Part 05 of 29: The DIALECTICA MANIFESTO -- The Dialectic of the Immanent Critique of Set Theory.

Full Title:  Part 05 of 29 --

The Dialectica Manifesto:

 

Dialectical Ideography and 

 

the Mission of F.E.D.

Dear Readers,

I am, together with F.E.D. Secretary-General Hermes de Nemores, and F.E.D. Public Liaison Officer Aoristos Dyosphainthos, organizing to develop a new, expanded edition of the F.E.D. introductory documents, for publication in book form, under a new title --

The Dialectica Manifesto:  Dialectical Ideography and the Mission of Foundation Encyclopedia Dialectica [F.E.D.]

-- and under the authorship of the entire Foundation collective.

Below is the fifth installment of a 29-part presentation of this introductory material, which the F.E.D. General Council has authorized for serialization via this blog over the coming months, as we develop the material.

I plan to inter-mix these installments with other blog-entries, including the planned additional F.E.D. Vignettes, other F.E.D.  news, my own blog-essays, etc.

Links to the earlier versions of these introductory documents are given below.

http://www.dialectics.org/dialectics/Primer_files/1_F.E.D.%20Intro.%20Letter,v.2,New_OCR.pdf

http://www.adventures-in-dialectics.org/Adventures-In-Dialectics/DiaRith/Intro/Dialectical-Ideography_An-Introductory-Letter.htm

http://www.dialectics.org/dialectics/Primer_files/2_F.E.D.%20Intro.%20Letter,%20Overall%20Prefactories%20%26%20Abstract,%20Supplements%20A.%20%26%20B.,%20v.2_OCR.pdf

http://www.dialectics.org/dialectics/Primer_files/3_F.E.D.%20Intro.%20Letter,%20Supplement%20A-1_OCR.pdf

http://www.dialectics.org/dialectics/Primer_files/5_F.E.D.%20Intro.%20Letter,%20Supplement%20B.,%20Prefatories,%20v.2_OCR.pdf

http://www.dialectics.org/dialectics/Primer_files/4_F.E.D.%20Intro.%20Letter,%20Supplement%20B-1,%20v.2_OCR.pdf

Unlike the typical blog-entry, this series will attempt to deliver an introduction to the Foundation worldview as a totality, in a connected account, making explicit many of the interconnexions among the parts.

Enjoy!!!

Regards,

Miguel

 

Part 05 of 29 --

The Dialectica Manifesto:

 

Dialectical Ideography and 

 

the Mission of F.E.D.

 

 

 

The Dialectic 
of the Immanent Critique 
of Set Theory.


Work by a growing cadre mathematicians, beginning from the mid-1800s, sought to establish a Platonian view of mathematical ideas, today often called “Mathematical Platonism”, or “Mathematical Realism” -- the doctrine that mathematical objects are not human mental constructs, but, instead, Real, immutable, objective but transcendental entities, not accessible to sensuous perception, only to noetic experiencing, and asserting the existence of an eternal realm of ‘‘‘mathematical «eide»’’’, but often without any inheritance of Platonian dialectic:  a kind of partially, logically ‘Aristotelianized Platonism’.

  
Much of this effort focused later on the development of “Set Theory”, taken to be a theory of the ultimate, “Platonically Real” objects of mathematics. 

 
This “Platonic” movement within mathematics was often tied to, or part of, a larger movement, sourced in an ‘intendedly’ non-Hegelian, non-dialectical logic, or even in a polemically and overtly anti-dialectical logic. 


This larger movement sought to ‘‘‘mathematicize’’’ Aristotelian logic, and to produce an extended Aristotelian “mathematics of logic” that would also be the “logic of mathematics”.


They then hoped to use that “mathematical logic”, “symbolic logic”, or ‘‘‘ideographical logic’’’ to establish a secure, “axiomatic” and ‘postulational’ “Foundation” for mathematics, á la Euclid’s five-postulate deductive system for just the classical geometry portion of mathematics as a whole.

