Eternal Return implies a Finite World, No Beginning of Time, and a single unique Eternity

Eternal Return implies 

a Finite World, No Beginning of Time, 

and a single, unique Eternity.

The construction involves taking into account the set of natural laws NL that govern our world - I assume there may be such NL. Define the set W:= “set of all possible states of affairs of the world at any given instant consistent with NL”.

Next we look at the "set of all possible sequences* with no repeats  consistent wit NL (these are our circuits/loops) of the elements of W" - call it Ws. The elements of Ws look like this for any element x of W: (x, y, z, r,…, s, t, x) if W is countable, or the union of [x, r] and [t, x], if W is uncountable, and none of y, z, r,…, s, t  are equal to x (*). Eternal Return demands that Ws is non empty.

It also insists that there are no infinitely long elements in Ws, since if it was otherwise it would be possible to enter a non-terminating sequence with no repeats (in it the ER view would be false) - contradiction. Hence all elements of Ws are finite

• Ws is finite: there are only finitely many distinct world histories with no return.
• Hence W is finite or some NL consistent states of affairs never obtain, which raises the question: what makes those states NL consistent? Wouldn’t NL consistency of a state w(a) demand that it can obtain? But then if w(a) can’t obtain, it is no longer NL consistent – contradiction.
• Hence W is finite.

Next, we model Eternity.

Also ER implies that either, there is no beginning of time, or if there was one we get an immediate contradiction, or if we’re lenient and look the other way we still will be left with an eternity consisting of an impoverished oscillation between states, which also could be considered empirically falsified since we perceive change. A detailed proof could be provided, but just consider the fact that the first instant of time is nor a return point of any predecessor except itself (contrary to ER), and any future instant of time will always lack sufficient predecessors to satisfy both requirements - change and the ER requirement. So we model Eternity to be isomorphic with the integers i.e. it extends into infinity both into the past and to the future.

The construction proceeds as follows - we take some element of W, w(k) i.e. w(k) is some state of affairs at a given instant. Next we construct the “an infinite concatenation* with neither beginning nor end, of all elements of Ws beginning with w(k)”, denoted ?E.w(k). We remove each of the doubling w(k)’s where each sequence joins (*).

However it may be the case that not all concatenations* of elements of Ws are consistent with NL, since there may be “jumps” (discontinuities) in some of them at the joins. So we define E.w(k) identically to ?E.w(k) with the proviso that all concatenations* therein are consistent with NL.

This construction yields an Eternity conditioned on Eternal Return to w(k), denoted E.w(k).

E.w(k) = … w(i) w(k) w(j) w(x)… w(y) w(z) w(k) w(p)… w(q) w(k) w(r)…

If we assume the Eternal Return view as true, then for all w(j), w(k) in W, E.w(j) = E.w(k), since if that wasn’t the case, some instants would not return to themselves contrary to the assumption – contradiction. 

• For all w(j), w(k) in W, E.w(j) = E.w(k) 

-       Hence there’s only one unique Eternity.

Some thoughts on William Blake's ..... AUGURIES OF INNOCENCE

‎To see a World in a Grain of Sand 
And a Heaven in a Wild Flower, 
Hold Infinity in the palm of your hand 
And Eternity in an hour.
[...]

Upon some reflection, one may start developing a sense of empathy for the somewhat romantic and existential character of that perennial plea uttered by humanity, ringing out with a tone of longing, it’s echo reverberating through the Platonic realm, and reflecting back here so softly that only the hearts of poets can hear it.

[...]
Every night and every morn
Some to misery are born,
Every morn and every night
Some are born to sweet delight.

Some are born to sweet delight,

Some are born to endless night.

[...]

God appears, and God is light,

To those poor souls who dwell in night;

But does a human form display

To those who dwell in realms of day.

