Dear Hegel-Marx Group Members,

INTRODUCTION. The quote below implicitly raises the issue as to whether the "Set
of All Sets" and the "Set Of All Objects" exhibit the same pattern of
"meta-<<monad>>-ization", or "meta-unit-ization" that we have hypothesized, in
our posts on "The Prelude to the Psycho-Historical Equations" ["A New Model of
the Universe, Using the Purely-Qualitative Dialectical Ideography"], to be
universal principle of ontological innovation in Nature, and as the concrete
meaning of the <<aufheben>> process, which we have hypothesized to be the
universal essence of dialectical process.

The purpose of this reply is to briefly address this issue.

For more depth on this issue, including as to how it applies for human-social
evolution, and meta-evolution, use the following link, to F.E.D. essays and
letters on the subject --

http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/DiaRith/Intro/D\
ialectical-Ideography_An-Introductory-Letter.htm#Example_5

http://www.dialectics.org/Briefs_files/F.E.D.-Brief2-part3-07DEC2008_OCR.pdf

-- and --

http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/MetaMonadology/\
\
\
MetaMonadology-entry.htm




PRELIMINARIES.


1. The Meaning of the <<Aufheben>> [Self-]Operation, as the Essence of
Dialectical Process.

This everyday old German word -- <<aufheben>> -- asserts a three-fold, and --
superficially -- incompatible bundle of meaning, e.g., to pick-up a stone from
the ground, and to raise it, e.g., to eye-level.

Thus, it means to change the position of the stone, i.e., (a.) to NEGATE its
position, and, concurrently, also, (b.) to PRESERVE the stone, in your grasp,
and, concurrently, also, to (c.) ELEVATE the stone as well.

<<Aufheben>>, in a more technical, 'scientifico-philosophical' usage, also
describes what the cosmos-content does to itself, in creating new, unprecedented
content; in creating, not just new beingS, but NEW KINDS of being; new
"ontology".

This usage is concretized, by F.E.D., in terms of the concept of
"[Self-]Meta-Monadization".



2. The Meaning of the Term "[Self-]Meta-<<Monad>>-ization", or of
"[Self-]Meta-Unit-ization".

In the posts here on the "The Prelude to the Psycho-Historical Equations", the
"self-meta-evolution" of the cosmos is described, in terms of <<aufheben>>
"self-metamonadization", as follows --

This self-<<aufheben>> of the content of the cosmos is accomplished by
"self-meta-<<monad>>-ization", that is, by "self-meta-unit-ization".

<<Monad>> is Greek for "unit".

The qualitative units, or <<monads>>, that our scientists call "quarks", et al.,
once their population, and its spatial concentration, grows to a sufficient
level, birth new populations, of a new kind of qualitative unit, a meta-unit –-
the "sub-atomic particle" unit –- i.e., to populations of "meta-quarks", e.g.,
protons, each one made up of a heterogeneous multiplicity of "quarks".

These new populations of sub-atomic "particle" units, once grown to sufficient
quantity & density, birth populations of "atom"-units, i.e., populations of
meta-sub-atomic "particle" units, each one made up out of a heterogeneous
multiplicity of sub-atomic "particle" sub-units.

These new populations of atom units, once they grow to sufficient quantity &
density, birth populations of molecule units, i.e., populations of meta-atoms,
meta-atomic units, each one made up out of a heterogeneous multiplicity of
atom-units.

These new populations of molecule units, once they grow to sufficient quantity &
density, birth populations of prokaryotic "living" cell units, i.e., to
populations of meta-molecules, meta-molecular units, each one made up out of a
heterogeneous multiplicity of molecule units. . . .

If you want to understand this progression of novelty -- of NEW KINDS of being;
of new "ontology" -- as achieved by means of "meta-unit-ization", by an example
closer to home, consider that part of human Nature which is written language.

Start from phonetic letters.

Each word unit is a "meta-letter" unit, each word typically made up out of a
heterogeneous multiplicity of letter–units.

Each sentence unit is a "meta-word" unit, each sentence typically made up out of
a heterogeneous multiplicity of word-units.

Each paragraph unit is a "meta-sentence" unit, each paragraph typically made up
out of a heterogeneous multiplicity of sentence-units.

