__Full Title__--

**Q**:

**What Drives the Dialectic of Arithmetics**?

**A**: "

**Unsolvable**"

**Equations Becoming Solved**.

Dear Readers,

Quoted below is a passage from the Background section of

**F**.

__.__

**E**__. Vignette__

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

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

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

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

-- on the driver of the

*'*of the Standard Arithmetics.

**Meta**-**System**atic**'**__Dialectical__**Presentation****"**

__The__

**‘‘‘**

__Driver__

**’’’**

__for the Progression of the__

**“**

__Standard Arithmetics__

**”:**

__The Solution of__

**“**

__Unsolvable__

**”****.**

__Equations__Despite those limitations of its scope, the ‘meta-model’ presented herein does capture

*‘*

*The Gödelian*__Dialectic__*’*of this progression of the axiomatic systems of the Standard Arithmetics.

That is, this ‘meta-model’ captures a pedagogically-selected, pedagogically-optimized progression of “Gödel formulae” analogues for this progression of arithmetics.

Each such Gödel formula “deformalizes” into a proposition asserting that a certain “diophantine” algebraic equation cannot be solved within the [“incomplete”]

*‘number-ontology’*, or

*‘number-kind ideo-ontology’*, of the given arithmetical axiomatic system, inside which it arises as a ‘well-formed equation’ of that system.

Also, as a ‘Gödel formula proposition’, asserting the unsolvability of such an equation in such an axioms-system, such a proposition would be true of the axioms-system within which it arises, but would be neither deductively provable nor deductively dis-provable -- i.e., would be

*“*decidable” -- from the axioms of that axioms-system, i.e., within the logical-language resources, syntactical and semantic, of that axioms-system.

__un__Derivation of such a well-formed unsolvable equation, and unprovable theorem, within such an axioms-system precipitates an immanent formal-logical «

**»,**

*aporia**impasse*,

*quandary*, or

*predicament*, for that axioms-system:

*a*

*logical*-- the

__in__adequacy*‘*

__a__*-*pore’, or

*‘*

__not__*-*pore’; the absence of a “passage”, of an “opening”, beyond the present “impasse”, which appears, at first, to be a hopeless dead end.

However, such an equation

*solvable within the consecutive*

__would be__*next*arithmetical axioms-system, the “successor system” of that “predecessor” arithmetical axioms-system, by means of the new ‘ideo-ontology’ -- by means of the new

*of*

__kind__*‘number*

*ontology’*-- which first fully arrives in that successor system.

That same equation would also be solvable in all subsequent arithmetical axioms-systems in this progression, using that new ‘ideo-ontology’, which is

*conserved*, as well as qualitatively/‘ideo-ontologically’

*further extended*, in all subsequent arithmetical axioms-systems in this ‘meta-system’, or systems-progression.

Such a ‘Gödel formula proposition’, asserting the unsolvability of that equation

*in*that predecessor system, will become

*provably*

*true*, via the expanded logical-language resources of that successor arithmetical axioms-system, as well as

*in*all of

*its*“successor systems”, in which those new logical-language resources are also both

*conserved*&

*determinately-negated/changed/elevated/further-extended*in a qualitative/‘ideo-ontological’ sense.

**"**A rendition together with examples of such "solved unsolvables" can be found here -

**-**

**http://www.dialectics.org/dialectics/Correspondence.html**

http://www.dialectics.org/dialectics/Correspondence_files/FEA0E1~1.PDF

-- and in the following sub-section of the Background section of Vignette

**:**

**#4****"**

__The Pedagogical Strategy Guiding System Order Choices for Our Presentational____.__

**‘****Meta-Model****’**

**F**

**.**

__E__**.**

__D__**.**

*could have*chosen to present the

**6**kinds of

*“unsolvable diophantine equations”*listed below in another possible order than the order in which they are listed below, which is also that in which our ‘meta-model’ actually presents them --

