Dear Readers,

On this

**19**th anniversary of Seldon’s April**7**th,**1996**breakthrough -- his sudden discovery of the_{N}__Q___*‘*, after years of [re-]searching for, and of**First**-__Dialectical__Arithmetic for contra**Boolean Algebra**’*slow*progress toward finding, a*“*-- the**mathematics of**”__dialectics__**F**.**.**__E__**. General Council has cleared, for public dissemination, its**__D__**eight***defining the*__speci__fications sheets*‘***meta**-**number**’**that we call***value**‘*, ‘**Full Zero**’**.**’ -- as distinct from ordinary zero,**0**, which we, in this context, call*‘*,**Empty Zero**’**0****-- and elaborating upon the candidate postulate(s) to govern the use of this****new***'*, this**-**__ideo__**ontology**'*new*__dialectical__*-*, within the Seldonian**ideographical symbol***, or***seventh***'*,**Mu**'*.*__dialectical__arithmetic
I have, in my dissertation-contribution to the

**Foundation**, for my induction-into-membership in the**Foundation**, entitled*“*[which is accessible via the following links, on the Vignettes Page, as Vignette**The Gödelian**”__Dialectic__of the Standard Arithmetics**#4**, Parts**0**,**I**, and**II**, at --
-- described the internal inadequacies

**‘self-incompletenesses’ of each successive***/**axioms**-*of the**system****-- how each***standard arithmetics***is marked by algebraic, “diophantine equations” which, grounding an***standard arithmetic**‘‘‘*, or**immanent critique**’’’*‘‘‘*, and an**self**-**critique**’’’*‘*, or**ideo**-**-**__intra__**duality**’*‘*, of each such**ideo**-**-**__self__**duality**’**by***system***, are --***itself**-- well-formed within that***syntactically****, but for which,***arithmetic**, no ‘semantification’ of the***semantically**__known,__**un****x**, of that algebraic equation, i.e., no*(***solution***) of that***s***, are available***equation****expressible***/*__that__**within***standard arithmetic**’s**axioms**-*, i.e.,**system**__the, partially tacit, ‘‘‘ideo-ontological commitments__**within****presumptions***/***self-limitations’’’ of that***/***.***system*
Thus, the

Thus, in a sense,

*equation***x****+ 1 = 1**is**solvable**__un____the__**within****of the so-called “***system of arithmetic***N**atural” numbers,**, wherein**__N__**N****= {1, 2, 3, ...}**, and indeed this**asserts a***equation**psychohistorical***for the concept of**__paradox__**native to***addition**that***.***system*Thus, in a sense,

__the limitations of the__**within**__N__**,***system***x****=****.**, although this equation of**x**and*to***/****.**must be considered a 'meta-arithmetical', '''meta-mathematical''' assertion, because**.**is not an element of -- is not a*"***number**"**within***--***N**.**,***this equation***x****+ 1 = 1**, marks the__present__ational*transition***the***from*__N__*system***the***to*__W__**, the***system**axioms**-*of the so-called “**system****hole” numbers,**__W__**W****= {0, 1, 2, 3, ...}**, wherein that*equation***readily solvable:**__is__**x****= 0**.
However, the

Thus,

*equation***x****+ 1 = 0**is**solvable**__un__**within***the*__W__**, and indeed asserts a***system**psychohistorical***for the concept of**__paradox__**native to***addition**that***.***system*Thus,

__the limitations of__**within**__W__**, in a sense,***system***x****=****.**[again, as a 'meta-arithmetical' assertion, because**.**is not an element of -- is not a*"*within --**number**"**W**] and this**marks the***equation*__present__ational*transition***the***from*__W__*system***the***to*__Z__**, the***system**axioms**-*of the so-called**system***“*--**integers**”**Z**

**= {..., -3, -2, -1, ±0, +1, +2, +3, ...}**

-- wherein that

*equation***readily solvable:**__is__**x****= -1**.
However, the

*equation***2x****= 1**is**solvable**__un____the__**within**__Z__**, and indeed asserts a***system**psychohistorical***for the concept of**__paradox__**native to***multiplication**that***.***system*
Thus,

__the limitations of the__**withi****n**__Z__**, in a sense,***system***x****=****.**[again, as a 'meta-arithmetical' assertion, because**.**is not an element of -- is not a*"*within --**number**"**Z**], and this**marks the***equation*__present__ational*transition***the***from*__Z__*system***the***to*__Q__**, the***system**axioms**-*of the so-called**system***“*--**rational numbers**”**Q**

**= {....-3/2...-2/1...-1/2...±0/1...+1/2...+2/1...+3/2....}**

-- wherein that

*equation***readily solvable:**__is__**x****= +1/2**.
However, the

Thus,

*equation***x**^{2}**= 2**is**solvable**__un____the__**within**__Q__**, and indeed implies a***system**psychohistorical***for the concept of**__paradox__**native to***exponentiation**that***-- that***system***x**must be either**odd and even, or***both***odd nor even.***neither*Thus,

__the limitations of the__**within**__Q__**, in a sense,***system***x****=****.**[again, as a 'meta-arithmetical' assertion, because**.**is not an element of -- is not a*"*within --**number**"**Q**], and this**marks the***equation*__present__ational*transition***the***from*__Q__*system***the***to*__R__**, the***system**axioms**-*of the so-called**system***“*__R__*eal numbers**”*--**R**

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

**pi**

**....-e....-**

**\/**

**2....±0.000.......+**

**\/**

**2....+e....+**

**pi**

**.....**

**}**

-- wherein that

*equation***readily solvable:**__is__**x****= ±****\/****2**.
However, the

Thus,

*equation***x**is^{2}+ 1 = 0**solvable**__un____the__**within**__R__**, and indeed implies a***system**psychohistorical***for the concept of**__paradox__**native to***inverse values**that***-- for***system***implies that, for that***that equation***x**, its**and its***additive inverse value**multiplicative inverse**must be***value****:***equal***-x = +1/+x**.Thus,