Each chapter unit is a "meta-paragraph" unit, each chapter typically made up out
of a heterogeneous multiplicity of paragraph-units.

Each book unit is a "meta-chapter" unit, each book typically made up out of a
heterogeneous multiplicity of chapter-units.

Each library unit is . . . –- well, you get the idea!



ISSUE RESOLUTION. Our answer is that -- YES! -- the dialectics of the "Set of
All Sets" and of the "Set of All Objects" also exhibit this structure of
"Meta-Monadology" -- synchronic "meta-fractal" structure that arises from the
diachronic "meta-fractal" process of "self-meta-monadization".

Thus, if our universe of discourse involves only two "logical individuals", only
two distinct, qualitative objects, such as an apple and an orange, denoted by a,
and b, then, if we take these two "idea-objects" to be two distinct, qualitative
units, because, certainly --

a ~< & ~= & ~> b

-- then the very formation of the "Universe of Discourse Set", here --

U = { a, b } = O-sub-0

-- is a "singular", or "non-multiplicitous", "meta-monadization": the two units,
a and b, are <<aufheben>> cancelled, conserved, and elevated, all three
concurrently, into a new, higher, single object / individual / unit, namely, the
set U.

In the next step, the sub-sets of the set U = { a, b } --

namely the sets {_}, {a}, {b}, and the "improper" sub-set of U,

{a, b} itself

-- are "added to" ["unioned with", denoted by u] U, to form --

U u 2^U = { a, b, {a}, {b}, {a,b}, {_} } = O-sub-1.

Once again, the "improper subset" of U, {a,b}, is -- when it occurs INSIDE the
set a O-sub-1 -- a "meta-U unit, made up out of a heterogeneous multiplicity of
the units of U, i.e., of the [sub-]units a and b.

The new "singleton" units, {a} and {b}, are also meta-U units, but each one is
made up out of a "singular NON-multiplicity" of the predecessor units of U.

Even more so, in the next step, most of the incremental contents, the sub-sets
of O-sub-1, of the next attempt at the "Set of All Objects", namely O-sub-2,
are, precisely, "meta-units of the predecessor, O-sub-1, units, each one made up
out of a heterogeneous multiplicity of the predecessor units", e.g., to cite
just 4 of the 64 new subsets of --

O-sub-1 u 2^[O-sub-1] = [U u 2^U] u 2^[U u 2^U] = O-sub-2 --

{ a, b, {a} };

{ a, b, {b} };

{ a, b, {a,b}, {_} };

{ a, b, {a}, {b}, {_} },

etc. --

and this pattern, of "metamonadization", holds even more true with every
subsequent self-iteration of each subsequent attempt at "The Set of All
Objects", in the self-induced, own-essence-induced, own-definition-induced
process that IS the "Set of All Objects", as modeled by the equation --

O-sub-T = [ O-sub-0 ]^(2^T).


