__Full Title__: Part

**05**of

**29**--

#
__The Dialectica Manifesto__:

__The Dialectica Manifesto__

##
__Dialectical ____Ideography__ and

__Dialectical__

__Ideography__

##
the Mission
of F.__E__.__D__.

__E__

__D__

Dear
Readers,

I am, together with

**F**.**.**__E__**. Secretary-General Hermes de Nemores, and**__D__**F**.**.**__E__**. Public Liaison Officer Aoristos Dyosphainthos, organizing to develop a new, expanded edition of the**__D__**F**.**.**__E__**. introductory documents, for publication in book form, under a new title --**__D__

__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

**installment of a***fifth***29**-part presentation of this introductory material, which the**F**.**.**__E__**. General Council has authorized for serialization via this blog over the coming months, as we develop the material.**__D__
I
plan to inter-mix these installments with other blog-entries, including the
planned additional

**F**.**.**__E__**. Vignettes, other**__D__**F**.**.**__E__**. news, my own blog-essays, etc.**__D__
Links
to the earlier versions of these introductory documents are given below.

Unlike
the typical blog-entry, this series will attempt to deliver an introduction to
the

**Foundation**worldview as a**, in a***totality***, making explicit many of***connected account***.***the interconnexions among the parts*
Enjoy!!!

Regards,

Miguel

Part

**05**of**29**--#
__The Dialectica Manifesto__:

__The Dialectica Manifesto__

##
__Dialectical ____Ideography__ and

__Dialectical__

__Ideography__

##
the Mission
of F.__E__.__D__.

__E__

__D__

*The*__Dialectic__

*of the Immanent Critique*

*of Set Theory*.Work by a growing cadre mathematicians, beginning from the mid-

**1800**s, 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,

**, immutable, objective but**

*Real***entities, not accessible to sensuous perception, only to noetic experiencing, and asserting the existence of an eternal realm of ‘‘‘mathematical «**

*transcendental***»’’’, but often without any inheritance of Platonian**

*eide***: a kind of partially, logically**

__dialectic__*‘*

*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’’’

**logic, and to produce an extended Aristotelian “mathematics of logic” that would also be the “logic of mathematics”.**

*Aristotelian*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__*, the*

**’**__theory__*“*

*ensembles**, the*

**theory**”*“*

*manifolds theory**”*, or of the

*“*

__set__*-*approach to attaining an axiomatic foundation for all of mathematics created a model, and a kind of

**”**__theory__**, for**

__metric__*‘‘‘*

*sets within sets**’’’*.

The process of forming

*‘‘‘*

*sets inside of sets**’’’*is a process which we of

**F**.

**.**

__E__**. describe as one of the**

__D__*‘*

__meta__*-*,

**monadizing**’*‘*

__meta__*-*«

**»’**

*monads**-*, or

**creating***‘*

*logical**, and thereby also as one of a*

**-**__meta__**individuals**-**creating**’

__neo__*-*«

*»-making,*

**arithmos**

__new__*‘*

__ideo__*-*making, qualitatively self-transforming

**ontology**’-*‘*

__self__*-*,

**’**__internalization__*‘*

__self__*-*,

__re__-__entry__’*‘*

__self__*-*,

__inclusion__’*‘*

__self__*-*,

__incorporation__’*‘*

__self__*-*, or

__containment__’*‘*

__set__*-*,

__containment__’

*of***.**

*sets*It is a process in which

**themselves become their own**

*sets**“*

*opposites**”*--

**of**

*elements***.**

*sets*This process of

*the becoming**-“*

*elements**”*; of

**of**“**sets**”**themselves**

**the becoming**-“

*elements**”*of

**[**

*set***idea**[

**l**]-]

*objects**, of entities which are*

__already__

*sets**-*, is a process which we recognize to be one of ‘«

**of**-**elements****»**

*aufheben*

__self__*-*, as well as being a process which turns out to be

**subsumption**’**to the attempts of ‘set-logicians’ to “reduce” all of mathematics to ‘set-logic’.**

__crucial__A

**for such**

__metric__

*set**‘*

__element__ization*’*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__*-*, which are

**objects**’’’**themselves**

__not__**, might be assigned to**

*sets**‘*

**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

**denoted**

*set***{**

**a**

**,**

**b**

**}**-- the “collecting” or “gathering together” of the two [idea-]

**into a single ideal**

*objects***-- is then of**

*unity*

**logical type**

**1**.

This

**, “enclosing”, or “containing”, both**

*set***a**and

**b**, thereby represents a more

*‘determinations-*,

**’**__reduced__*‘characteristics-*, “more

**impoverished**’

*abstract**”*[

*idea**-*]