*the limitations of the***within**__R__**, in a sense,***system***x =****.**[again, as a 'meta-arithmetical' assertion, because**.**is not an element of -- is not a*"*within --**number**"**R**], and this**marks the***equation*__present__ational*transition***the***from*__R__*system***the***to*__C__**, the***system**axioms**-*of the so-called**system***“*__C__*omplex numbers**”*--**C**

**= {R + Ri}**

-- wherein that

*equation***readily solvable:**__is__**x****= ±i**. And so on**. . .**.
However, notice also that, in

**of these**__NONE__**-- [***systems**not**in***],**__N__*not in***,**__W__*not in***,**__Z__*not in***,**__Q__*not in***,**__R__*not in***, ... -- is**__C__**workable; is an***division by ZERO***of the form***equation***x = c/0****[in the***solvable*__N__**, such an***system***is not even ‘‘‘well-formed’’’, because the number***equation***0**is not even part of the*‘*-- of either the syntax, or the semantics -- of**ideo**-**ontology**’*that***].***system*
This internal, immanent inadequacy and ‘‘‘incompleteness’’’
of

__ALL__**in the***systems***of the***progression***of the***systems***is evidently of a far deeper sort than the inadequacies and ‘‘‘incompleteness’’’ that drive that***standard arithmetics***, and that were***progression***in that***progressively solved***, as outlined above.***progression*
The [implicitly-

**]**__dialectical__*first**-*order-logic, Peano**of the “***axioms system***atural” numbers, which**__N__**denotes by**__Encyclopedia Dialectica__**, and sees as being**__N___**interpreted as a***standardly**“***purely**”-__quant__itative**, is the***arithmetic***, «***first***»***arché**category***of***/system***in the Seldonian***arithmetic***of***progression*__non__*-*,**standard****.**__dialectical__arithmetics
The Seldonian

*‘*[explicitly-]**First**__Dialectical__Arithmetic*’*, which**denotes by**__Encyclopedia Dialectica___{N}**, and interprets as a**__Q___*“***purely**”-__qual__itative*ordinal***, is the***arithmetic**second**category***of***/system***in the Seldonian***arithmetic***of***progression*__non__*-*,**standard****, the**__dialectical__arithmetics*first**‘***contra**-**category**’*/**‘*, in that**contra**-**system**’**.***progression*
The

*seventh***of***system***in that Seldonian**__dialectical__arithmetic**, which***progression***connotes by**__Encyclopedia Dialectica___{R}__q____{M}_{Q}_{N}__=___{R}__q____{M}_{U}__=___{R}__m__**-- the**_____*second*__uni__*-*of**category system****-- arises naturally as the first**__dialectical__arithmetic__non__*-“*, fully-ideographic, fully-algorithmic**syncopated**”**for***arithmetic**“*”.__dimensional____analysis__
I that

*seventh***, questions leading to the***system**‘***Full Zero***meta**-*’ concept also arise naturally, and yield,**number****, an**__at____long____last__**in which***arithmetic***appears to become***division by zero*__non__*-*problematic.
As a result of that ‘‘‘rectification’’’ and ‘‘‘regularization’’’
of

**, dynamical***division by zero**“*,**singularities**”**manifesting as**__present__ly*‘*, i.e., as**infinity****residuals**’**, in the predictions of [especially***infinite errors***]**__non__linear**, including of those which represent***dynamical differential equations**this*__h__*umanity**’***s**__present__ly*most advanced**scientific**-***consensus****of the “laws” of***expressions***, can be ‘semantified’ by correct solution-values, under intuitively satisfying new axioms, which can be stated, briefly, as:***Nature**“ ‘*.

**’ “times” a**__Empty__Zero**metrological unit**yields ‘__qualifier__**’ ”**__Full__Zero
-- and --

*“ ‘*

**Full Zero**’ is operationally dominant, in multiplication and division, with respect to all other ‘[meta-]number’ values in this

_{R}

__m__

_____

*system**, that is, multiplication and division operations if they involve ‘*.

**Full Zero**’, yield only ‘**Full Zero**’ ”
-- or --

**0**

__m__

^{o}**=**

**.**.

-- and --

**[ for all**

__m__

^{o}**in**

_{R}

__m__

_____**][ [**

**.**

**x**

__m__

^{o}**=**

__m__

^{o}**x**

**.**

**=**

**.**

**] & [**

**.**

*/*

__m__

^{o}**=**

__m__

^{o}

*/***.**

**=**

**.**

**] ]**.

*in the cases of the systems of arithmetic -- of the*

__Un__like__,__

**N****W**,

__,__

**Z****Q**,

**R**, and

__, ..., systems of arithmetic -- considered earlier, above, in this case, the present case, the case of the system of arithmetic,__

**C**

_{R}

__m__**,**

_____**.**

__, finally, an element of the set -- is, at last, a__

**IS**

**number**

__the__

**within***'''*--

**number**-**space**'''

_{R}**. Thus --**

__m__**0**

__m__

^{o}**=**

**.**.

-- is no longer a '''meta-mathematical''' assertion.

The classic published rendition of an earlier version of
this theory is available via --

-- on pages

**A**-**7**through**A**-**21**of the latter.
I have posted the

*sheets of the new***eight***‘***Full Zero**’**, below, for your convenience.**__speci__fication
May you much enjoy this deeper glimpse into the world-historical
fruition of these

*arithmetics of*__dialectic__*!*
Regards,

Miguel

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