‘Dialectic of the Standard Arithmetics’
--
‘Dialectogram’:
‘The Meta-Systematic Dialectic of the Domain of the Standard Arithmetics’,
First Triad.
Dear
Reader,
This paradigmatic case of the ‘Goedelian Dialectic’, treated,
in this blog entry, in its first triad of axioms-systems[-fragments]
for standard arithmetics, can serve as yet another example of the failure
of the contemporary mathematical sub-culture to recognize the ‘dialecticality’ of
its primary thought-products, already latent and unconsciously present
in its history, and in its present-day proofs as well.
The Triadic, Platonian-format ‘dialectogram’ below describes, in
systematic order -- in consecutive, simplest to more complex order -- the first
three axioms-systems, or Lakatosian ‘‘‘counter-example’’’ axioms-systems-fragments,
as categories, needed for a systematic, dialectical categorial-progression
method of presentation of the contemporary Domain of the “Standard” arithmetics.
This Domain is thereby presented
as a ‘synchronic meta-system’, made up out of a heterogeneous multiplicity of
all-present predecessor-successor axioms-systems, i.e., as a
presentational dialectical progression of axioms-systems for the “Standard” arithmetics,
Domain, D = #.
The first triad of arithmetical axioms-systems,
and of ‘counter-exemplary’ axioms-system-fragment(s), connotes,
in sum, the axioms-system of the “Whole” numbers, W,
by means of the following, equational-definitional, ‘non-amalgamative sum’ of progressively-evoked
arithmetical systems/categories --
W = N + a + qaN.
In the above, “purely”-qualitative, “purely”-categorial
equation, the term W denotes the axioms-system of the “Whole”-numbers
“Standard” system
of arithmetic, such that
W = { 0, 1, 2, 3,
... }.
The term N denotes the axioms-system
of the “Natural”-numbers “Standard” system of arithmetic, such that N = { I, II, III,
... }.
The term a denotes the axioms-system-fragment of
the ‘aught’-numbers, i.e., of the ‘‘‘zeros’’’, or
of the set of all ‘self-subtractions’ of all “Natural”
numbers.
The latter set is the “solution-set” for solving the simple,
N-paradoxical,
N-unsolvable “diophantine”
algebraic equation 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 “Natural” numbers, arises immanently, inside the algebra of the
“Natural”
numbers, that has no solution within the “Natural” numbers
“space”. This fact already intimates
Goedel’s proof about the ‘‘‘self-incompleteness’’’ of the “Natural”
arithmetic’s axioms-system.
This equation thus points to a Lakatosian ‘‘‘counter-example’’’
to any notion that the ‘Natural” numbers exhaust the totality of possible “Standard” number-kinds. Furthermore,
it implies the possibility
of a specific kind of “Standard” number which is a non-cardinal,
non-counting
kind of
number -- a ‘no-counts’
kind of “Standard” number.
The term qaN denotes the axioms-system-fragment for
the integration, synthesis, or complex unity of the
“Standard Natural”
numbers with the ‘aught’-numbers, with the
latter as “placeholders”, in a “place-value” system of now “Whole”
number ‘numeralics’.
FYI: Much of the work
of Karl Seldon, and of his collaborators, including work by “yours truly”, is
available for your
free-of-charge download via --
Regards,
Miguel Detonacciones,
Member, Foundation Encyclopedia Dialectica
[F.E.D.],
Officer, F.E.D.
Office of Public Liaison
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