F.E.D. Dialectics: The ‘Categoriality’ of Typical Axioms-Systems.

Wednesday, August 03, 2022

The ‘Categoriality’ of Typical Axioms-Systems.

The ‘Categoriality’ of Typical AxiomS-Systems –

Is via the ‘‘‘Family’’’ of Multiple, Variant Axioms-Systems ‘‘‘Inside’’’ Each Typical, Single ‘Ideo-Ontological’ Category, or ‘Ideo-«Arithmos»’, of Axioms-System ‘Ideo-«Monads»’ or Ideo-Units, each of those units being a distinct axiomatization, family-related to each other axioms-system unit in that family/category.

Dear Reader,

In our ‘evolute’, [meta-]systematic-dialectical, and historical-dialectical categorial progression models, which are also systems-progression models, of [meta-systematic and historical] axioms-systems progressions, e.g., of mathematical, arithmetical axioms-systems’ Gödelian-incompleteness-driven [i.e., locally-unsolvable diophantine equation-driven] progressions, we represent each term, each category, each «arithmos» of a given such progression of categories as an assemblage of more than one axioms-system -- of a multiplicity of ‘‘‘family-related’’’ axioms-system variations -- by way of single [ideo-]ontological category, via a single ‘category-symbol’ interpreted from the _NQ arithmetic for dialectic.

Any such “axioms-system” category thus “refers to” or ‘“contains”’ multiple axioms-system units – two or more – thus qualifying it also as an «arithmos» of such units.

For example, an axioms-system defining the arithmetic of a given, “standard” number-system will typically have multiple variants; will typically be but one of a variety of such axioms-system units/«monads».

Perhaps the best known case within this example is that of the “standard” second-order-logic axiomatization(s) of the “number-space” of the “Natural” Numbers, N.

We denote the category of those second-order axioms-systems by the singly-underscored symbol for that number-space, namely N.

Historically, there is an axiomatization for --

N = {1, 2, 3,…},

-- and, later, another axiomatization for --

N = {0, 1, 2, 3,…}.

In some cases, the axiomatic variability ‘“contained”’ in an ‘ideo-ontological’ category standing for a given [family of] arithmetic(s), call that category X, for a given number-space call, it X, might arise by way of including in that category both first-order-logic and second-order-logic axiomatizations for arithmetics over the number-space or “numbers-set” X.

Because we typically denote first-order axiomatizations for a “numbers-set” X, by X_, and second-order axiomatizations of X by X or X_, we sometimes denote a category that includes both first-order and second-order axiomatizations of the same numbers-set by _X_.

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