I'm sure there may be numerous ways of interpreting this poem, and maybe even some of them could be considered as better than others? But it seems that the content of the poem itself opposes the existence of any hierarchy which should serve as the absolute measure of any interpretation's alleged objective adequacy. This, perhaps could be an interesting direction of inquiry in itself, but not one which I presently wish to pursue. Instead I will share some of my reflections inspired by the poem.
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What I've noticed, prompted by reflections meandering toward the blurry boundaries of the poem's subject matter, is the apparent content-relational continuity between the first, and the last few stanzas, and the message their interplay carries. It's plausible that the author intended this symmetry, if not throughout the entire poem then at least between its first and the last stanzas.
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What it reveals, if one entertains this observation as one not entirely devoid of merit, is the conditioning of one's conceptual repertoire to the attributes (nature) of the "realm" they inhabit and experience.
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Those who are "doomed" - "to endless night", which is some otherworldly-Heavenly domain - and united with God's immanent presence, are perchance privileged having such direct and immediate access to the divine and transcendental nature of reality. Being familiar with that realm, they know too well how it permeates everything, and that it should be evident - "seen" - even in a wildflower - a worldly entity. Sadly, such a particular form of being is absent from the realm of the universal which they dwell in. (Note the mysterious and uncrystallized nature of that which cannot be "seen" i.e. during the "night", and consider the magnitude of that mystery with "endless" predicated of it.)
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As they dream of seeing (a blank concept, although entertainable, yet devoid of any experiencial referent), they imagine seeing that which they only know - "heaven" - in an object (the wild flower) endowed with a particular, grounded yet unique nature. If only they were granted that possibility - to experience, if only an infinitesimal fraction of the "realm of day" - no larger than a grain of sand would suffice.
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But alas! - their gaze invariably transcends the flower's earthly, finite form, failing to appreciate its pure and simple beauty.
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Those who dwell in the "sweet delight", are "blessed" with the clarity offered by the world bathed in the "light of day", and unperturbed by that which cannot be seen, can only relate to the divine in terms of the spatiotemporal boundaries, imposed on them by the "realm" they inhabit, and cannot help but reduce the transcendental to such a conceptual confinement: “Hold Infinity in the palm of your hand, and Eternity in an hour”.
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This is also why, despite persistently harbouring hopes and dreams - which resemble restless chimeras that rummage anxiously through the most elusive depths of the soul, where mystery, doubt and wonder seek refuge from the syrupy tentacles of the hedonistic bliss - they occasionally experience fleeting suspicions with regard to the "all revealing" "sweet delight" - whether there may not be some paradoxical, and double edged nature to it.
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Being conditioned to the "explicated", worldly realm, which overwhelms the eyes with demonstrative, verging on obtrusive, clarity (hence a presumed "absolute clarity"), and crystallized, unambiguously delineated finite forms, one experiences God accordingly - in a familiar and non hesitant manner.
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A still further stretched interpretation, on a somewhat relatively deeper reading, would allow seeing those diverse realms, presented in the poem, as analogues of certain attitudes which we're all free to adopt - or maybe not entirely free, hence the somber tone and categorical emphasis on the phrases "born to ..."? 
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Upon adopting, or tentatively veering toward some preferred set of attitudes, we progressively develop, and subsequently reinforce, the positive feedback driven interdependence of our experiential and conceptual repertoire - their character in turn, ultimately presents us with a corresponding manifestation of Being.

Linguistic Meme Dynamics

First of all, I need to explain the distinction between linguistic, and the most common conceptual memes. The later are purely semantic ones and owe their existence to culture and possibly historical contingencies. A good example may be the meme God. Linguistic memes carry no semantic weight in their own right. Instead they play the role of grammatical devices which serve as cues to the listener how the context where they appear should be interpreted. They resemble the function of case markers.

One common meme is the term "like". From my observation it serves (in most cases) the role of pre-empting a description.

The scientific method algorithm, computational complexity and TOE's.

I see science, or more precisely the method by which science proceeds (the scientific method) as an algorithm which asks progressively more complex questions, and subsequently receives answers of proportional complexity. Roughly, the science (theory) algorithm goes something like this:

Here we take the chain "(large)-(large with branches)-(large with leaves on some branches)-..." as an analogue for a chain of concepts of increasing complexity.

Science: Is the object General Sherman (largest tree in the world) larger (taller) than 10 metres?

Nature: Yes.

Science: Hoorray!

Science: Does General Sherman have branches (defined in one way or another)?

Nature: Yes.

Science: Hoorray!

Science: Do some of General Sherman's branches have feathers (defined in one way or another)?

Nature: No.

Science: Damn! Back to the drawing board.

Science: Do some of General Sherman's branches have leaves (defined in one way or another)?

Nature: Yes.

Science: Hoorray!

Science: ...?

Nature: ...

Science: ...!

     :
     :
     :

Possibly ad infinitum.

A corollary of this model is, that it may be the case that our theories, as they evolve in discrete steps, one superseeding the previous one, will grow to arbitrary large complexity as they accommodate phenomena of gradually increasing complexity.

It is a theorem of Algorithmic Information Theory that the complexity of certain stings/objects/concepts put a lower bound on the complexity of the theory/program that generates them. Thus one cannot hope for a complete theory of everything (TOE) without some harboured assumptions about the complexity of the universe and the phenomena within it. It would certainly be presumptuous to harbour such assumptions since as far as I know the question whether the universe (multiverse) is finite or infinite, and hence it's complexity, remains an open question.