**1**

**.**

**[**

**n**

**+**

**x1**

**=**

**n**

**]**,

**for**

**n**in

**N**, posing the paradox, for

__N__**’s notion of number ‘**

__#__

__N__

*atural**-*

**ness**’, of

*'*

__n____on__*-*

**i**ncreasive addition*’*. Solution[-set]:

**x1**

**=**

**0**, or

**x1**is contained in

**a**is contained in

**W**.

This equation jumps us from

__N__**to**

__#__

__W__**;**

__#__**2**

**.**[

**w**

**+**

**x2**

**=**

**0**;

**x2**,

**w**

**~=**

**0**], for

**w**

**>**

**0**in

**W**, posing the paradox, for

**’s notion of number ‘**

__W__

__W__

*hole**-*

**ness**’, of

*‘*

__de__

**creasive addition***’*.

Solution-set:

**x2**is contained in

**m**is contained in

**Z**. This equation jumps us from

__W__**to**

__#__

__Z__**;**

__#__**3**

**.**

**[**

**|**

**x3 x**

**z|**

**<**

**|z|**;

**x3**,

**z**

**~=**

**±**

**0**

**]**, for

**z**in

**Z**, posing the paradox, for

__Z__**’s notion of number**

__#__*‘‘‘*

**[**

*integ***e**]

*r**-*ity’’’, of

*‘*

__de__*creasive multiplication*

*’*. Solution-set:

**x3**is contained in

**(**

**±**

**0**

**,**

**+**

**1)**

*****is contained in

**f**is contained in

**Q**. This [in]equation in

__Z__**jumps us into**

__#__

__Q__**;**

__#__**[**

*****__Note__: ‘

**(**

**±**

**0**

**,**

**+**

**1)**’ means all

**f**ractions

*strictly**between***±**

**0**

**&**

**+**

**1**, i.e., excluding

**±**

**0**

**&**

**+**

**1**].

**4**

**.**

**[**

**x4**

**^2**

**-**

**p**

**=**

**±**

**0**

**]**,

**±**

**0**

**<**

**p**in

**Q**a

**Q**

**rime number, posing the paradox, for**

__p__

__Q__**’s notion of number**

__#__*‘*

*ratio**nality’, of*

**-***“*

__in__

**commensurability***. Solution-set:*

**”****x4**is contained in

**{**

**±**square-root of

**p**

**}**is contained in

**d**is contained in

**R**. This equation jumps us from

__Q__**to**

__#__

__R__**;**

__#__**5**

**.**

**[**

**x5**

**^2**

**+**

**1**

**.**

**0**

**...**

**=**

**±**

**0**

**.**

**0**

**...**

**]**, or

**[**

**-**

**x5**

**=**

**(**

**+**

**1**

**.**

**0**

**...**

**)**

**/**

**x5**

**=**

**x5**

**^-1**

**]**, posing the paradox, for

__R__**’s notion of number ‘**

__#__

__R__

*eal*

**-****i**ty’, of

*‘the*

__i__*unit’s*

*additive*

*inverse*

*/*

*multiplicative inverse*

*equality*

*or*

*identity*

*’*. Solution-set:

**x5**is contained in

**{±**square-root of

**-**

**1**} such that

**±**

**is contained in**

__i__**i**is contained in

**C**. This equation jumps us from

__R__**to**

__#__

__C__**;**

__#__**6**

**.**

**[**

**+**

**x**

**6**

**y**

**6**

**=**

**-**

**y**

**6**

**x**

**6**;

**x**

**6**,

**y**

**6**

**~=**

**0**

**]**, posing the paradox, for

**C**’s notion of number ‘

__C__

*omplex**ity’, of*

**-***“*

*multiplicative*

__anti__

**-commutativity***”*, or of

*sign*

**-**

*reversal as a result of multiplicative factor*

**-**

*reversal***.**... . This equation jumps us from

__C__**to**

__#__

__H__**.**

__#__Equation

**2**. is inexpressible in

__N__**, because it involves**

__#__**0**, not in

**N**. It would take us next to

__Z__**.**

__#__If we started with equation

**3**., which is not well-formed in

__N__**, because it uses the useless- or meaningless-in-**

__#__

__N__**“absolute value” operator, ‘**

__#__**|...|**’, it would take us next to

__Q__**.**

__#__Equation

**4**. is inexpressible in

__N__**, because it involves**

__#__**0**, not in

**N**. It would takes us next us to

__R__

__#__**.**

Equation

**5**. is also inexpressible in

__N__**, because it involves**

__#__**0**, not in

**N**. It would takes us next us to

__C__**.**

__#__If we started with equation

**6**., which is not well-formed in

__N__**, because it uses the useless- or meaningless-in-**

__#__

__N__**“signs”, ‘**

__#__**+**’ and ‘

**-**’, it would take us next to

__H__

__#__

*!***F**

**.**

__E__**.**

__D__**.**came to the conclusion that

__N__

__#__**,**

__W__

__#__**,**

__Z__

__#__**,**

__Q__

__#__**,**

__R__

__#__**,**

__C__

__#__**,**

**&**

__H__

__#__**...**were the right «

**» for the «**

*species***» of [counting] number, the best progressive partitioning of the generic “Standard” number concept, the best division [«**