  
Some within this movement even sought to “reduce” all of mathematics outright to formal logic alone! 


In either case, this would require the formulation, in the “artificial language” of that new “symbolic logic”, of a few -- supposedly “self-evident”, uncontrovertible -- premises, from which all of [the rest of] mathematics potentially could be, and then, slowly and painstakingly, would actually be, rigorously deduced.


Tendentially, as these efforts were pursued, the development of “Set Theory”, of the new “Mathematical Logic”, and of the ‘aspirationally’ secure logical “Foundation” for all of mathematics, became increasingly convergent and intertwined.


Protagonists of these movements included Boole, Peirce, Cantor, Frege, Peano, Russell, and Gödel -- names which will arise again in the course of our outline, in this manifesto, of a dialectical, immanent critique of “Modern Mathematics”, paralleling, in many ways, Marx’s dialectical, immanent critique of “Modern [i.e., capital-epoch] Economics”. 

Crucially -- for their story, and for our story, and for our immanent critique -- their mathematical logic of the ‘[proto-]«arithmoi» theory’, the [proto-]‘totality theory’, the “ensembles theory”, the “manifolds theory”, or of the “set-theory” approach to attaining an axiomatic foundation for all of mathematics created a model, and a kind of metric, for ‘‘‘sets within sets’’’. 


 
The process of forming ‘‘‘sets inside of sets’’’ is a process which we of F.E.D. describe as one of the ‘meta-monadizing’, ‘meta-«monads»’-creating, or ‘logical meta-individuals-creating’, and thereby also as one of a neo-«arithmos»-making, new ‘ideo-ontology’-making, qualitatively self-transforming ‘self-internalization’, ‘self-re-entry’, ‘self-inclusion’, ‘self-incorporation’, ‘self-containment’, or ‘set-containment’, of sets.


It is a process in which sets themselves become their own “opposites” -- elements of sets.


This process of the becoming-“elements” of “sets” themselves; of the becoming-“elements” of set [idea[l]-]objects, of entities which are already sets-of-elements, is a process which we recognize to be one of ‘«aufheben» self-subsumption’, as well as being a process which turns out to be crucial to the attempts of ‘set-logicians’ to “reduce” all of mathematics to ‘set-logic’.

  
A metric for such set ‘elementization’ is embedded in a theory called the theory of logical types.


A set-representation which “contains” only representations of “logical individuals”, e.g., of ‘fundamental objects’, or ‘‘‘ur-objects’’’, which are not themselves sets, might be assigned to ‘logical type 1’.


Thus, for example, if a and b denote two such “thought-concrete”, or “determinations-rich”, ‘base-[idea-]objects’ [perhaps, at root, idea-representations of physical, sensuous objects], then the set denoted {a, b} -- the “collecting” or “gathering together” of the two [idea-]objects into a single ideal unity -- is then of logical type 1.

This set, “enclosing”, or “containing”, both a and b, thereby represents a more ‘determinations-reduced’, ‘characteristics-impoverished’, “more abstract” [idea-]object, because it is defined as denoting only those determinations, characteristics, qualities, or “predicates” which a and b both exhibit; which they “have in common”.


A set of logical type 2 would then be a set that includes sets of ‘base-objects’ among its elements, such as the set denoted by:

{ a, b, {a}, {b}, {a, b} }.

The ‘‘‘logical type’’’ of a set, per the definition of ‘‘‘logical type’’’ given above, can be determined directly by counting the number of ‘‘‘opening braces’’’ -- ‘{‘ -- or of ‘‘‘closing braces’’’ -- ‘}’ -- to their deepest, or maximal, level within the set whose ‘‘‘logical type’’’ metric is to be evaluated.


Notice that the contents of the set {a, b} are also [«aufheben»] contained/conserved within the contents of the set { a, b, {a}, {b}, {a, b} }, but also that { a, b, {a}, {b}, {a, b} } is a kind of not-{a, b} --

        {a, b}     ~=    { a, b, {a}, {b}, {a, b} }.

