Dear Readers,

On this **19**th anniversary
of Seldon’s April **7**th,
**1996**
breakthrough -- his sudden discovery of the _{N}__Q___
*‘***First **__Dialectical__
Arithmetic for contra-**Boolean Algebra**’,
after years of [re-]searching for, and of *slow* progress toward finding, a *“***mathematics of **__dialectics__” -- the **F**.__E__.__D__. General Council has cleared,
for public dissemination, its **eight **__speci__fications sheets defining the *‘***meta**-**number**’ *value* that we call *‘***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** *'*__ideo__-**ontology**', this *new *__dialectical__*-***ideographical
symbol**, within the Seldonian **seventh**, or* '***Mu**', __dialectical__ arithmetic.

I have, in my dissertation-contribution to the **Foundation**, for my induction-into-membership
in the **Foundation**,
entitled *“***The GÃ¶delian **__Dialectic__ of the Standard
Arithmetics” [which is accessible via the following links, on
the Vignettes Page, as Vignette **#4**, Parts **0**, **I**, and **II**, at --

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

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

Thus, in a sense, **within** the limitations of the __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* *from* the __N__ *system* *to* the __W__ *system*, the *axioms**-***system**
of the so-called “__W__hole” numbers, **W**** = {0, 1, 2, 3, ...}**,
wherein that *equation*
__is__ readily solvable:
**x**** =
0**.

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

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

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

-- wherein that *equation* __is__ readily
solvable: **x**** = -1**.

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

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

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

-- wherein that *equation* __is__ readily
solvable: **x**** = +1/2**.

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

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

**R**** = {.....-****pi****....-e....-****\/****2....±0.000.......+****\/****2....+e....+****pi****.....****}**

-- wherein that *equation* __is__ readily solvable:
**x**** =
±****\/****2**.

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

** **

**-x = +1/+x**.

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

**C**** = {R + Ri}**

-- wherein that *equation* __is__ readily solvable:
**x**** = ±i**. And so on **. . .**.

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

This internal, immanent inadequacy and ‘‘‘incompleteness’’’
of __ALL__ *systems* in the *progression* of the *systems* of the *standard arithmetics* is evidently of a far deeper sort
than the inadequacies and ‘‘‘incompleteness’’’ that drive that *progression*, and that
were *progressively solved*
in that *progression*,
as outlined above.

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

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

The *seventh*
*system* of __dialectical__ arithmetic in that
Seldonian *progression*,
which __Encyclopedia Dialectica__
connotes by _{R}__q____{M}_{Q}_{N} __=__ _{R}__q____{M}_{U} __=__ _{R}__m_______ -- the *second *__uni__*-***category system** of __dialectical__ arithmetic -- arises
naturally as the first __non__*-“***syncopated**”,
fully-ideographic, fully-algorithmic *arithmetic *for *“*__dimensional__ __analysis__”.

I that *seventh*
*system*, questions
leading to the *‘***Full Zero** *meta**-***number**’
concept also arise naturally, and yield, __at__ __long__ __last__,
an *arithmetic *in
which *division by zero*
appears to become __non__*-*problematic.

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

*“ ‘*__Empty__ Zero’ “times” a **metrological unit **__qualifier__ yields ‘__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}*/***.**** = ****.****] ]**.

__Un__like in the cases of the systems of arithmetic -- of the **N**, **W**, **Z**, **Q**, **R**, and **C**, ..., systems of arithmetic -- considered earlier, above, in this case, the present case, the case of the system of arithmetic, _{R}__m_______, **.** **IS**, finally, an element of the set -- is, at last, a **number**** ****within **the *'''***number**-**space**''' -- _{R}__m__. Thus --

**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 **eight** sheets of the new *‘***Full
Zero**’ __speci__fication, below,
for your convenience.

May you much enjoy this deeper glimpse into the world-historical
fruition of these *arithmetics
of* __dialectic__*! *

Regards,

Miguel