their "__

**R**__maginary" numbers axioms sub-system__

**i**

*counter**-*

**, refuting any claim of the "**

*example*__eal" Numbers System to '''completeness''' regarding [Goedel: "diophantine"] algebraic equation-solving capability;__

**R**The Axioms-System denoted by

**C**__--__

**#****s**:

__#__= 6

**C**

__#__~+~

__h__**#**__The "__

**=**__omplex" Numbers Arithmetic, plus their "__

**c**__amiltonian quaternionic" numbers axioms sub-system__

**h**

*counter**-*

**, refuting any claim of the "**

*example*__omplex" Numbers System to '''completeness''' regarding [Goedel: "diophantine"] algebraic equation-solving capability; . . .__

**c****. . .**

__Final Overview Depictions__: 'The

**Meta**-**Systematic**__Dialectical__*of the*

**Method of**__Present__ation*'''*of the

**Dialectical****Progression**'''*--*

**Standard Arithmetics**'Tony Smith on Hegelian

*[*

**'**__dialectical__categories*]*

**meta**-*--*

**kinds**"...Hegel's procedure is not really

*ad hoc*at all. To see this we have first to consider what a category is. It is a principle (a universal) for unifying a manifold of some sort or other (different individuals, or particulars). A category thus articulates a structure with two poles, a pole of unity and a pole of differences. In Hegelian language this sort of structure, captured in some category, can be described as a unity of identity in difference, or as a reconciliation of universal and individuals. From this general notion of a category we can go on to derive three general types of categorial structures. In one the moment of unity is stressed, with the moment of differences implicit. In another the moment of difference is emphasized, with the moment of unity now being only implicit. In a third both unity and difference are made explicit together."

"Hegel's next claim is that there is a systematic order connecting these three categorial structures. A structure of unity in which differences are merely implicit is simpler than one in which these differences are explicitly introduced; and one in which both unity and differences are explicit is yet more complex still. Similarly, the first sort of structure is the most abstract, while the other structures are successively more [M.D.: thought-]concrete."

"Yet another way of speaking about the immanent connections here is through the idea of a dialectical contradiction."

"Hegel's views on contradiction have been quite controversial. But at least in the context of constructing a systematic theory of categories he appears to have meant something fairly straightforward. "

"If a category is in general a principle that unifies a manifold, then if a specific category only explicates the moment of unity [M.D.: e.g., category

**N**__, connoting the__

**#***of the "*

**system of standard arithmetic****N**atural" Numbers, as given above], leaving the moment of difference implicit [M.D.: all of the subsequently-

**present**

**ed**

**different***of*

**kinds***of the*

**standard numbers**

**s***are still merely implicit and invisible in*

**tandard arithmetics**

**N**__, especially__

**#**

**/***those that involve the*

**next**

**number****0**], then there is a "contradiction" between what it inherently is

*qua*category [M.D.: and there is also a

__between the__

**discrepancy****of**

__D__*omain**in-*

**standard number**-

**gene***, denoted by*

**ral**__, and__

**#**

**N**__as a claim to exhaustively__

**#**__that__

**present**

__D__*omain**] (a unifier of a manifold) [M.D.: in our case,*

__ex__plicitly

**N**

**#****denotes the "unity" of the "manifold", or <<**

*>>, of the potentially- infinitely-many number-individuals known collectively as the "*

**arithmos****N**atural" Numbers], and what it is explicitly (the moment of unity alone) [M.D.: as well as a

**Goedelian***of the*

__in__completeness

**N**

**#***-- it's incapability to solve semantically all of the "diophantine" algebraic equations which it can well-form syntactically]."*

**arithmetic**"Overcoming this contradiction requires that the initial category be "negated" in the sense that a second category must be formulated that makes the moment of difference explicit [M.D.: e.g., in our case, the

**second***is*

**category**__, connoting the axioms-sub-system for the number-space__

**a#****a = {I-I**,

**II-II**,

**III-III**,

**... }**, the number-space of the

*'*of the "

**self**-**differencing**'**N**atural" Numbers, describing the different kind of number(s), which is(are) missing in the

**N**number-space, namely,

**0**(s)]."

"But when this is done the moment of difference will be emphasized at the cost of having the moment of unity made merely implicit [M.D.: in our case, and as per Hegel, the former most[ only]-emphasized, vanguard, or 'meta-meristem' of stage

**0**our categorial progression -- the former right-most

*, connoting the*

**category**-**symbol***of "simple[st] unity" in our case, namely, the*

**category**

**N**__, in__

**#**

**N**

**#****^(2^0)**=

**N**

**#****^1**=

**N**__-- will be__

**#***"*[Hegel] to a

**demoted**"*[ary]-status position, namely, to the position to the left of, or*

**second***'''*, the

**BEHIND**'''*right-most, most-emphasized, vanguard, 'meta-meristemal'*

**new***, namely, the stage*

**category**-**symbol**

**1**

**category**-**symbol**__, arising from the__

**a#***, or*

**Goedelian**__immanent__critique*, as connoted by*

__-__**self****critique**

**N**

**#**

**N**__, of__

**#**

**N**__as comprehending the totality of its__

**#****,**

__D__*omain*__:__

**#**

**N**

**#****^(2^1)**=

**N**

**#****^2**=

**N**

**#**

**N**__=__

**#**

**N**

**#****~+~**

__]."__

**a#**"Once again, there is a contradiction between what a category inherently is and what it is explicitly [M.D.: and, in our case, there is also a

