###
__Un__*solvable Algebraic Equations
of the *Seldonian* *_{X}__Q___* **Arithmetic*__s__ *for *__Dialectics__.

__Un__

_{X}

__s__

__Dialectics__

Dear Readers,

__. This blog-entry --__

**Caveat**__like many others, typical within this blog -- is__

**un**__about__

**not****, or**

*interpretations**'*

*meta*

*-*

*model*

*meta**-***'***equation***, of the**

*solutions*

__gene__ric*'*

*that inhere in the*

*meta**-***'***numeral*__s__

_{X}__Q___**, wherein**

*algebra*__s__

__X__**one of the**

*represents*

*standard***, and wherein the '[pre-]subscriptization' of**

*arithmetic*__s__**indicates the**

__X____of that__

**subsumption**

*standard***by one of the**

*arithmetic*

__non__*-*,

**standard**

__dialectical__arithmetical

__Q___

*axiom*__s__*-*, and the

**system**__s____of the set or space of__

**subsumption***,*

*numeral*__s__**X**, by the set or space of

*'*

*,*

*meta**-***'***numeral*__s__**{**

__q__

_{X}**} =**

**.**

_{X}__Q__On the contrary, this blog-entry is about the

*"*interpreted", or, more accurately, is about the

__un__*'*interpreted',

__-__**minimally***'*

*, and*

*meta**-***'***number*__s__

*their***and**

*arithmetic*__s__**,**

*algebra*__s__**, i.e., in the**

*themselves*

__gene__ric**of these**

*form*__s__*'*

*,*

*meta**-***'***number*__s__**, and**

*arithmetic*__s__**-- about the new**

*algebra*__s__**'ideo-ontology' that**

*mathematical**our*

**undergirds***'*

*meta*

*-*

*model*

*meta**-***'***equation*__s__**&**

*their*

*interpretations*

**/****.**

*solutions*As in the

*first*text module pasted-in below, we have noted, in this blog, before, the expectation, related to Gödel’s epochal “incompleteness theorem”, that any

*axiom*__s__*-*for a

**system**

*standard***that encompasses at least the so-called “**

*arithmetic***atural” numbers**

__N__**will permit the formation, within its rules of syntax, of (one or more)**

*arithmetic***(**

*algebraic equation***), thus “well-formed”, which are**

*s*

__un__

*solvable***that**

__within__**, and one of which will the “deformalization” of the “Gödel formula”,**

*arithmetic***G**, for that

**.**

*arithmetic***G**is a formula for a proposition that asserts

**that it cannot be**

*of itself***from the**

*proved***of that**

*axioms*

*axiom*__s__*-*for this

**system**

*standard***. Hence, if that proposition can be shown, non-deductively, to be true, then that**

*arithmetic*

*axiom*__s__*-*is

**system***“*, and the

**formally**”__in__complete**of this**

*algebra*

*standard***is capable of forming at least one syntactically “legal”**

*arithmetic***,**

*equation*

*whose***(**

*solution***) do(es) not exist**

*s***this**

__within__**, and which is an**

*standard arithmetic***that is tied to**

*equation***G**.

If, on the contrary,

**G**, in fact,**be***can**formally***from those***deduced***, then that***axioms**axiom*__s__*-*for this**system***standard***is***arithmetic**“*, i.e., is**formally**”__in__consistent*‘“*-- i.e., a false proposition can be**formally**-__self__**contradictory**’’’**from its***deduced***.***axioms***How much can we rightfully expect this expectation to hold also for the**

*¿*

*axiom*__s__*-*of those

**system**__s__

__non__*-*

**standard**

*system*__s__of**that are the Seldonian**

*arithmetic**‘*

**arithmetic**’__s__for__dialectics__**?****the Seldonian**

*¿*__Must__*‘*also ‘‘‘contain’’’

**arithmetic**’__s__for__dialectics__**, tied to**

*algebraic equations*

*their***versions of**

*own***G**, that are “well-formed” but

__un__

*solvable*

__within__**, out of**

*them*

*formal**-*

**logical necessity****?**
Let us grapple,

*, with a easier question:***first****the Seldonian***¿*__Do__*‘*also ‘‘‘contain’’’**arithmetic**’__s__for__dialectics__**that are “well-formed”***algebraic equation*__s____within__*them***, but that are also**__un__*solvable*__within__*them?*
Well, let’s start by looking at the

