‘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-Ã -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^St
-- 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 Dialectic’
itself.”
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.
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