Indeed, { a, b, {a}, {b}, {a, b} } is qualitatively unequal to — not merely quantitatively unequal to — {a, b}  [using the sign ‘~’ to stand for the word “not”] --

{a, b}    ~>   { a, b, {a}, {b}, {a, b} }

AND

{a, b}    ~=   { a, b, {a}, {b}, {a, b} }

AND

{a, b}    ~<   { a, b, {a}, {b}, {a, b} }

THEREFORE

{a, b}  

is not quantitatively unequal to, or quantitatively equal to 

 { a, b, {a}, {b}, {a, b} }

ERGO 

{a, b}   is qualitatively unequal to   { a, b, {a}, {b}, {a, b} }

ERGO 

{a, b}   '[ ~>  &  ~=  &  ~< ]'   { a, b, {a}, {b}, {a, b} }

-- wherein the new, '''non-standard''' relation-symbol, '[ ~>  &  ~=  &  ~< ]', enables us to summarize, in a single statement, the ‘negated trichotomy’ of the conjunction of the three statements --

‘{a, b}  is not greater than  { a, b, {a}, {b}, {a, b} }’, and
‘{a, b}  is not equal to        { a, b, {a}, {b}, {a, b} }’, and 
‘{a, b}  is not less than       { a, b, {a}, {b}, {a, b} }’.

What we are saying, in other words, is that mathematics immanently needs to recognize, and distinguish, [at least] two qualitatively distinct «species» of the «genos» — denoted ‘~=’ — of inequality.

 One «species» is already recognized, and denoted herein -- given typographical limitations that exclude the use of the conventional symbol -- by the ideographical symbol   ‘[ > OR < ]’. 

The other «species» is currently, in general, unrecognized in conventional mathematics, and is denoted, herein -- given typographical limitations that exclude the use of the F.E.D. standard symbol for this relation -- by the “compound” ideographical symbol, and ‘neogram’ --

'[ ~>  &  ~=  &  ~< ]'.

This dialectical <<diairesis>> within the category of the mathematical relation of inequality is illustrated below --

 

 
Notice also that the ‘successor-set’, { a, b, {a}, {b}, {a, b} }, differs, ‘contentally’, from the ‘predecessor-set’, {a, b}, in that it contains — together with the ‘predecessor-set’ itself, {a, b} — also [most of] the [“standard”] “sub-sets” of that ‘predecessor-set’. 

That is, ‘the successor-set’, { a, b, {a}, {b}, {a, b} }, contains [most of] the elements of most of the [“standard”] “set of all sub-sets” — i.e., the elements of [most of] the so-called “power-set” — of the ‘predecessor-set’, {a, b}, ‘‘‘plus’’’ [or “Union”, denoted ‘∪’] that ‘predecessor-set’ itself.

The [“standard”] “sub-sets” of {a, b} include the “improper” subset of {a, b} — none other than the whole of set {a, b} itself — so that the ‘successor-set’, { a, b, {a}, {b}, {a, b} }, results from, in part, a ‘self-internalization’ of the previous whole / entire set, or ‘‘‘totality’’’, {a, b}, which ‘‘‘now’’’ becomes a ‘‘‘mere’’’ [new] part inside of the new, expanded, ‘ideo-ontologically’ richer whole / ‘‘‘totality’’’,  

{ a, b, {a}, {b}, {a, b} }.

Thus, the ‘successor-set’, here, is the ‘predecessor-set’ itself, ‘‘‘plus’’’ the elements of [most of] the “power-set” of that ‘predecessor’ set.

