Excerpts from the summary section of the essay "The Goedelian Dialectic of the Standard Arithmetics" are presented below, with adaptations to the typography available here.
The full essay is available via the following URLs --
http://www.dialectics.info/dialectics/Welcome.html
http://www.dialectics.info/dialectics/Vignettes.html
http://www.dialectics.info/dialectics/Vignettes_files/v.1,Part_II_of_II,Miguel_Detonacciones,F.E.D._Vignette_4,The_Goedelian_Dialectic_of_the_Standard_Arithmetics,posted_20SEP2012.pdf
"Summary & Prospect: Narrative Commentary on this Solution-Presentation as a Whole.
As the immediately-above-rendered ‘qualifier-series’, ‘qualifiers-sum’, or ‘cumulum’, whose most compact canonical forms are --
C# ~+~ h#, or )-|-(#6, or N^(2^6)
-- expresses, our solution-presentation is one which progresses in «aporia», that is, in system-of-arithmetic-versus-counterexample pairs.
The self-reflection, or immanent critique -- i.e., the self-critique -- of our «arché» system of arithmetic, N#, yields our first ‘counter-system’, #qNN = a#, which is also our first system-of-arithmetic-versus-counterexample pair/«aporia»: N# ~+~ a# [---) q1 + q2.
From there on, every new ‘counter-exemplary counter-system’ -- every new ‘contra-thesis’ category of arithmetic -- every new «aporia»-sum pairing, arises by the ‘‘‘self-reflection’’’ -- i.e., by the immanent critique -- of its immediate predecessor ‘counter-example’, ‘counter-system’, ‘contra-thesis’ category, starting with #qaa = m#.
And then, next, every such ‘contra-thesis’ category -- in the stage/step right after the one in which it first forms -- ‘‘‘conquers’’, ‘‘‘converts’’’, ‘‘‘takes possession of’’’, ‘‘‘appropriates’’’, ‘‘‘assimilates’’’, ‘‘‘uplifts’’’/elevates, ‘‘‘adjusts’’’, or ‘‘‘combines with’’’ -- every previously-precipitated category, in forming its representation of the next full system of arithmetic.
Finally, in each phase, stage, or step of this ‘self-dialogic’ / ‘self-argumenting’ / ‘self-augmenting’ progression, for every such ‘contra-thesis’ category, it is none other than the ‘self-conquering’ -- the ‘self-conversion’, the critical ‘self-possession’, the immanently critical ‘self-appropriation’, the «aufheben» ‘self-assimilation’, the «aufheben» ‘self-elevation’, the ‘self-corrective self-adjustment’, the ‘meta-monadic self-combination’ -- of that ‘contra-category’ which creates, or concretizes, the next ‘counter-exemplary counter-system’, the next ‘contra-thesis’ category -- and, thereby, the next «aporia».
This progression of «aporia» is potentially-infinite in Aristotle’s sense.
That is, it has no known stage/step of intrinsic self-termination in terms of possibility.
It thus exhibits a potential-infinity version of the Gödelian “Inexhaustibility of Mathematics”.
However, actually, historically, to-date, this progression does end -- e.g., it ends as a widely-diffused human-phenomic achievement. Presently, perhaps, it ends with G#, the Grassmannian-dialectical, n-dimensional ‘‘‘arithmetic of geometries’’’, in terms of that part of the potential infinity of arithmetical possibility that Terran humanity has actualized, so far.
If we reflect upon the course of this ‘meta-systematic dialectical’ presentation of the standard arithmetics as a whole, in terms of its representation by the progression of ‘‘‘number-spaces’’’, the progression of arithmetical analytical geometries by which we have ‘thematized’ it, we see the following.
The first few phases -- ending in phases/stages 2 and 3 -- of this movement, after the initial positing of N#, in stage 0, and its own immediate self-critique, in stage 1, yielding/constructing W#, in stage 2,and including the self-critique of the «aporia» of W# and m#, yielding/constructing Z#, in stage 3, were, in terms of number-spaces, about the “filling-in”/‘explicitization’ of ‘‘‘number-points’’’ to the left of theN ‘‘‘number-points’’’, to the left side ‘lessor-side’ of the ‘‘‘number-point’’’ labeled I.