For it may be the case that our theories can only approach a limit which would be a TOE, but necessarily never reach it.

The above ruminations have been primarily inspired by the work of Gregory Chaitin.

Constructing a set disjoint from, and equinumerous with any other set.

A while ago I needed (a lemma in a larger proof) to construct for any arbitrary set A, another set that is disjoint from A and of the same cardinality as A (without the use of higher machinery of ordinals). The task is harder than it may seem. One may choose to attempt a proof by contradiction. One of my friends has in fact proceeded that way. I came up with the following construction which I’m quite fond of.

First, let me state a theorem (which is a direct consequence of Russell’s paradox) that I helped myself to.

Theorem. For each set S, the set

Is not a member of S.

For if we assume otherwise, an immediate contradiction ensues, for

Now for the construction. Let’s take some set A, and construct a set B that is disjoint from it, and equinumerous with it. Consider the set

Theorem: For any set A the following set B is disjoint from A and equinumerous with it:

Less formally

It is clear that B has the same cardinality as A. (post-Clarification: just to be pedantic, it's not just the triple x, y, z that is in A, but rather I should have written "for x, y, z,... in A". I forgot the ellipsis.)

It is also clear that,

since,

which is impossible by construction of R_A . Hence B is equinumerous to, and disjoint from A. QED

Thoughts

I don’t think that it is too much of a presumption to claim that our conceptual repertoire developed thought human intellectual history is largely an outgrowth (and conditioned by) of the world we happen to occupy and the form we happen to posses.

We have had some success and seen a development of intellectual brevity which in effect made it possible to surpass our perceptual and rational limitations, leaving them behind as hindering properties of our human form (example: existence of the electromagnetic spectrum - our senses do not perceive X rays for instance, and peculiar existential nature of quantum entities - it seems that the distributivity theorem of propositional calculus fails at the quantum level ) . Nevertheless it is that very form that was the starting point, and persistently forces it’s limitations on our creativity and ingenuity.

Furthermore it should be noted that the universe we happen to occupy is a rather narrow experimental and severely conditioning ground – why, it just happens to be a world where sentient beings are possible. Empiria seems to suggest that other worlds are physically possible, and logic certainly doesn’t demand that possible worlds necessarily contain sentient beings.

One may find oneself wondering whether the universe (or the multiverse in its entire spatio- temporal and/or Hugh Everett’ian sense, or modal realism sense after David Lewis) is finite or infinite. If the latter than we should , with mature humility, accept epistemic defeat, in the “let’s ask nature what she is” game. If the former, then the problem doesn’t disappear but becomes more interesting when we consider questions of information, self-reference and computational complexity. 
But notice that I have introduced a rather presumptuous classical dichotomy which in itself is an effect of intellectual conditioning to particular type of logic. Why not finite and infinite simultaneously (?), or neither finite nor infinite? And if simultaneously finite and infinite, then it appears that, again, I’m  introducing crude notions concerning what we call time, yet have no conclusive answer to what it is or even whether it is a fundamentally coherent notion at all. What we refer to as time may just be a phenomenal manifestation of  some more fundamental properties (Sean Carroll), peculiarities if you will, of the universe we happen to find ourselves in. So again one is reasoning from possibly perceptual limitations, making any inquiry fundamentally unsound.

So we continue on what we supposedly do best – conjure up new concepts about what is, and what could be, with more lucid moments realizing that this is just a peculiar property of some peculiar (what we call, sentient and intelligent) entity occupying some peculiar universe – a property motivated by the desire and hope of finding an alibi for our existence, or maybe not even motivated at all, but it sure sounds more reassuring that such behavior is intended.

Super Intelligent Beings.

Imagine yourself, perceiving a plant, say a tree - its incredibly slow metabolic rate and predictable "behaviour" i.e growth. Also in some sense the plant is defenceless against your whim, whatever it may be - to cut it down, burn it, or eat (parts of) it. Better still imagine a stone - its practically unchanging crystal structure, and unique combination of minerals shaped by geological processes over billions of years. Again - the stone is a passive, and for all practical purposes (by definition!), a lifeless object. Now imagine, that a super-intelligent being would perceive YOU in a manner analogous to your perception of that tree, or rock.     

In the time that you would take one breath, it would seem to such a being that eons have passed during which its leaps and strides of insight would dwarf the totality of our (human) current intellectual achievements.