Regards,

Miguel


--- In hegel-marx@yahoogroups.com, "detonacciones" <ProfitSci@...> wrote:
>
> Dear Hegel-Marx Group Members,
>
> Note that the self-movement of the "Set of All Objects" is, precisely, a
DIALECTIC, an <<AUFHEBEN>> <<auto-kinesis>>; a set self-progression; a
self-driven categorial progression of sets as categories:
>
> If we use the shorthand symbols 'A -C B' to stand for 'set A is wholly
contained within set B', then we have the <<aufheben>> condition that --
>
> Osub0 -C Osub1 -C Osub2 -C Osub3 -C Osub4 -C Osub5 -C Osub6 -C...
>
> -- and, in general, for every T in the set of Natural Numbers, N --
>
> OsubT -C Osub(T+1).
>
>
> Notice that the self-propelled movement --
>
> OsubT ---> Osub(T+1) = [OsubT]^2
>
> -- wherein --
>
> Osub(T+1) = [OsubT] union [Power-Set of OsubT]
>
> -- is an <<aufheben>> self-negation of OsubT.
>
>
>
> When OsubT "reflects/reflexes/refluxes" upon itself --
>
> OsubT x OsubT = OsubT "of" OsubT = [OsubT]^2
>
> -- the resulting process --
>
> OsubT union 2^OsubT
>
> -- exhibits all of the moments of the <<aufheben>> operation:
>
> (1) "conserving" OsubT within Osub(T+1),
> (2) "elevating" OsubT to a one unit higher Russellian "logical type"
level, qualitatively changing OsubT into a qualitatively different/higher
Osub(T+1), and
> (3) "determinately [self-]negating" OsubT, converting it[self] into
Osub(T+1), because Osub(T+1) is a DETERMINATE case of NOT-OsubT.
>
>
>
> Thus, [OsubT]^2 = OsubT x OsubT = OsubT <<AUFHEBEN>> OsubT.
>
>
>
> The drive behind --
>
> OsubT ---> Osub(T+1)
>
> -- is INTERNAL, IMMANENT to OsubT itself. It is not an
externally-caused/imposed transition.
>
> The symbol '--->' within the symbol-complex --
>
> OsubT ---> Osub(T+1)
>
> Means NOT so much that OsubT "IMPLIES" Osub(T+1), but that OsubT "BECOMES"
Osub(T+1); "TRANSFORMS ITSELF INTO" Osub(T+1), "CAUSES ITSELF TO CHANGE INTO"
Osub(T+1), for its own, INTERNAL, IMMANENT reasons.
>
>
>
> The self-movement expressible as --
>
> Osub0 ---> Osub1 ---> Osub2 ---> Osub3 ---> Osub4 ---> Osub5 ---> ...
>
> is a "self-necessary", "neccessitarian" one. Since OsubT is a mental,
idea-object, rather than a physical, <<physis>>-object, the necessity of its
self-movement is a mental, ideative, conceptual, semantic necessity, rather than
a physical necessity.
>
>
> The set Osub0, or U, may start out as representing a collection of physical
objects, of <<physis>> objects. However, from Osub1 on, the expanded universe
set includes set-theroetical, extensional representations of Predicates of
various orders, of Qualities, of "Intensions" such as are perceived/consturcted
by human subjects, by human cultures, by human "Phenomes".
>
> Thus, this model captures something which is of the essence of Marx's
world-historical breakthrough in human cognition, in human science, in human
knowing, evident already in his "Theses On Feuerbach"; of Marx's
negation/transcendance/synthesis/complex unification of both one-sided
mechanical materialism and 'other-sided' dialectical idealism; his founding of
what F.E.D. terms "PSYCHO-Historical Materialism", in which the human Phenome
and its Meme-Pool -- in which human ideology and human science -- are not
denied;, are not declared non-existent, or outside of the "material" universe,
but in which they are seen as "matter" too; as part of a <<Genos>> of matter
which comes in two primary <<species>> --
>
> <<species #1>>: <<physis>>-matter; "physical" matter;
>
> <<species #2>>: idea-matter.
>
>
>
> The formula for "the Set of All Objects" --
>
> OsubT = [Osub0]^(2^T) = U^(2^T)
>
> -- is not an [EXO-]"Iteration", like
>
> XsubT = (F^T) "of" (Xsub0).
>
>
> In the case of OsubT, the "F" is replaced by OsubT itself.
>
> Therefore, F.E.D. calls --
>
> OsubT = [Osub0]^(2^T)
>
> -- a "SELF-Iteration", an "ENDO-iteration".
>
>
>
> In the symbol complexes --
>
> [OsubT]^2 = OsubT "times" OsubT = OsubT "of" OsubT
>
> -- we express a "Self-[Reflexive ]Function", or "Self-Reflexion" -- a Bending
["Flex"] Back ["Re"] Upon "Self" -- of OsubT, and also a "Self- Refluxive
Function", or "Self-Refluxion" -- a Flowing ["Flux"] Back ["Re"] upon/to "Self"