The next two phases of this movement, stages 4 and 5, were about “filling-in”/‘explicitizing’ new ‘‘‘number-points’’’, in-between the Z ‘integ[e]ral’ ‘‘‘number-points’’’, via the self-critique of the Z# & f# «aporia» of stage 3, yielding/constructing the Q# & d# «aporia» of stage 4, followed by its own self-critique, yielding the R# & i# «aporia» of stage 5, in the process first ‘densifying’, and then ‘solid-ifying’, the emergent, single, “solid” “number-line” analytical geometry of '“Real”' number.
The final phase of this movement covered by our presentation herein, that of the transition from stage 5 to stage 6, driven by the self-critique of the R# & i# «aporia», opened a new, and to-be-enduring, «modus operandi» of the ‘ideo-ontological extent-tion’ of number-kinds and of their arithmetics, no longer adding just new “points”, but evoking/’explicitizing’ implied entire new “number-lines” -- whole new ‘‘‘dimensions’’’ of number(s) -- from out of their former invisible implicitude in ‘‘‘number-space’’’.
In the part of this story that we have recounted herein, this new “m.o.” is just starting, by the adjunction -- and by the “Cartesian product” interaction, with the “real” ‘number(s)-dimension’, with the ‘r-axis’, with R-- of the so-called “imaginary” ‘number(s)-dimension’, of the‘“i-axis”’, of I -- the ‘number(s)-axis’ with i as its unit[y], a whole new hyper-“dense”, or “solid”, number-line -- to form the ‘number(s)-plane’ of the standard “Complex Numbers”, C = C^1 ---[ C# [using '---[' to denote the phrase 'is contained in'].
Something of the further course of this new «modus operandi», in our interpretation of the Gödelian Dialectic‘Meta-Equation’ beyond stages s# = 0 through s# = 6, recounted herein, can be summarized as follows.
If we relegate/recede/demote the 4 ‘pre-R’ stages/steps of our ‘meta-systematic dialectical’ presentation -- i.e., those 4 «aporia» which first exhibit the ‘‘‘number-spaces’’’ of the N#, W#, Z#, & Q# arithmetics -- to a kind of collective ‘sub-Real’ class, then we have R# as a kind of ‘neo-«arché»’, with the following arithmetic-system succession, ‘thematized’ via adds of number-space axes:
w Dimensionality Range, Name, & Dimensionality Character of the wth ‘‘‘Number-Space’’’
0: 2^0 = 1 (---) R: "Reals"; 1single, fixed “continuous” “number-line” dimension/axis;
1: 2^1 = 2 (---) C: "'Complexes''';2 fixed, “continuous”, mutually-perpendicular “number-line” axes;
2: 2^2 = 4 (---) H: "[Hamilton] Quaternions"; 4 fixed, “continuous”, perpendicular “number-lines”/axes;
3: 2^3 = 8 (---) O: "Octonions"; 8 fixed, “continuous”, perpendicular “number-line” dimensions/axes;
4: 2^4 = 16 (---) S: "Sedenions"; 16 fixed, “continuous”, perpendicular “number-line” dimensions/axes;
. . .
{n}: N (---) K: "'Clifford arithmetics"'; for any fixed dimensionality n, “continuous”, perpendicular “number-lines”;
n^: N (---) G: Grassmann numbers. Multiplication of 2 distinct Grassmann numbers of dimensionality n produces a Grassmann number of dimensionality n + 1; variable,varying, escalating dimensionality. Squaring a Grassmann number produces “nil” [ Grassmann numbers are "nilpotent" ];
w^: W (---) WQ: Seldonian ‘Ontological-Qualifier meta-numbers’. Self-multiplication of any single WQ ‘meta-number’ unit-qualifier, qn, is represented analytical-geometrically as a unit-length, one-dimensional, ‘impartable’ directed line-segment, or ‘meta-vector’, and yields --
qn + q2n | n is in W - {0}
-- as its product, i.e., the positive-square-root-of-2-length diagonal of the [2-dimensional] square parallelogram formed by the mutually-perpendicular, unit-length ‘meta-vectors’ qn and q2n, joined at their q0 origin.
If q1, the «arché» of the WQ ‘meta-numbers’, is self-multiplied, and, successively, recursively, each product of such self-multiplication is, in its own turn, self-multiplied, as in the ‘Dyadic Seldon Function’, then this yields ‘[hypo-/hyper-]unit-cubic spaces’ of successive dimensionalities --
1, 2, 4, 8, 16, 32, 64, ...
-- variable, self-varying, self-escalating dimensionalities; capable of modeling the foregoing succession of systems of arithmetic; . . ."
Regards,
Miguel
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