Part 01: Dialectics and Self-Reflexive Functions Series. ‘Self-Re-Entry’ of the ‘Metafinite’ Set of All Sets: A Universal Metaphor for the Dialectics of Nature.

 

Part 01:

 

Dialectics and Self-Reflexive Functions Series.

 

 

‘Self-Re-Entry’ of the ‘Metafinite’

Set of All Sets:

 

A Universal Metaphor for the Dialectics of Nature.

 

 

 

 

 

 

 

Dear Reader,

 

G. Spencer Brown once wrote, long ago, regarding certain expressions of the super-parsimonious arithmetic-algebra that he had developed, that “...the whole expression…can…be regarded as re-entering its own inner space…” [G. Spencer Brown, Laws of Form, George Allen and Unwin, Ltd., 1969, p. 56].

 

He also wrote, in the same source: “The question of whether or not functions of themselves are allowable has been discussed at...length by many authorities since Principia Mathematica was published.  ...if an operation can take its own result as a base, the function determined by this operation can be its own argument.” [ibid., p. 97, emphases added by M.D.].

 

To whatever degree that Brown’s description of ‘self-re-entry’ may be apt for the syntactic/computational process that Brown was, in the quote above, describing, his description is more clearly and more concretely apt for another process, that he did not there address – namely, for the ‘self-iterating’ process of the finitary [‘metafinite’] ‘“Set of All Sets”’ for a given Domain/universe-of-discourse.

 

Since that ‘“Set of All Sets”’ is inadequate to itself as soon as it is formed, thereby requiring yet another iteration to attempt to reach its ever-receding, ever-elusive objective, we will denote this ‘ideo-object’ by the symbol S_t, with t serving as this set-eventity’s ‘self-iterations counter’, its count of the consecutive ‘tries’ at achieving the ‘“Set of All Sets”’.

 

We will further define the process of this set-eventity by what we will call the ‘«aufheben» evolute product of sets’:  

S_t+1  =  S_t²  =  S_t ´ S_t = S_t(S_t) = 

S_t È 2^St, wherein 2^St denotes the “power set”, or “set of all subsets” of the exponentiated set, here S_t.  As an operator, or operation, upon itself, S_t(S_t) or S_tS_t  – indeed, as an «aufheben» self-operation – this product rule defines this ‘“Set of All Sets”’ function as a “Function of Itself”, as a ‘function/argument identical’, in our sense.  

Both of the syntactic forms S_t(S_t) and S_tS_t also connote, for us, the self-reflection by, the self-inspection by, and the self-evaluation by S_t, its evaluation of itself as to whether or not it qualifies as the ‘“Set Of All Sets”’ for its [finitary] universe of discourse.  That is, S_t(S_t) and S_tS_t connotatively assert that a generic human mind, ‘mentally-embodying’ S_t, knowing/‘mentally-embodying’ the content of S_t, spontaneously performs S_t(S_t) or S_tS_t as an immanent critique, or self-critique, of S_t as a claimant to being the ‘“Set Of All Sets”’ for its Domain. 

 

This ‘‘‘self-reflexion’’’ and ‘‘‘self-reflection’’’ critique yields S_t È 2^St, or ‘S_t + DS_t’, as its positive fruition, by noting that something more, something additional, DS_t or, more specifically, 2^St, is still needed to get [further] to[ward the ever-unreachable] S, this Domain’s ‘“Set Of All Sets”’.    

 

For every finite value of t, the set S_t will always lack the containment inside itself of its very own subsets, hence every such S_t fails to be its Domain’s ‘“Set of All Sets”’, and thus definitionally self-requires yet a further ‘self-iteration’, to S_t+1, and so on. 

 

Now, note that, for each ‘self-iteration’/stage t, the entire set, S_t, reappears inside what was its [former] ‘“self”’, S_t, as its own “improper” subset, and as a new element within set S_t+1, together with all of the other, “proper” subsets of S_t.

 

For example, consider the set {x, y, z}.  Its set of subsets is of cardinality –

 2^{|{x, y, z}|} = 2³ = 8, and 2^{{x, y, z}}  =

{ {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}, {_} },

wherein {_} denotes the “empty set”.