The various parts of the ‘successor-set’, { a, b, {a}, {b}, {a, b} }, might, for example, be interpreted as follows:  ‘a’ names a concrete, complex, ‘full-determinations’ ‘prior-to-sets’ ‘‘‘ur-object’’’, as does ‘b’, for a qualitatively distinct / other such object; ‘{a}’ names a predicate formulated to express, as a univocal, singular quality / ‘‘‘in-tension’’’, the total ‘‘‘nature’’’ /- content / ‘‘‘predicate’’’ unique to ‘a’; ‘{b}’, in turn, names a predicate formulated to express, as a singular quality / ‘‘‘in-tension’’’, the total ‘‘‘nature’’’ / content unique to ‘{b}’, and; ‘{a, b}’ names a predicate formulated to express, as a singular quality / ‘‘‘in-tension’’’, just those qualit(y)(ies) shared in common by ‘a’ and ‘b’ alone among the totality of ‘‘‘ur-objects’’’ that constitute the base of the universe[-of-discourse] being modeled. 

The set-succession — or «aufheben» set-progression — partially depicted here is thus one which models what we term a ‘predico-dynamasis’, or ‘qualo-dynamasis’, progressively conceptualizing — or lifting out of ‘‘‘chaotic’’’ and ‘‘‘inchoate’’’ implicitude; progressively ‘explicitizing’ — more and more predicates, so as to articulate ever-more distinctly and ever more concretely, ‘‘‘for-themselves’’’, the richness of the determinations of that universe’s    ‘‘‘ur-objects’’’, ‘‘‘in-themselves’’’.

Thus, in summary, the ‘predecessor-set’ / logical-type, above, is «aufheben»-conserved, and also, simultaneously, «aufheben»-elevated [in logical type, as well as being expanded in contents-ontology], and thus also «aufheben»-negated/annulled/canceled/qualitatively-transformed, by this ‘«aufheben» self-product’, or ‘Power-Set Evolute Self-Product’, of sets.

If we denote by T, and also by S₀, the “universal set”, the set of All ‘‘‘logical individuals’’’, or, i.e., the ‘‘‘Totality’’’ of ‘‘‘ur-objects’’’ that are part(s) of a given universe of discourse, and if we denote by s[ T ] the ‘successor universe-set’ of the ‘predecessor universe-set’, T, and if P[ T ] denotes the “set of all subsets”, or “Power-set”, of the set T, then the formula for the product-rule just named above can be stated as follows [using the sign '=' to stand for the phrase 'is equal to by definition'] --

s[ T ]    =    T × T    =   T²    =    T + Δ[ T ]    =     

T   ∪   P[ T ]

-- or --

s[ S₀ ]    =    S₀ x S₀    =    S₀²    =    S₀ ∪ ΔS₀     =      

S₀  ∪  P[ S₀ ]      =      S₁

-- or, more generally, for the variable t successively taking on the values 0, 1, 2, 3, 4, ..., as --

s[ S_t ]   =   S_t+1   =   S_t x S_t    =     


S_t²    =   S_t  ∪    ΔS_t   =    S_t  ∪  P[ S_t ]

-- or --

s^t[ S₀ ]   =   S_t   =   S₀^{2^t}

 
-- wherein 2^t  ≡  2^t.

The resulting «aufheben»-progression of sets — namely, the set-sequence-containing the sets denoted by   { S_t } as t successively takes on the values 0, 1, 2, 3, 4, ... — i.e., for the “Natural” ordinality, or order of progression, of the “Whole” Number value, t, provides, especially for ‘‘‘realistic’’’, finite, ‘‘actually-constructed’’’ successive universes of discourse, a propositionally non-self-contradictory, non-paradoxical model of the most central, most crucial [idea-]object in all of set theory as such, the ‘‘‘set of all sets’’’.

This ‘‘‘set of all sets’’’ — since it is set-theory’s own, native definition of the “set” itself, the set-theoretical, or “ex-tension-al”, definition of the ‘‘‘in-tension’’’ of the “set” concept itself — is the central idea-object of set-theory, though, ironically, and tellingly, it is suppressed in “Standard” Set Theory. 