-- of OsubT, in which OsubT semantically, as meaning, as definition -- and as
sentential SUBJECT, in this context -- confronts itself as sentential OBJECT,
and, finding that its OBJECT-ive existence in its Tth stage falls short of its
meaning/nature/definition, expands itself by the set-ontological increment
denoted by 2^OsubT; by the generation, and by the inclusion within itself, of
all of its sub-sets, thus transforming its own self/identity, into that of a NEW
set, with a DIFFERENT power-set.
>
> It finds that its Tth self, as sentential OBJECT, contradicts its self as
sentential SUBJECT, and responds by expanding itself set-qualitatively;
set-ontologically.
>
>
>
> When one mentally simulates the "Set Of All Objects", when one mentally
behaves AS IT; constructs it in one's own mind, one sees that this set is
"forced" to expand itself -- by adding its own sub-sets to itself; is forced to
do so by its own nature; by its own essence; by its own definition, by its own
name.
>
>
> This process of its continuing qualitative self-expansion potentially never
ends, because each time it changes itself, changes what it contains, its
power-set also therefore changes; its "new" self means that its "new" power-set
is unprecedented, qualitatively different from all of the previous power-sets of
its previous selves in its series/sequence/succession/progression of selves.
>
> Regards,
>
> Miguel
>
> --- In hegel-marx@yahoogroups.com, "detonacciones" <ProfitSci@> wrote:
> >
> > Dear Hegel-Marx Group Members,
> >
> > RE: A Set-Theoreical Model of the Dialectical <<Logik>> of the [Hegelian +]
Dialectic,
> >
> > On which is also an IMMANENT Critique of Set Theory,
> >
> > and a Set Theoretical Model of the "Meta-Numbers" of the N\Q
> >
> > Dialectical Symbolic Logic, or "Symbolic Dialectics".
> >
> >
> >
> > The "Finitary Set of All Sets", and, even more so, the "Finitary Set of All
Objects", as mental constructs which are "self-moving", and, moreover,
"self-developing", due to their essences/definitions alone -- constituing
therefore two "idea-object" examples of Plato's <<autokinesis>> -- interest me,
from a Hegelian perspective, because, for example 'O', denoting the "Set of All
Objects", proxies Hegel's "Science of <<Logik>>" category of "BEING", and
beyond, in interesting ways.
> >
> > In set theory, the "INtension" of a given set -- the meaning, the predicate
which, as an "EXtension", it denotes -- is the Quality shared by every one of
the members/objects contained in that set.
> >
> > Thus the "Natural Number" 3 has been defined/modeled, logically, or
set-theoretically, as the intension of the set of all sets that contain exactly
three members/elements, because "three-ness" is therefore the one quality that
all of these -- otherwise often quite disparate -- sets have in common.
> >
> > If the <<arche'>>, or beginning point/state, of the "Set of All Objects", as
a self-expanding set, is the "Universe Set", or "Universal Set" of a given
[finite] Universe[-of-discourse], denoted by U, so that --
> >
> > O_subscript_Zero = U = Osub0, for short
> >
> > -- then the Quality shared by all members of U, the implicit connotation, or
"INtension", which that set, as an explicit "EXtension", represents, is the
common Quality of BEING -- common to all beings which make up that Universe of
discourse.
> >
> > If that U "universe" of Objects is already "large" and "diverse", then the
"common denominator" Quality represented by U will be something highly
attenuated -- almost "NOTHING", in terms of any specificity, or
> > "determinateness".
> >
> > Even if the membership of U is not "large" and "diverse' to begin with, the
membership of O will get "larger and larger", and simultaneouly, "more and more
diverse", as U = Osub0 "ingests" all of its subsets -- whose collection is
usually denoted by 2^[Osub0] -- including that subset of itself which is itself
as a whole [it's own "improper" subset], as Osub0 transforms itself into --
> >
> > Osub1 = Osub0 union 2^[Osub0]
> >
> > -- in its Sisyphosean attempt to include within itself the "objects" which