**, because it is defined as denoting**

*object***those determinations, characteristics, qualities, or “predicates” which**

__only__**a**and

**b**

**exhibit; which they “have in common”.**

__both__A set of

__logical____type__**2**would then be

*a set that includes*

__sets____of__*‘*, such as the set denoted by:

**base**-**objects**’ among its**elements****{**

**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 [«

**»] contained/conserved within the contents of the set**

__aufheben__**{**

**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**

**}**

**}**.

**{**

**a**

**,**

**b**

**,**

**{**

**a**

**}**

**,**

**{**

**b**

**}**

**,**

**{**

**a**

**,**

**b**

**}**

**}**is

*—*

__qual__itatively__un__equal to

__not__merely

__quant__itatively*—*

__un__equal 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**

__quant__itatively__un__equal**to**,

**or**

*to*__quant__itatively__equal__**{**

**a**

**,**

**b**

**,**

**{**

**a**

**}**

**,**

**{**

**b**

**}**

**,**

**{**

**a**

**,**

**b**

**}**

**}**

**ERGO**

**{**

**a**

**,**

**b**

**}**

**is**

__qual__itatively__un__equal**to**

**{**

**a**

**,**

**b**

**,**

**{**

**a**

**}**

**,**

**{**

**b**

**}**

**,**

**{**

**a**

**,**

**b**

**}**

**}**

**ERGO**

**{****a****,****b****}****'[**

**~****>**

**&**

**~**

**=**

**&**

**~**

**<**]'

**{**

**a**

**,**

**b**

**,**

**{**

**a**

**}**

**,**

**{**

**b**

**}**

**,**

**{**

**a**

**,**

**b**

**}**

**}**

-- wherein the new, '''

*-*

__non__*''' relation-symbol, '[*

__standard__

**~****>**

**&**

**~**

**=**

**&**

**~**

**<**]', 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

*needs to recognize, and distinguish, [at least]*__immanently__*qualitatively distinct «*__two__**» of the «***species***» — denoted ‘***genos***’ — of****~****=****.***inequality*
One «

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

**» is currently, in general,***species**recognized in conventional mathematics, and is denoted, herein -- given typographical limitations that exclude the use of the*__un__**F**.__.__**E**__. standard symbol for this relation -- by the “compound” ideographical symbol, and ‘neogram’ --__**D**
'[

This

**~****>****&****~****=****&****~****<**]'.This

__<<__**dialectical***>> within the category of the mathematical relation of inequality is illustrated below --***diairesis**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 “__U__nion”, denoted ‘**∪**’] that ‘predecessor-set’ itself.
The [“standard”] “sub-sets” of

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****}**include the*“*subset of*”***improper****{****a****,****b****}**— none other than*of set***the whole****{****a****,****b****}**itself — so that the ‘successor-set’,**{****a****,****b****,****{****a****}****,****{****b****}****,****{****a****,****b****}****}**, results from, in part, a ‘__self__*-*’ of the previous whole / entire set, or ‘‘‘totality’’’,__internalization__**{****a****,****b****}**, which ‘‘‘now’’’ becomes a ‘‘‘mere’’’ [new]__part__*of the new, expanded, ‘ideo-ontologically’ richer whole / ‘‘‘totality’’’,***inside****{****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’’’*to*__unique__**‘****a****’**; ‘**{****b****}**’, in turn, names a predicate formulated to express, as a singular quality / ‘‘‘in-tension’’’, the total ‘‘‘nature’’’ / content*to ‘*__unique__**{****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****’***among the totality of ‘‘‘ur-objects’’’ that constitute the*__alone__**of the universe[-of-discourse] being modeled.**__base__
The set-succession — or «

Thus, in summary, the ‘predecessor-set’ / logical-type, above, is «

**» set-progression — partially depicted here is thus one which models what we term a**__aufheben__*‘***predico***-*, or*’*__dynamasis__*‘*__qualo__*-*, 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’’’.*’*__dynamasis__Thus, in summary, the ‘predecessor-set’ / logical-type, above, is «