Hence, also, the ‘‘‘set of all sets’’’ is the central locus of a dialectical, immanent critique of that set theory.





This ‘‘‘set of all sets’’’ is a ‘contra-Parmenidean’ mental ‘eventity’; a mental ‘‘‘self-movement’’’; an ‘ideo-auto-kinesic’, ‘[ideo-onto-]dynamical’, ‘ideo-onto-logic-ally’ self-expanding ‘‘‘idea-object’’’, and one which, for appropriate universes of discourse, implicitly contains all of the wherewithal for ‘The Gödelian Dialectic’ [see next section].

But why is this ‘‘‘set of all sets’’’ a ‘self-changing’ ‘‘‘idea-object’’’; an ‘‘‘idea-object’’’ that itself induces change in itself; an ‘‘‘idea-object’’’ that itself causes itself to expand, qualitatively, ‘ideo-ontologically’; an idea-object that is also an ‘idea-subject’, or agent of change, with respect to itself; an ‘idea-entity’ that “won’t stand still” in your mind, in any human mind, once that mind constructs it, and lends that mind’s ‘subject-ivity’ to that mental construct; an ‘idea-entity’ that forces itself to grow, and that is, thus, an ‘idea-eventity’, a mental process object “made of” ‘ideo-«auto-kinesis»’?

This [finitary] ‘‘‘set of all sets’’’ is ‘‘‘forced’’’, in an attempt to fulfill its own definition, the definition of its very self, i.e., to attempt to “be[come]” what it “is” — viz., that it contains “All” sets — indeed, forces itself into continual expansion of its contents, of its ‘‘‘elements’’’, of its ‘‘‘membership’’’ — forces itself into continual qualitative, ideo-ontological, ‘predicatory’ self-expansion, not by the ‘internalization’ of anything ‘‘‘external’’’ to it, because it already contains all of the '''ur-objects''' / "logical individuals" that found and base the entire universe of discourse in question, but, rather, on the contrary, via the continual ‘self[-and-other-subsets]-internalization’, the ‘internalization’ of what is already ‘‘‘internal’’’ to it, of what it already ‘‘‘contains’’’ implicitly; the ‘internalization’ of itself as a whole — of its own “improper subset” — as well as of all of the “proper subsets” of itself. 

This ‘‘‘set of all sets’’’ is ‘‘‘forced’’’ to do so, to continually re-‘‘‘internalize itself’’’ by its own nature / essence / ‘essence-iality’ / essentiality / logical necessity; by its own ‘‘‘self’’’; by its own name/description/definition, i.e., by the ‘intra-duality’, or ‘self-duality’, or ‘indivi[sible]-duality’, of its every momentaneous ‘‘‘state’’’ of existence in the mind — because it always, in every “moment”, “still” excludes those very sets which constitute its own “power set”, its own subsets, among which is that set which is its own “improper” subset, namely, none other than itself. 

But this ‘‘‘set of ALL sets’’’, as that, as such, is not, per its very name/definition, supposed to exclude any [finite, ‘‘‘constructible’’’] sets at all. 

Yet, each time it ‘internalizes’ all of its subsets, including itself, it thereby transforms itself into a new, qualitatively different, ‘ideo-ontologically different’, qualitatively expanded, ‘ideo-ontologically’ expanded, set, with yet a NEW, different set of subsets — a qualitatively different “power-set” — all of whose subsets are thus not yet included in itself, among its “elements”. 

Therefore, it must, each time it tries to [re-]form itself, ‘internalize’ its own subsets, including itself, again. 

But, in so doing, each time, it changes itself again, thus bringing a new, different set of [its] subsets -- a new, more rarefied set of ‘extensional predicates’ -- into [potential] existence. 

And so, it must actualize that potential existence, by self-/power-set-‘internalizing’  again... .