it still excludes/lacks, and which, hence, makes its name a lie; makes its
momentary existence [always and forever] violate its own essence/definition.
> >
> > Its "self-motion", its "self-change", is driven by this [ontological]
"self-contradiction", this "self-opposition". this "self-disagreement".
> >
> > The "subsets" which it will thereby "add" to it's [successor-set's]
membership will extensionally represent new intensions, new Predicates -- i.e.,
more specific Qualities, mostly qualities shared by smaller sub-groups of
"logical individuals", or "objects", that are grouped together in, or contained
in, the original universe of discourse set, denoted by U.
> >
> > Thus, the "Set of All Objects" will, simply by trying to "BE"[come] itself,
"auto-construct" successive waves of more and more subtle Predicates --
Predicates of Predicates; Predicates of Predicates grouped together with the
"base"/"Ur" Objects of U, which are NOT sets; Predicates of Predicates of
Predicates,. . ., etc., as --
> >
> > Osub0 becomes Osub1 becomes Osub2 becomes Osub3 . . ..
> >
> > Thus, even an initially "small" and "non-diverse"
> >
> > Osub0 = U
> >
> > will quickly become "attenuated" in terms of the overall, unified
"intension" of its growing "extension", and will thus -- as a whole, entire set
-- more and more approximate an "intension" or "Predicate" of "NOTHING" as the
common Quality of all of the "BEING(s)" in its Objects-realm / content.
> >
> > So, as --
> >
> > Osub0 ---> Osub1 ---> Osub2 ---> Osub3 ---> . . .
> >
> > -- We also have it that, for the totality of the BEING represented by OsubT
as T escalates --
> >
> > BEING ---> NOTHING.
> >
> > Might it not be interesting to further scrutinize our mental experience when
we mentally simulate this subjective object, O^(2^T), to see if that experience
replicates other Concepts/"determinations" of the categorial progression which
Hegel claims for his <<Logik>>?
> >
> >
> > First, to make this discussion of "The Set Of All OBJECTS" a little more
concrete, for those of us who are not steeped in modern set theory:
> >
> > Suppose we consider a "Universe of Discourse" consisting of only two
"objects", or "logical individuals" -- an "apple", denoted by a, and an
"orange", denoted by b.
> >
> > Then we have --
> >
> > U = {a,b} = Osub0, and;
> >
> > 2^U = the Set of all Sub-Sets of U = { {a}, {b}, {a,b}, {_} },
> >
> > -- wherein '{_}' stands for the "Empty Set", which is automatically an
element of every power-set per Modern Standard Set Theory.
> >
> >
> > If |U| denotes the "cardinality" of U, i.e., its [cardinal] number of
members, its "elements-count", then --
> >
> > |U| = 2 = |{a, b}|
> >
> > -- and therefore the cardinality of the set 2^U is --
> >
> > |2^U] = |{ {a}, {b}, {a, b}, {_} }| = 4 = 2^|U| = 2^2 = 4.
> >
> >
> > Then we have --
> >
> > Osub1 = Osub0 union 2^[Osub0| = U union 2^U =
> >
> > {a, b} union { {a}, {b}, {a,b}, {_} } =
> >
> > { a, b, {a}, {b}, {a,b}, {_} }
> >
> > -- wherein the new elements -- this time SETS-as-elements, note well --
represent the "extensions" of the following "intensions", or "Predicates" --
> >
> > {a} extensionally stands the intension/predicate/quality of this apple --
"[this-]apple-ness"?;
> >
> > {b} extensionally stands the intension/predicate/quality of this orange --
[this-]"orange-ness"?;
> >
> > {a, b} extensionally stands the intension/predicate/quality shared by both
this apple and this orange -- "[this, partial ]fruit-ness"?
> >
> > -- and --
> >
> > {_} extensionally stands for the intension/predicate/quality shared by no
thing(s) [in this universe of discourse], or shared by [any]thing(s) which exist
in this universe of discourse, but which are unmentioned by / unknown to /
unrecognized by its present discoursers / observers / participants.
> >
> >
> >
> > Now, for anybody brave enough:
> >
> > What is the content of the next set in this self-propelled
set-self-progression; the membership of, namely, Osub2?
> >
> >
> >
> >
> > Note that, already, just one step beyond set Osub0, at set Osub1, we have --
> >
> > Osub1 = { a, b, {a}, {b}, {a,b}, {_} }