**in the Seldonian***first*__ex__plicitly__dialectical__arithmetic**, the***progression of*__dialectical__arithmetic__s__**atural-numbers,**__N__**N**, subsuming**,**__dialectical__arithmetic**.**_{N}__Q___
The

_{N}__Q___*system**of***encompasses, through its***arithmetic**product rule***,***axiom**axiom*§**9**, an*additions**-*excerpt from**only****atural-numbers**__N__**, if only within***arithmetic***subscripts level.***its*
And, indeed, the following

**of***equation***’s**_{N}__Q___**is,***algebra***the**__within___{N}__Q___*axiom*__s__*-*, well-formed but**system**__un__**: for every***solvable***n**in**N**,__x__**=**__q__**is**_{x}**in**__not__**, if --**_{N}__Q__

__x__*****

__q__

_{n}**=**

__q__

_{x}*****

__q__**=**

_{n}

__q__**.**

_{n}
And, as also with the

*‘*of the**Gödelian**’__Dialectic____standard__*system*__s__of**, for***arithmetic**its**systematic***['***progression***---)**']*of number**-***spaces**--**N**

**---)**

**W**

**---)**

**Z**

**---)**

**Q**

**---)**

**R**

**---)**

**C**

**---)**...

**.**

-- the very

**of the***next*__non__*-*,**standard****, in**__dialectical__arithmetics**of the***one**two***of the Seldonian***directions***-- the***progression***which follows the***direction**‘*of the**Gödelian progression**’*standard**systems of***, just given, in terms of the***arithmetic**next**standard***that is subsumed by the***arithmetic***version of the***next*__Q_____non__*-***standard****.***arithmetic*__s__
In this case, it is the

_{W}__Q___**, that renders**__dialectical__arithmetic**,***solvable***, that**__within__itself**. That***equation***is***equation***by setting***solvable***q**_{x}**=****q****, and by applying the**_{0}*‘*«**double**-**conservation****»***aufheben***,**__e__volute product rule*axiom*§**9**[see*second*module below] --**q**

**is in**

_{x}**, for every**

_{W}__Q__**w**in

**W**, if --

**q**

_{x}*****

__q__**=**

_{w}

__q__**;**

_{w}**q**

_{0}*****

__q__**=**

_{w}

__q__**+**

_{w}

__q__

_{w}

_{+}**=**

_{0}

__q__**+**

_{w}

__q__**=**

_{w}

__q__**, so**

_{w}**q**

_{0}*****

__q__**=**

_{w}

__q__**, including --**

_{w}**q**

_{0}*****

**q**

_{0}**=**

_{}**q**

_{0}**+**

_{}**q**

_{0}

_{}

_{+}**=**

_{0}**q**

_{0}**+**

_{}**q**

_{0}**=**

_{}**q**

_{0}**, so**

_{}**q**

_{0}*****

**q**

_{0}**=**

_{}**q**

**[thus, by**

_{0 }

*its*

**subsuming****W**, the

*pattern rises up again within the*

**Boolean**

__gene__rally*'*

__contra__*-*

**Boolean**'**, in the form of the exceptional value,**

__Q___**q**

**].**

_{0}

_{}**q**

**is**

_{0 }__in__

**not**

_{N}__Q__**&**

**q**

**is in**

_{0}**.**

_{W}__Q__
However, also as with the

*“***standard***systems of**arithmetic**”*, this**,***next arithmetic***, has**_{W}__Q___*its**own*__un__*solvable***(***equation***) -- in this example, an***s***not even expressible -- not “well-formed” -- within***equation***, because**_{N}__Q___**q****is not an element of the ‘meta-number space’**_{0}**--**_{N}__Q__
for every

**w**in**W**,__q__**is**_{x}**in**__not__**, if --**_{W}__Q__

__x__*****

__q__

_{w}**=**

__q__

_{x}*****

__q__**=**

_{w}**q**

**.**

_{0}
In this case, it is the

_{Z}__Q___**, that renders**__dialectical__arithmetic**,***solvable***, the**__within__itself**immediately above***equation***. That****is***equation***by setting***solvable***q**_{x}**=****q**_{-}**, and by applying a new version of**_{w}*axiom*§**9**, e.g., the*‘***-**__meta__**heterosis**__con__volute product rule*’*, as follows --

