*‘*

**’ --**__Dialectic__of the Standard Arithmetic__s__*‘*:

**’**__Dialectogram__*‘*

**The**-__Meta__**Systematic**__Dialectic__of the

__D__*omain of the Standard Arithmetic*__s__*’*,

*.*

**First**__Tri__ad
Dear

**,***Readers*
This paradigmatic case of the ‘Goedelian Dialectic’, treated,
in this blog entry, in its first triad of axiom

__s__-system__s__[-fragment__s__] for standard arithmetic__s__, can serve as yet another example of the failure of the contemporary mathematical sub-culture to recognize the ‘dialecticality’ of its primary thought-product__s__, already latent and unconsciously present in its history, and in its present-day proof__s__as well.
The Triadic, Platonian-format

*‘*below describes, in systematic order -- in consecutive, simplest to more complex order -- the first three axiom**dialectogram**’__s__-system__s__, or Lakatosian ‘‘‘counter-example’’’ axiom__s__-system__s__-fragment__s__, as categor__ies__, needed for a systematic, dialectical categorial-progression method of presentation of the*contemporary***of the**__D__*omain**“*arithmetic**Standard**”__s__.
This

**is thereby presented as a ‘synchronic meta-system’, made up out of a heterogeneous multiplicity of all-present predecessor-successor axiom**__D__*omain*__s__-system__s__, i.e., as a presentational dialectical progression of axiom__s__-system__s__for the*“*arithmetic**Standard**”__s__,**,**__D__*omain***D =**.__#__
The first triad of arithmetical axiom

__s__-system__s__, and of ‘counter-exemplary’ axiom__s__-system-fragment(__s__), connotes, in sum, the axiom__s__-system of the “**hole” number**__W____s__,**W**, by means of the following, equational-definitional, ‘non-amalgamative sum’ of progressively-evoked arithmetical system__s__/categor__ies__--

__W__**=**

__N__+__a__+

__q__

_{a}**.**

_{N}
In the above, “purely”-qualitative, “purely”-categorial
equation, the term

**denotes the axiom**__W____s__-system of the “**hole”-number**__W____s__*“*system of arithmetic, such that**Standard**”**W**

__=__**{**

**0**,

**1**,

**2**,

**3**,

**... }**.

The term

**denotes the axiom**__N____s__-system of the “**atural”-number**__N____s__*“*system of arithmetic, such that**Standard**”**N**__=__**{****I**,**II**,**III**,**... }**.
The term

**denotes the axiom**__a____s__-system-fragment of the ‘**ught’-number**__a____s__, i.e., of the ‘‘‘zero__s__’’’, or of the set of all ‘__self__-subtraction__s__’ of all “**atural” number**__N____s__.
The latter set is the “solution-set” for solving the simple,

**-**__N__**,***paradoxical***-**__N__**“diophantine” algebraic equation**__un__solvable**x****+ n = n**, for every**n**in**N**. The latter equation transposes to --**x**

**= n - n**.

This equation shows that a syntactically-correct algebraic
equation, over the “

**atural” number**__N____s__, arises**,***immanently***the algebra of the “***inside***atural” number**__N____s__, that has no solution within the “**atural” number**__N____s__“space”. This fact already intimates Goedel’s proof about the ‘‘‘self-incompleteness’’’ of the “**atural” arithmetic’s axiom**__N____s__-system.
This equation thus points to a Lakatosian

*‘‘‘*to any notion that the ‘**-**__counter__**example**’’’**atural” number**__N____s__**the***exhaust***of***totality**possible**“*number-**Standard**”**. Furthermore, it implies the***kind*__s__**of a***possibility*__speci__fic**of***kind**“*number which is a**Standard**”__non__*-*,**cardinal**__non__*-***counting****of number -- a***kind**‘***-**__no__**count**’__s__**of***kind**“*number.**Standard**”
The term

__q___{a}**denotes the axiom**_{N}__s__-system-fragment for the integration, synthesis, or complex unity of the*“*

**Standard****atural” number**

__N____s__with the ‘

**ught’-number**

__a____s__, with the latter as “placeholder

__s__”, in a “place-value” system of now “

**hole” number ‘numeralic**

__W____s__’.

FYI: Much of the work
of Karl Seldon, and of his collaborators, including work by “yours truly”, is
available for

**free-of-charge download via --***your*
Regards,

Miguel Detonacciones,

Member,

**Foundation****[**__Encyclopedia Dialectica__**F**.**.**__E__**.],**__D__
Officer,

**F**.**.**__E__**. Office of Public Liaison**__D__
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