**»-**__aufheben__*, and also, simultaneously, «*__conserved__**»-**__aufheben__*[in logical type, as well as being*__elevated__*in contents-*__expanded__*], and thus also «*__ontology__**»-**__aufheben__*/annulled/canceled/*__negated__**itatively-transformed, by this ‘«**__qual__**» self-product’, or ‘Power-Set Evolute Self-Product’, of sets.**__aufheben__
If we denote by

**, and also by**__T____S__**, the “universal set”, the set of**_{0}**‘‘‘logical individuals’’’, or, i.e., the ‘‘‘**__All__**otality’’’ of ‘‘‘ur-objects’’’ that are part(s) of a given universe of discourse, and if we denote by**__T____s__**[**__T__**]**the ‘**uccessor universe-set’ of the ‘predecessor universe-set’,**__s__**, and if**__T__**P[**denotes the “set of all subsets”, or “__T__]**ower-set”, of the set**__P__**, then the formula for the product-rule just named above can be stated as follows [using the sign '**__T____' to stand for the phrase 'is equal to by definition'] --__**=**

__s__**[**

__T__

**]**

**=**

__T__**×**

__T__

**=**

__T__^{2}

**=**

__T__**+**__Δ__[

__T__

**]**

**=**

__T__

**∪**

**P**

**[**

__T__**]**

-- or --

__s__**[**

__S___{0}_{ }

**]**

**=**

__S__

_{0}**x**

__S__

_{0}

**=**

__S__

_{0}

^{2}

**=**

__S__

_{0}**∪**

__Δ__

__S__

_{0}

**=**

__S__

_{0}

**∪****P**

**[**

__S___{0 }

**]**

**=**

__S__

_{1}
-- 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__

_{t}

^{2}

**=**

__S__

_{t}**∪**

__Δ__

__S__

_{t}

**=**

__S__

_{t}**∪**

**P**

**[**

__S__

_{t}**]**

-- or --

__s__

^{t}**[**

__S__

_{0}_{ }

**]**

**=**

__S__

_{t}**=**

__S__

_{0}

^{2^}

^{t}-- wherein

**2**

**^****t**

**≡**

**2**

**.**

^{t}The resulting «

**»-progression of sets — namely, the set-sequence-containing the set**

__aufheben____s__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’’’,

*, ‘‘actually-constructed’’’ successive universes of discourse, a*

__finite__*propositionally*

**-self-contradictory, non-paradoxical model of the most central, most crucial [idea-]object in all of set theory as such, the**

__non__*‘‘‘*

**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

This

*‘‘‘**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__*-*; an**movement**’’’*‘***ideo***-*, ‘[*-*__auto__*’*__kinesic__**ideo***-*]*-*__onto____dynamical__*’*,*‘**ideo-onto-logic-ally’*__self__*-*‘‘‘idea-object’’’, and one which, for appropriate universes of discourse, implicitly contains all of the wherewithal for**expanding***‘*__The____Gödelian____Dialectic__*’*[see next section].
But

*is this*__why__*‘‘‘**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-*that is also an ‘idea-*__object__**’, or**__subject____agent____of__*, 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-*__change____ev__ent*ity**’*, 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*“*sets — indeed,*”*__All__*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***forces**__itself__*‘*__internalization__*’*of anything*‘‘‘**external**’’’*to it, because it already contains all of the '''ur-objects''' / "logical individuals" that**and**__found__**the entire universe of discourse in question, but, rather, on the contrary, via the continual**__base__*‘*__self__*[-and-other-subsets**]-*__internalization__*’*, the ‘*’ of what is already ‘‘‘internal’’’ to it, of what it already ‘‘‘contains’’’ implicitly; the*__internalization__*‘*__internalization__*’*of**— of its own***itself as a whole**“*proper subset” — as well as of all of the__im__*“*subsets” of itself.__proper__
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/**, i.e., by the***definition**‘**intra**-*, or*’*__duality__*‘***self***-*, or*’*__duality__*‘***indivi***[sible**]**-*, of its*’*__duality__**momentaneous ‘‘‘state’’’ of existence in the mind — because it always, in every “moment”, “still”***every**those very sets which constitute*__excludes__*“power set”,***its own***subsets, among which is that set which is***its own****its own***“*__im__*proper”*subset, namely, none other than*.*__itself__
But this