__between the__

**discrepancy****of**

__D__*omain**in-*

**standard number**-

**gene***, denoted by*

**ral**__, and__

**#****as a claim to exhaustively**

**N****#****~+~**__a#____that__

**present**

__D__*omain**, and, in particular, an*

__ex__plicitly__of the explicit__

**omission***,*

**axioms**-**element**

**q#****aN**, which reconciles and re-unifies

**and**

**N****#****in the new system of arithmetic of the "**

__a#____hole" Numbers:__

**W**

**W**

**#****=**

**N**

**#****~+~**

**a#****~+~**

**q#****aN**, which is contained in

**presenta**

**tion**__tage__

**s****2**--

**N**

**#****^(2^2)**=

**N**

**#****^4**=

**[**

**N****#****~+~**__][__**a#****=**

**N****#****~+~**__]__**a#**

**N**

**#****~+~**

**a#****~+~**

**q#****aN**

**~+~**

__=__

**m#**

**W**

**#****~+~**

__]."__

**m#**"Overcoming this contradiction demands that the second sort of category also be negated, and replaced [M.D.: here Tony Smith, more explicitly than earlier in this quoted discourse, makes the typical, but

*, fall into a '''convolute''' interpretation of*

**vitiating**__; the key to many__

**dialectic****F**.

__.__

**E**__.__

**D**__breakthroughs is the '''evolute''' interpretation of__

**Dialectics***, which is also clearly called for by Hegel, in his supreme methodological prescription, in the final section of his <<*

__dialectical__categorial progression*>>: "In the absolute method the Notion*

**Logik***maintains*itself in its otherness, the universal in its particularization, in judgement and reality; at each stage of its further determination it raises the entire mass of its preceding content, and by its dialectical advance it not only does not lose anything or leave anything behind, but carries along with it all it has gained ..."] with a category in which both poles, unity and difference, are each made explicit simultaneously [MD: in our case, that

*[sort of]*

**third***is connoted by the*

**category**

**category**-**symbol**

**q#****aN**.]."

"Hegel is well aware that "contradiction" and "negation" are not being used here in the sense given them in formal logic. Following a tradition that goes back to Plato, he asserts that in the above usage "contradiction" and "negation" are logical operators for ordering categories systematically, as opposed to logical operators for making formal inferences. The logic with which we are concerned here is

*dialectical*logic." ...

"... Since a category of unity-in-difference on one level can prove to be a category of simple unity from a higher level perspective, thereby initiating another dialectical progression from unity through difference to unity-in-difference, we can construct a systematic theory of categories by employing the dialectical method. In this sort of theory we move in a step-by-step fashion from simple and abstract categories to those that are [M.D.: more] complex and [M.D.: more thought-]concrete, with dialectical logic providing the warrant for each transition. ..."

"... At the conclusion of the linear progression of categories we once again arrive at the initial starting point [M.D.: in our case, back to

*at the*

**/**

__D__*of '''[*

**omain***]*

**standard***in-*

**number**-*, denoted by*

__gene__**ral**'''__]. But it has now been apprehended in thought [M.D.: i.e., has been__

**#**

__system__atically*,*

**analyzed into its component species***, for all who have '''followed''' the*

**and thereby comprehended***,*

**-**__system__atically**ordered**

**categorial progression**

**present***, by means of the*

**ation**

__dialectical__theory*that*

**constituted by***]."*

**progression**"If dialectical logic is rigorously adhered to, the move from one category to the next [M.D.: or,

__,__

**evolutely**__the__

**from**

**preceding**

__-__**sub****progression**

**/***of*

**cumulum***,*

**categories**__that again,__

**to**__the__

**plus***,*

**new**

**additional***that arise from the*

**categories***of that*

**-**__s____elf__**critique**

**preceding***, together forming the*

**cumulum of categories**

**new***that must, in its own turn, be*

**cumulum***...] is not*

__-__**self****critiqued***ad hoc*. The linear progression from a category of immediate unity to [M.D.: or,

__,__

**evolutely**

**from**

**that**

**to**

**that**__; in our case, from__

**plus**

__N#__

**to**

__N#__~+~__] one of difference [M.D.:__

**a#**__], and from there to [M.D.: or,__

**a#**__,__

**evolutely**__from__

**the**-__non__**amalgamtive, heterogeneous**__SUM__of those first two categories,

__to__

**them**__; in our case,__

**plus**

**from**

__N#__~+~

**a#**

**to**

**N**

**#****~+~**

**a#****~+~**

**q#****aN**] a category of unity in difference [M.D.:

**q#****aN**], is not a mere formal schema imposed by Hegel externally. It is instead "the absolute method . . . [M.D.: which] does not behave like external reflection but takes the determinate element from its own subject matter, since it is itself that subject matter's immanent principle and soul."... In this manner the object realm [M.D.: in our case, a realm of

*'''*; of "[

__-__**idea****objects**'''*]*

**standard***and their*

**numbers***] of experience [M.D.: in our case, an experience of partly '''internal''', mental, introspective, 'intro-objective', 'inter-subjectively objective',*

**arithmetics***'*

**historical material**__psycho__*'*] has been reconstructed in thought [M.D.: and, thereby, uplifted and transformed, from "a chaotic conception of the whole" [Marx], to

*and*

**-**__system__atically**ordered**

**subjectively comprehended**

**scientific**__, for those who have '''followed''' this__

**knowledge**

**categorial****progression***]."*

__present__ation of theory[Tony Smith,

__:__

**The Logic of Marx**'**s Capital***, State University of New York Press, [Albany, NY:*

**Replies to Hegelian Criticisms****1990**], pp.

**5**-

**8**].

Regards,

Miguel

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