Wednesday, August 01, 2018

Dialectic of the Standard Arithmetics’ -- ‘Dialectogram’: ‘The Meta-Systematic Dialectic of the Domain of the Standard Arithmetics’, First Triad.

Dialectic of the Standard Arithmetics’ --

Dialectogram:The Meta-Systematic Dialectic of the Domain of the Standard Arithmetics, First Triad.

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