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.
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