> >
> > -- such that the set Osub1, again, represents, EXtension-ally, a connotation
/ INtension / Quality / Predicate of our SELF-expanded Universe of Discourse.
> >
> > What is that Predicate, that describes/connotes the Quality that all of the
(6) "beings" or "objects" or "logical individuals" in our expanded
Universe/Universal Set have in common?
> >
> > I would describe this Predicate/Quality of "All BEING" [for our Universe] --
the "INtension" of Osub1 -- as being already pretty "attenuated" toward
"NOTHING" -- wouldn't you?
> >
> >
> >
> > For more on this Subject, see slides 28 through 32 of --
> >
> >
http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/Dialectical_Pic\
\
> > tography/Dialectical_Pictography.htm
> >
> >
> > A one paragraph description, plus a depiction, of the Set of All Sets
> > self-movement can be found here:
> >
> > http://www.dialectics.org/Glossary_files/OnePageGlossary_OCR.pdf
> >
> >
> >
> >
> > [NOTE: There is an error in the above-linked rendition,later/elsewhere
> > corrected by F.E.D., in its assertion that --
> >
> > Ssub0 = U
> >
> > -- wherein U denotes the Universal Set, or set of all logical individuals of
the[finite] Universe[-of-discourse], whereas, the correct assertion is that --
> >
> > Ssub0 = 2^U
> >
> > -- where 2^U denotes the "power-set", or set of all subsets, of U.
> >
> > Otherwise, the resulting "Set of all SETS" will contain the "Ur"-objects, or
"base objects" belonging to U, which are themselves NOT sets.
> >
> >
> > Almost all of the readily-available conceptual tools of our present
mathematical and scientific communities exhibit the "Parmenidean over-hang", or
the "Parmenidean hang-over", of "Onto-STASIS", of a "statical, ANTI-dynamical
ontology" -- even Hegel, in the mere circle / <<Kyklo>> of his <<En Kyklo
Paidea>> of the Philosophical Sciences: the great circle of --
> >
> > <<Logik>> ---> <<Natur>> ---> <<Geist>> ---> <<Logik>>.
> >
> > The only extant conceptual-mathematical, scientific tools that I am
presently aware of, that transcend this crypto-Parmenidean "onto-stasis"; that
accommodate a non-reductionist, and "dynamical, ontology", are those of F.E.D..
> >
> > These tools provide an Arithmetic+ whose "set-theoretical model" is not a
mere domain of statical relations, but the "auto-onto-dynamasis" and
"Predico-Dynamasis" of the finitary "set of all sets", and of the finitary
"universal set", or "set of all Objects" [including of idea-objects, such as
extensional representations of predicates, ordered pairs, relations, etc.], both
of which sets must continually, "ideo-auto-kinesically" expand themselves
QUALITATIVELY, ONTOLOGICALLY, because they always exclude those sets which are
> > the elements of their own power-sets, or set-of-all-subsets; those
"idea-objects" which are the elements of their own power-sets.
> >
> > The formula for the finitary "set of all Sets", for a finite
> > universe[-of-discourse], denoted by U, is, given Ssub0 = 2^U --
> >
> > SsubT = Ssub0^(2^T) = (Ssub(T-1))^2
> >
> > -- where (Ssub(T-1))^2 = Ssub(T-1) UNION 2^(Ssub(T-1)).
> >
> > The formula for the finitary "set of all Objects" is,
> > given Osub0 = U --
> >
> > OsubT = U^(2^T) = (Osub(T-1))^2
> >
> > -- where (Osub(T-1))^2 = Osub(T-1) UNION 2^(Osub(T-1)).
> >
> > The N\Q arithmetic can be comprehended as one which models the
> > "ideo-auto-onto-kinesis", and "ideo-auto-onto-dynamasis", of these two
> > ideo-<<autokinesis>>-exhibiting sets, which thus form its "set-theoretical
foundation/definition", analogous to the way in which set cardinality forms a
foundation for the purely-quantitative / onto-statical "Standard Natural
Arithmetic", of --
> >
> > N = {1, 2, 3, ... }.
> >
> > This N\Q "Meta-Natural Meta-Numbers" arithmetic, of --
> >
> > N\Q = { q-sub1, q-sub2, qsub3, . . . }
> >
> > -- then provides a "contra-Boolean" -- a Dialectical -- algebra of logic,
capable of modeling the diachronically self-expanding ontology that we observe
in our reconstructions of the cosmological self-evolutions, punctuated by
self-revolutions, of this <<Kosmos>>.
> >
> > Regards,
> >
> > Miguel
> >
>