__q__**is in**

_{x}**, for every**

_{Z}__Q__**z**in

**Z**, if --

**q**

_{x}*****

__q__**=**

_{z}**q**

**;**

_{0}

__q__

_{-}

_{w}*****

__q__

_{+}**=**

_{w}

__q__

_{+}

_{w}_{ }

_{+}_{ }

_{-}**=**

_{w}

__q__

_{w}

_{-}**=**

_{w}**q**

**, so --**

_{0}

__q__

_{-}

_{w}*****

__q__

_{+}**=**

_{w}**q**

**, and, in general,**

_{0}

__q__

_{-}

_{z}*****

__q__

_{+}**=**

_{z}

__q__

_{+}

_{z}*****

__q__

_{-}**=**

_{z}**q**

**, for every**

_{0}**z**in

**Z**.

__q__

_{-}

_{w}**is**

_{ }__in__

**not**

_{W}__Q__**&**

__q__

_{-}**is in**

_{w}**.**

_{Z}__Q__
However, also, and, again, as with the

*“***standard***systems of**arithmetic**”*, this**,***next arithmetic***, has**_{Z}__Q___*its**o**wn*__un__*solvable***(***equation***) -- in this example, an***s***involving the operation of***equation**‘ “*--**purely**”-**,**__qual__itative**categorial****division**’

__q__**is**

_{x}**in**

__not__**, if, for all**

_{Z}__Q__**a**,

**b**, and

**c**in

**Z**, such that

**a**

**≠**

**b**

**≠**

**c**

**≠**

**a**--

__x__**=**

__q__

_{x}**=**

__q__

_{a}

*/***(**

__q__

_{b}**+**

__q__

_{c}**)**.

The operation of

*‘ “*, '**purely**”-**,**__qual__itative**categorial****division**’**_****', is***/*_**defined for the**__not___{N}__Q___**, so that the***arithmetic***immediately above would not be “well-formed”***equation*__within__**.***it*
The operation of

*‘ “*be defined**purely**”-**,**__qual__itative**categorial****-**__self__**division**’**can****the**__within___{W}__Q___**, such that, for every**__dialectical__arithmetic**w**in**W**,__q___{w}*/*__q___{w}**=**__q___{w}*****__q___{-w}**=**q**.**_{0}__restricted binary__

**Un***‘*

__qual__itative**division**’**defined for the**

__is__

_{Z}__Q___**, e.g., via the**

__dialectical__arithmetic*‘*

**-**__meta__**heterosis**

__con__volute product rule*’*version of

*axiom*§**9**, such that --

__q__

_{a}

*/*

__q__

_{b}**=**

__q___{a}

*****__q__

_{-}_{b}**=**__q__

_{a}_{ }

_{+}_{ }

_{-}**=**

_{b}

__q__

_{a}

_{-}**.**

_{b}
However, even in

**, and in**_{Z}__Q___**, and in**_{Q}__Q___**, the**_{R}__Q___*‘*__qual__itative*division**’*of a**by a***category***of**__cumulum__**--***categories*

__q__

_{a}

*/***(**

__q__

_{b}**+**

__q__

_{c}**)**

-- is

**a value that is contained in any of**__not__**‘meta-number spaces’.***their*
One has to

**in the***progress*__other__**of the Seldonian***direction*__double__*-*[see bottom-most**progression****modules, below] of***two***of***systems**arithmetic*__s__*for***, to find an**__dialectics__**in which a value***arithmetic**of a**form similar to that*of --

__q__

_{a}

*/***(**

__q__

_{b}**+**

__q__

_{c}**)**

-- is contained, namely,
in the form

**.**__b__*/*(__a___{1}+__a___{2})
That

**, one that accomodates***arithmetic**"***purel**y"__qual__**,***itative**'*__qual__o*-**'''fractions''' of that form, is the***fractal**'**24**th**in that Seldonian***arithmetic***of***progression*__dialectical__**--***arithmetic*__s__

__q__

_{BA}**[---)**

__q__

_{24}
-- and that

For more regarding these issues, see the

www.dialectics.org

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

http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

-- pages

**24**th*system of***is***arithmetic***of***capable***to***interpretation**represent**two**-*level, e.g., «**»-***genos***-«**__over__**»,***species**‘*, trans-Platonian «**ideo**-**taxonomic**’**»***arithmoi eidetikoi***.**__dialectics__For more regarding these issues, see the

**F**.__.__*E*__. Primer by__**D****F**.__.__*E*__. Secretary-General Hermes de Nemores --__**D**www.dialectics.org

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

http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

-- pages

**A-15**through**A-17**.
Regards,

Miguel

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