Indeed, one obtains an augmented version of the same ‘ideo-«auto-kinesis»’ result, if one simply defines the “universal set” itself as “the set of ALL OBJECTS” [of the universe in question], provided that one grants that the more ‘‘‘rarified’’’, more abstract mental objects  — that the ‘idea-object’ that is each subset, i.e., each “extensional predicate”, denoted “extensionally”, per set theory, by the set of all objects that share the quality denoted by that predicate — are included among the “objects” referenced by the sub-phrase “ALL OBJECTS” -- 

One obtains, all over again, but this time in a deepened, more comprehensive form, a mental process object characterized by self-expanding ‘ideo-onto-dynamasis’, in the form of an ‘extensional-predicates-dynamasis’, or ‘predico-dynamasis’].

The formulae for the ‘‘‘set of all sets’’’ and the ‘‘‘set of all objects’’’ are also the prototypes for the [Dyadic] Seldon Function, which is the primary vehicle for the ‘dialectical meta-models’ of Encyclopedia Dialectica, including the F.E.D. ‘Psychohistorical Dialectical Meta-Equations’, when such 'meta-models' are expressed in the algebraic language of the F.E.D. ‘First Dialectical Arithmetic’.

 The ‘‘‘set of all sets’’’ is, thus, a logical/conceptual/mental ‘self-force’ that [en]forces the continual, mounting, self-«aufheben» ‘self-internalization’ of itself and of all of its [other] subsets, thus driving its qualitative self-expansion, in an open-ended, “potentially infinite” progress.   

The ‘‘‘set of all sets’’’ is, therefore — 

(1) The very object which expresses and stands for the “essence” / ”quality” that all sets have in common, per set theory’s immanent way of expressing such qualities, such that, e.g., the number two is represented by the set of all sets which have exactly two members, and the color “green” is represented by the set of all objects that look green to human perception.  However, contrary to the onto-statical proclivities of most “Standard” set-theorists, that quality turns out to be none other than an that of an uninterrupted movement of self-inclusion, of self-subsumption, of self-involution, of self-«aufheben» ‘self-internalization’;

(2) The vehicle of an immanent critique of [Parmenidean] set theory itself, via a «reductio ad absurdum» refutation of Standard Set Theory’s implicit ‘Parmenidean Postulate’ — the belief that sets, and their elements, and, indeed, that all mathematical, idea-objects, must be characterized by eternal «stasis», or changelessness;

(3) a set-theoretical model of the ‘dialectic’ itself; of a generic ‘Meta-Monadology’; of what we will come to call, below, an ‘auto-kinesic’, ‘ideo-onto-dynamical’, ‘Qualo-Peanic’, ‘ideo- meta-fractal’- constructing, ‘meta-finite’ ‘self-progression’; an ‘archeonic consecuum-cumulum’, driven by a succession of self-«aufheben» ‘self-internalizations’ which are also ‘self-meta-«monad»-izations’.

Sets of logical type 3 contain at most sets of sets of base objects, e.g. --

{ a, b, {a}, {b}, {a, b}, {{a}}, {{b}}, { {a, b} }, { {a}, {b} }, ..., { {a}, {a, b} }, { {b}, {a, b} } }.

Those elements of the latter set denoted by --

{ {a}, {a, b} } and { {b}, {a, b} }

-- are called “ordered pairs”, also written --

<a, b>  and  <b, a>

-- respectively, because for them, unlike for sets in general, order of listing matters -- 

{a, b}   =   {b, a}

-- but --

{ {a}, {a, b} }  ≡  <a, b>     ≠     <b, a>   ≡   { {b}, {a, b}}

-- in fact, in general —

<a, b>    ‘[ ~>  &  ~=  &  ~< ]’    <b, a>

--wherein ‘≡’ denotes ‘is equal to by definition’.


Thus, if we take “natural” numbers to be our ‘base [idea-]objects’, then sets or “classes” “of” or “containing” such numbers would be of logical type 1, classes “of” or “containing” classes [of such numbers] would be of logical type 2, and classes of classes of classes [of such numbers] would be of logical type 3, and so on.