*‘‘‘**set of*__ALL__sets*’’’*,**,***as that***, is***as such**, per its very name/definition,*__not__*[finite, ‘‘‘constructible’’’]***supposed to exclude**__any__*.***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**, different set of subsets — a qualitatively different “power-set” — all of whose subsets are thus***NEW**in itself, among its “elements”.***not yet included**
Therefore, it must,

*it tries to [re-]form itself,***each time***‘*__internalize__*’*its own subsets, including itself, again.
But, in so doing,

*, it changes itself again, thus bringing a new, different set of [its] subsets -- a new, more rarefied set of ‘extensional predicates’ -- into [potential] existence.***each time**
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***-*»’ result, if one simply defines the**kinesis***“*itself as*”***universal set***“*[of the universe in question], provided that one grants that the more ‘‘‘rarified’’’, more abstract mental objects — that the*”***the set of**__ALL____OBJECTS__*‘**-*__idea__**object***’*that is each subset, i.e., each*“**extensional**, denoted “extensionally”, per set theory, by the set of all objects that share the quality denoted by that predicate — are included among the**”***predicate***“*__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***-*, in the form of an

*-***onto***’***dynamasis***‘*

*extensional-*-

__predicates__*’, or*

**dynamasis***‘*

__predico__*-*].

*’***dynamasis**The formulae for the

*‘‘‘*

*set of all sets**’’’*and the

*‘‘‘*

*set of all objects**’’’*are also the prototypes for the [

**]**

*Dyadic***, which is the primary vehicle for the**

*Seldon Function**‘*

__dialectical__*of*

**meta**-**models**’**, including the**

__Encyclopedia Dialectica__**F**.

**.**

__E__**. ‘**

__D__

*Psychohistorical*__Dialectical__Meta*-*, when such 'meta-models' are expressed in the algebraic language of the

**Equations**’**F**.

**.**

__E__**.**

__D__*‘*

*First**.*

__Dialectical__**Arithmetic**’The

*‘‘‘*

*set of all sets**’’’*

**, thus, a logical/conceptual/mental**

__is__*‘*

**self***-*that [en]forces the continual, mounting,

*’***force**

__self__*-*«

*»*

**aufheben***‘*

__self__*-*of itself and of all of its [other] subsets, thus driving its qualitative self-expansion, in an open-ended, “potentially infinite” progress.

*’*__internalization__
The

*‘‘‘**set of all sets**’’’*is, therefore —**(**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-

**1**)*proclivities of most “Standard” set-theorists, that quality turns out to be none other than an that of*

__statical__

**an uninterrupted movement of self**-**inclusion***,*of

**self**-**subsumption***,*of

**self**-**involution***,*of

__self__*-*«

*»*

**aufheben***‘*

__self__*-*;

*’*__internalization__**(**The vehicle of an immanent critique of [Parmenidean] set theory itself, via a «

**2**)*» 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 «*

**reductio ad absurdum***», or changelessness;*

**stasis****(**a set-theoretical model of the

**3**)*‘*

__dialectic__*’*itself; of a generic ‘Meta-Monadology’; of what we will come to call, below, an ‘auto-kinesic’, ‘ideo-onto-dynamical’, ‘

*-Peanic’, ‘ideo-*

**Qualo***-*

**meta***’- constructing, ‘*

**fractal***-*

**meta***’ ‘self-progression’; an*

**finite***‘*

*archeonic*

*consecuum*-

*cumulum*

*’*, driven by a succession of

__self__*-*«

*»*

**aufheben***‘*

__self__*-*which are also

*’***internalizations***‘*

__self__*-*«

**-**__meta__*»-*

**monad**

*izations**’*.

Sets of

**logical type****3**contain at most

__sets____of____sets__of**, e.g. --**

__base__objects**{**

**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****and**

**b}****}****{**

*,*

**{b}***,*

**{a**

**b}**

**}**

-- are called “

*ordered pairs*”, also written --

**<a**,

**b>**and

**<b**,

**a>**

-- respectively, because for them,

__un__*like*

*for*,

__in general__**sets***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**”**to be our ‘base [idea-]objects’, then sets or “classes”**

*numbers**“*or “containing” such numbers would be of

**of**”

__logical____type__**1**,

__classes__*“*or “containing”

**of**”**[of such numbers] would be of**

__classes__

__logical____type__**2**, and

**[of such numbers] would be of**

__classes__of__classes__of__classes__

__logical____type__**3**, and so on.

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