 

If we designate {x, y, z} to be S₁, then –

S₂  =  S₁²  =  S₁ ´ S₁ = S₁(S₁) =

 

S₁ È 2^S1 =  {x, y, z}  È  2^{{x, y, z}}  = 

{ x, y, z, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}, {_} }

– whose cardinality is 11.

 

For each stage t, all of S_t reappears inside what was S_t, along with the other subsets of S_t, and thereby creates S_t+1.  Moreover, all of 2^St ‘‘‘re-enters’’’ into the “inner space” of S_t, thereby changing S_t, into S_t+1.  Thus S_t changes itself, by an ‘ideo-auto-kinesis’ [cf. Plato; cf. Marx]. 

 

Indeed, S_t «aufheben»s itself, in a classic set-theoretical model of ‘the reflexivity paradigm of dialectics’; and a classic model of, dialectical, ‘self-developing process’.

This ‘‘‘self-motion’’’ or ‘‘‘self-movement’’’ of S_t is driven by the very definition of S_t, and even by the very name of S_t, that is, by the very nature of S_t.  

 

That is, the set S_t determinately negates itself, as S_t, by becoming ‘not-S_t’ in the determinate form of S_t+1, which is qualitatively unequal to S_t. 

 

The set S_t also conserves itself inside the set S_t+1, by becoming an element of the set S_t+1. 

 

The Set S_t also elevates itself, by becoming S_t+1, a set of cardinality higher than that of S_t, and also of higher Russellian-Gödelian “logical type”, and also a set of expanded qualitative, ‘ideo-ontological’ content, containing additional, new “intensions” – new qualities – as represented by new “extensions”; the new elements that are the ‘extensional-predicate’ subsets of set S_t.

So, the ‘“Set of All Sets”’ self-operation/self-operator/-self-function thus fulfills all three “moments” of the classic «aufheben», or dialectical, process – determinate negation, conservation, and elevation.  

 

The general solution to the ‘quadratically nonlinear finite difference set-equations’ –

 S_t+1  =  S_t²  =  S_t È 2^St 

– is the ‘meta-exponential’ equation –

S_t  =  S₀^{2^t}

– in which the ‘mere exponent’, 2, has a higher exponent of its own, its ‘meta-exponent’, t.  In this solution function, the form of the ‘Dyadic Seldon Function equation’ is already presaged.

 

Typically, S₀ would be the “Universal Set” for the Universe of discourse in question, call it U, so that S₁ would be S₀^{2^1}   =  S₀² = U² = U È 2^U – the “Universal Set” of the Domain, ‘“plus”’ all of its subsets.  

However, for a Domain whose present elements-content is the result of an historical ‘meta-genealogy’, in which all of the rest of its elements arose from a single ‘«arché»-element’ ultimate ancestor, S₀ might be chosen to have just that ‘«arché»-element’ as its sole explicit content.  

In that case, the successive, consecutive steps of that Domain’s ‘“Set Of All Sets”’ would recapitulate the ‘meta-genealogical’, ‘element-ary’ history of that Domain, to-present – but, possibly also, those later steps might predictively ‘pre-construct’ future elements of that ‘meta-genealogy’.  

 

This, ‘“Set of All Sets”’, ‘self-developing [ideational] process’ thus provides an abstract metaphor for each stage, each epoch, of the physical process of ‘“Natural History”’; of the ‘self-meta-evolution’ of our cosmos – in short, of the ‘“Dialectic(s) of Nature”’, each stage ‘modelable’, in the _NQ ‘arithmetic for modeling dialectics’, as –

q_x ® q_x <´> q_x  =  q_x²  =  q_x(q_x)  = 

q_x <+> q_xx [see ‘dialectogram’ diagram below] 

– and it was the metaphor that led to Karl Seldon’s discovery of the ‘self-reflexive functions’ that we have named ‘‘‘The Seldon Functions’’’.

 

 

 

 

 

 

For more information regarding these Seldonian insights, and to read and/or download, free of charge, PDFs and/or JPGs of Foundation books, other texts, and images, please see:

 

www.dialectics.info

 

 

 

 

 

 

 

 

 

 

 

 

For partially pictographical, ‘poster-ized’ visualizations of many of these Seldonian insights – specimens of ‘dialectical art’ – as well as dialectically-illustrated books published by the F.E.D. Press, 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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