*genos***»] of that «**

*diairesis***» of number into «**

*genos***» of number, the best ‘‘‘speciation’’’ of number-kinds for ready assimilation by those to whom we planned to present the “Standard Arithmetics”, given the contemporary view of the standard arithmetics, and of standard mathematics in general, and given the total psychohistorical / phenomic / ideological cognitive**

*species**context*of ‘recent-modern’ humanity.

**F**

**.**

__E__**.**

__D__**.**came to the further conclusion that the sequence given above was the right sequence of presentation of these number-«

**», representing the right simplicity-to-complexity, abstractness-to-thought-concreteness gradient, with the right “consecutive**

*species***tep-sizes”, in terms of the ‘‘‘sizes’’’ of the qualitative increments in ‘ideo-ontology’, for optimal ease of assimilation.**

__s__The inspiration for the

*order of presentation*that

**F**

**.**

__E__**.**

__D__**.**has selected for this Standard Arithmetics’ systems-progression is partly pictorial.

It is the perceived coherence of the

*“*

*order of filling*

*-*

*in*

*”*of ‘Standard-Numbers-Space’, as expressed by the ‘spaces-progression’ of the diagrams/depictions of the ‘number-spaces’, or the ‘‘‘analytical geometries’’’, of the “Standard” systems of arithmetic, as shown via the graphics below ... .

__Note__: We are herein rendering explicit throughout, in both our ideographic renderings of the individual numbers/numerals of the various numbers-systems, and in our depictions thereof... certain key features [e.g., the “leading zeros” place-holders in

**0...01**within

**, for example], i.e., various**

__W__*syntactic*attributes -- signifying

*semantic*attributes of the concepts of those numbers -- which are usually, in common uses of these numbers, left implicit, abbreviated-out, or ignored, but which are of crucial conceptual significance in tracking the cumulative changes in ‘number ideo-ontology’ from each

**tage and predecessor system of arithmetic to its successor**

__s__**tage and system, throughout the entire progression of systems of “Standard Arithmetic”.**

__s__We also, in the depictions below, use ‘

**(---)**’ as ‘assignment sign’, or ‘interpretation sign’, from one “pure” ‘ideo-system’ to another.

We note, in passing, one further dimension of the gradient -- the gradual gradations, escalating from the «

**»’s maximal relative simplicity, to the maximal relative complexity of the final ‘counter-supplement’**

*arché**‘*

*culminant*

*’*that we consider, that runs through the entire

__clusive ‘qualitative interval’,__

*in***[**

__N__

__#__**,**

__h__

__#__**]**, in the progress from

__N__**to**

__#__

__h__**.**

__#__This further so-‘gradated’ dimension is that of the increasing relative thought-complexity of the

*arithmetical*‘aporialized’, or ‘paradoxicalized’, in the succession of “unsolvable” diophantine algebraic equations that catapult our core section discourse from arithmetical system to next higher arithmetical system.

__operations__Equations

**1**

**.**

**&**

**2**. involve the paradoxes of the arithmetical

*operation.*

**addition**Equation

**3**. involves the paradoxes of the arithmetical

**operation, definable, in the**

*multiplication*

__E__**.**

__D__**.**

__of such operations, as the__

**dialectic***‘*operation.

__-__**meta****addition**’Equations

**4**. and

**5**. involve the paradoxes of arithmetical

**, definable, in that same**

*exponentiation*__of arithmetical operations, as the__

**dialectic***‘*operation.

**-**__meta__**multiplication**’Together with

*equation***6**

**.**,

**equations****4**. and

**5**. involve

*quadratic*__non__linearity**--**

**2**nd degree

__algebraic__**equation****[albeit**

__non__linearity*not*

**].**

__integrodifferential__equation__non__linearity

**"**Regards,

Miguel

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