Wednesday, February 23, 2022

Set Theory and Dialectics. -- Part 08: Seldon’s Secrets’ Series.

 



The Dialectic of Set Theory

and

The Set Theory of Dialectic.

 

-- Part 08: Seldon’s Secrets Series.

 

 

 

 

 

Dear Reader,

 

 

 

It is my pleasure, and my honor, as an elected member of the Foundation Encyclopedia Dialectica [F.E.D.] General Council, and as a voting member of F.E.D., to share, with you, from time to time, as they are approved for public release, by the F.E.D. General Council, key excerpts from the internal writings, and from the internal sayings, of our co-founder, Karl Seldon.

 

This eighth release in this new such series is posted below [Some E.D. standard edits have been applied, in the version presented below, by the editors of the F.E.D. Special Council for the Encyclopedia, to the direct transcript of our co-founder’s discourse].

 

In this 8th installment, Seldon responds to an interlocutor regarding the ‘set-theoretical pathway’ that put Seldon onto the trail that eventually led him to the NQ ‘arithmetic/algebra for modeling dialectics’, as both a Gödelian non-standard model of the Natural Numbers”, and as a ‘contra-Boolean algebra’, transcending Boole’s formal-logical “Fundamental Law of Thought” or “Law of [exo-]Duality” -- 

x^2 = x

-- via Seldon’s ‘Fundamental Equation of Dialectics’, or ‘ “Law” of Intra-Duality ’ -- 

x^2 = x + delta(x).

 

 

Interlocutor –

Now, [after]…, I think I can appreciate much better some of the conundrums and solutions you told me about 20ish years ago: sets of all sets including themselves, aufheben, q-arithmetic operations, etc.

 

Seldon –

Yes.  The Set of All Sets’ ‘ideo-phenomena, in particular, became paradigmatic in the process that led, for me, to the NQ arithmetic, as both one of the non-standard models of the “Natural Numbers”, as predicted by the joint implications of Gödel’s completeness and incompleteness theorems, and as a ‘contra-Boolean arithmetic’ with a ‘contra-Boolean algebra’, based upon a strong negation of the Boolean Fundamental Law of Thought”.”  

 

“The one-sentence definition of ‘‘‘dialectics’’’ that emerged, for me, from this ‘set of all sets’ paradigm, among others, is the following.”

 

Dialectics is the theory of «aufheben» processes, and of the «aufheben» relations to which «aufheben» processes give rise.”

 

“The «aufheben» concept unifies the, dialectical, concepts of ‘‘‘negation’’’, “[self-]contradiction” [internal duality; ‘intra-duality’; ‘self-duality’], ‘quantitative change becoming qualitative, ontological change’, and ‘‘‘negation of [the] negation’’’.”

 

“The prevailing ideology permeating modern mathematics wonts Cantorian, infinitary “Set Theory” to be the inner foundation, the very core, of all of mathematics.”

 

“A dialectical, immanent critique – really a self-critique – of this ideology, that permeates and vitiates modern mathematics, finds the «aufheben» process at the very heart of [finitary] set theory, as the very set-theoretical, “extensional” definition of the very “set” concept itself.”

 

“Soon, we plan finally to finish and to publish volume 1 of A Dialectical “Theory of Everything”, whose individual-volume [sub-]title is Four Converging Pathways to the Encyclopedia Dialectica Ideographies for Dialectics.”

 

We had rough-drafted the first four volumes, volume 0 through volume 3, even before we published volume 0.  And we started on readying volume 1 for publication years ago, right after we published volume 0.  We have continued to work on volume 1 ever since.”

 

“But we gave the publishing of volume 1 low priority, because we felt that it was more important to establish, first, the practical value of the first-explicitly-dialectical, NQ arithmetic/algebra, through demonstrating many of its applications, than to document the “biography” of its discovery.”

 

Our report of the results from our study of the “Set of All Sets” ‘ideo-phenomenology’ will form the first section, and the first of the four “pathways”, to be presented in volume 1.”

 

“If the standard “Natural Numbers” and their arithmetic can be derived from set theory via the “cardinality” of sets, then the non-standard “meta-Natural meta-Numbers”, and their, NQ, ‘arithmetic for dialectics’, can be understood as derived from set theory via the immanent ‘ordinality’ of the immanently ‘ideo-self-dynamical’ nature of the [finitary] ‘‘‘set of all sets’’’.”


“Reasons why the [finitary] ‘‘‘set of all sets’’’ concept proved so paradigmatic for me for the discovery of the NQ arithmetical/algebraic models of dialectic, and, thereby, for the discovery also of the other, richer, including ‘qualo-quantitative’ systems of dialectical arithmetic/algebra that follow from it, dialectically, when it is applied to itself, include –

 

1.  Inherent 'Ideo-Dynamism'.  Despite the Parmenidean proclivities of many set theorists, a finitary set of all sets for a given domain, is an inherently, ineluctably ‘qualo-dynamical’, ‘ideo-onto-dynamical’ conceptual object.”


“To the inquisition of the “eternal, immutable”, statical orthodoxy of set theory, I must echo Galileo: Eppur si muove.”

 

“Actual infinity” is not actual.  It is a figment of the Cantorian-theistic, mystified, ideological-mathematical imagination.”

 

“The “power set”, or “set of all subsets”, of the finitary “universal set”, the “universal set” that expresses a given domain, launches the set of all sets for that domain, or “universe of discourse”, as defined by that universal set.”

 

“But this first attempt at the set of all sets is inadequate to that set’s name/definition/meaning.  This is because it still excludes the power-set of itself, the set of all of its own subsets, the latter set being qualitatively different, higher in logical type, higher in ‘set-theoretical scale’, and larger in its “cardinality”, its number of ‘ideo-ontologically’ new, different elements, than that initial-attempt set.”

