F.E.D. Dialectics

Monday, March 24, 2025

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 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^S^t, wherein 2^S^t 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.

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^S^t ‘‘‘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’.

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.

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.

YOU are invited to post your comments on this blog-entry below!

Sunday, March 23, 2025

Part 02: Seldon’s Slogans Series. My Take on Slogan 1.

Part 02:

Seldon’s Slogans Series.

My Take on Slogan 1.

GLOBAL STRATEGIC HYPOTHESES.

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, Encyclopedia Dialectica key themes of Seldonian Theory.

The 2nd text 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].

Seldon’s Slogan #1 --

“Dialectic is the Cause of Time.”

M.D.: My take on this slogan is as follows. I see it as analogous to Seldon’s point about the ‘intra-duality’ of stars.

A typical “main sequence” star is a complex unity of a colossal thermo-nuclear self-explosion, issuing from its Hydrogen core – i.e., from its H+, protonic core – in opposition to a colossal self-gravitational self-implosion.

Now I want to indulge you, for a moment, in a counter-factual supposition. Such counter-factuals are not realistic. They are thought experiments, imagining something which abstracts from, and omits, integral aspects of reality. And yet, such counter-factual thought experiments can sometimes reveal real insights.

When a typical stars forms, self-gravitational self-implosion precedes the ignition of its, answering, core thermonuclear self-explosion.

Imagine a star from which the thermonuclear self-explosion side of its ‘intra-duality’ has been suspended.

Also imagine a star from which the self-gravitational self-implosion side of its ‘intra-duality’ has been removed.

In both cases, as you may well notice, the life-duration of the imagined star would be over momentaneously – probably in mere nanoseconds.

In the first case, a colossal, supernova-like self-explosion would instantly end the life of the star as such.

In the second case, a colossal and unimpeded self-implosion would instantly produce a [‘metafinite’] “singularity” – a “black hole” – as we hypothesize it, a minute ‘holonium’ core.

So also is it with the origin of space-and-time, per our ‘self-resurging big surge’, or ‘regenerist’, hypothesis – conjectured as an improved replacement for both the one-time-only “Big Bang” theory, and the “inflaton” cosmological “inflation” theory: our hypothesis of the epochally-recurring thermodynamic self-regeneration of our cosmos, driven by “Dark Energy”.

Per that – ‘regenerist’ – hypothesis, the early “inflationary” epoch of our cosmos was not due to any, hypothetical, never-yet-empirically-isolated “inflaton” field, but to “Dark Energy” itself, to the hyper-accelerated expansion of physical space in the absence of any restraint by “Dark Matter”.

It is, per Seldon’s view, Dark Energy’s incessant expansion of physical space that is the substance of universal, cosmological time, with increments of physical space representing also increments of time, albeit inconstantly so.

But he supposes that “Dark Energy” pre-exists the birth of “Dark Matter”, and, indeed, causes that birth.

In the absence of “Dark Matter”, he holds, “Dark Energy” would cause an unchecked, “inflationary” hyper-accelerating explosion of physical space.

If one believes that there must be a finite limit to the rate of spatial self-expansion – however far beyond the speed of light that finite limit might be, then, Seldon hypothesizes, that limit was reached in the form of a ‘self-damage’ to the ‘spandetron’ units that Seldon conjectures to be the ‘monads of physical space’.

In that ‘self-damage’, vast clouds of some of those ‘spandetrons’ lost their capability to self-replicate – the incessant self-replication capability that drives the “Dark Energy” accelerating self-expansion of physical space. Those clouds formed the substance of “Dark Matter”, which restrained and slowed down Dark Energy’s self-expansion of space.

In the continued absence of “Dark Matter”, the TIME born of “Dark Energy” would last only perhaps nanoseconds, before the demise of such a “Dark Energy-only” cosmos, due to the unlimited excession of its spatial expansion acceleration, perhaps resulting in a ‘“Big Rip”’ ‘disexistentiation’ of the cosmos, long before any self-aware, cosmos-aware humanoids could come into existence within it.

In the presence of the “Dark Matter”, born out of the finite limit to that excession, time could stretch, instead, as it has, for billions of years, and continuing still.

It is thus, per hypothesis, the opposition, the ‘intra-duality’ of our cosmos as a whole, of “Dark Energy” versus “Dark Matter”, that gifts us our ‘TIME of long duration’.

Without the dialectical antithesis of “Dark Energy” versus “Dark Matter”, TIME would be almost instantaneously gone, ‘‘‘in no time at all’’’ – per this, yet another counter-factual thought experiment.

For more information regarding these Seldonian insights, please see --