 

“So that first-attempt “set of all sets” must, to satisfy its name/definition/meaning, self-expand, by [«aufheben»-]internalizing every element of its own power-set, the set of all of its own subsets [which process, by the way, constitutes also a model of the «aufheben» process, which is the heart of all dialectic].”

 

“This process, of power-set internalization, is self-demanded, at ever-higher scales, by every subsequent attempt at the set of all sets, because every such attempt always still excludes the power-set specific to that attempt-set, a power-set that is qualitatively different and bigger for each attempt-set vis-à-vis every earlier such attempt-set.”

 

“If you define a set-product rule, using standard power-set notation, in which ‘2^S’ denotes the power-set of a set denoted by S’, then, defining that set-product rule by, with ‘t’ denoting an [ordinal] “whole number –

St x St  =  St^2  =  St+1  = 

St U 2^St

-- wherein ‘U’ denotes the standard set-theoretical operation of “Union”, then the finitary set of all sets can be expressed, not as some kind of “impossible” statical idea-object, but, instead, as the ‘ideo-auto-dynamical self-movement described by the set equation –

 

St = S0^(2^t)

 

-- for S0 = 2^u, wherein ‘u’ denotes the universal set for the given “universe of discourse”.”


“Thus we have, for me, one of the earliest appearances of the form of what many of my colleagues like to call “The Seldon Function”, but what I prefer to call The Dyadic Self-Reflexive Function’, or ‘The Dyadic Dialectical Function’.



“Note that the set-product rule that engenders this Dyadic Self-Reflexive Function form also exhibits the contra-Boolean’ logical-algebraic form --

generically, x^2  =  x + delta(x) -- in the specific form of --

 

St^2  =  St U delta(St)  =  St U 2^S 


-- in contrast to the Boolean logical-algebraic form of Boole’s “Fundamental Law of Thought”, or “Law of [exo-]Duality”, x^2 = x^1 = x

 

“The ‘self-dynamism’ of the finitary set of all sets, for each finitary “universe of discourse”, describes the ‘self-«aufheben»’, ‘ideo-ontologically dynamical’, qualitative, immanent self-evolution of that “universe of discourse”.”

 

“In particular, it describes predico-dynamasis – the cognitive ‘ordinality’ of the defining of the ever-subtler predicates for that universe of discourse.”

 

“This is because the subsets generated by each round of this power-set process are the set-theoretical, “extensional” representations of the ever-subtler predicates describing the ever-subtler qualities shared in common among the subsets of each previous round of subsets generation, and, ultimately, by the non-set idea-objects which are the elements of the universal set for each given universe of discourse.”

 


2.  Fundamentality of the Set of All Sets to Set Theory.  The finitary set of all sets is the fundamental, defining conceptual object of finitary set theory, as an “extensional” theory, or “theory of [the] extensions” [of “intensions”].”

 

“That is, set theory defines the meanings or “intensions” of conceptual objects, e.g., of the “predicates” representing qualities, by their “extensions”, that is, by sets – by the set of all objects which share a given quality/predicate in common.”

 

“ “Red” means “the set of all red things”, the set of all objects which the human eye perceives as exhibiting the color named “red”.  That set “points to” the quality which all of its elements exhibit, even if those elements are otherwise qualitatively disparate.”

 

“Thus, the set-theoretical, “extensional” definition of the meaning/quality of the very concept of “set” itself is the set of all sets.”

 

“And yet the set of all sets is a set and a concept which is banned from standard set theory; which is virtually outlawed as “impossible” and made “unmentionable” in standard set theory discourse.”

 

“This ‘self-outlawing’ of the very heart-concept of set theory within standard set theory itself is a glaring, egregious symptom of an ideological, mystical blockage of scientific rationality, and is grounds for a dialectical, immanent critique – really a ‘self-critique’ – of standard set theory.”

 

“And, because the set of all sets self-movement, or ‘ideo-auto-kinesis’ [cf. Plato], is also an «aufheben» process par excellence, this immanent-critique/self-critique of set theory yields, as its positive fruition, a mathematical theory of dialectics itself.”

 

“In particular, the set of all sets, as the root idea-object of set theory, is seen to constitute ‘The Dialectic of Set Theory’, and gives rise to ‘A Set-Theory of Dialectic’, the sought-after dialectical-algebraic modeling of which helped to lead me to the NQ arithmetical/algebraic model of dialectics.”                                                                                                                                                    

“Thus, the immanent critique of “Set Theory” reveals, via the Set-Theoretic definition of the “Set” concept itself – i.e., via the [finitary] “Set of All Sets” – The Dialectic of the Set Concept, and A Set-Theory of the Dialecticitself. 

 

 

 

 

 

 

 

 

 

 

 





 

 

 

 

 

 

 

 

For more information regarding these Seldonian insights, please see --

 

www.dialectics.info

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights -- specimens of dialectical art -- see:


https://www.etsy.com/shop/DialecticsMATH

 

 

 

 

¡ENJOY!

 

 

 

 

 

 

 

Regards,

 

 

Miguel Detonacciones,

 

Voting Member, Foundation Encyclopedia Dialectica [F.E.D.];

Elected Member, F.E.D. General Council;

Participant, F.E.D. Special Council for Public Liaison;

Officer, F.E.D. Office of Public Liaison.

 

 

 

 

YOU are invited to post your comments on this blog-entry below!

 

 

 

 

 

 

 

 

 

 

 




No comments:

Post a Comment