*Games! [Draft]

An introduction to the standard surreal numbers

WARNING: The following maths are hyper experimental — proceed w/ caution..

Nonstandard Surreal Numbers

We shall begin during the 0th day, at the epsilon-th moment:

the epsilonth moment

Proceeding as in the standard surreals, we eventually arrive at the omega-th epsilon moment:

the omegath epsilon moment

Following this construction, we arrive at omega, however we have done so in a hyper fashion:

omega in nonstandard surreal notation

Continuing on:

nonstandard surreals

We can push this notation incredibly far:

epsilon zero
How To Count Past Inifinity

Can we go further?

“Of course we can. This is math, not science.”

It’s true because we say it is

“We are creating this universe, ourselves.”

Let there be ..

The Universe of *Games!

Let the universe (V) of games (G) be defined as:


A simple model of infinitely deep infinite time

(Slightly modified from original version )

We typically represent time by real numbers, possibly extended to include an idealized point in the infinite past (“minus infinity”) or infinite future (“plus infinity”). However, we can also conceive of orderings of time in which there isn’t just one time in the infinite past, but entire timelines in the infinite past as well.

  • Define an epoch as an infinitely long period of time, but where any two points of time in the same epoch are a finite amount of time different from one another.
  • Time can consist of a sequence of “epochs”, just as we mark out time as a sequence of instants, or hours, or years, or centuries.

We might represent time then by a pair of labels: a label E for the epoch, and a label t for the time within the epoch (as measured from some fixed event during that epoch). Most importantly, we must be able to describe an order on the labels of the epochs: for instance, we may label them by numbers. These labels might be drawn from

  • a finite set, such as {0,1,2,3}, so that time is infinite but divided into finitely many epochs;
  • all natural numbers {0,1,2,3,4,…}, so that there is a first epoch but no final epoch;
  • all integers, so that there is no first or last epoch, but each epoch is separated from any other epoch by a finite number of epochs;
  • or all real numbers, so that any two different epochs are separated by an (uncountable!) infinity of other epochs.

Times are given by ordered pairs (E,t), where (E,t) < (E’,t’) if either E < E’, or E = E’ and t < t’. (In the latter case, the difference in time between the events is t’t; but in the former case, the first event precedes the second by an infinite duration governed by the difference between the epochs.)

On causality in infinite stretches of time

As to cause and effect, there is a reasonable question as to how an event in one epoch affects events infinitely further in time. The very existence of these infinitely old creatures across the epochs is one example of that: the father, for instance, may reasonably be construed as causing (among other things) his own continued existence. The notion of time that I’ve put forward doesn’t give any description of what a continuous sort of causality would look like which ties together the epochs; perhaps if these infinitely aged creatures went into a sort of dormant state, and then awoke in another epoch, you could simply define the continuity of their existence in terms of convergent states of their behaviour agreeing with each other going into the infinite future of one epoch and the infinite past of another. This implicitly invokes a notion of topology both on the timeline stretching across epochs, but also on the behaviours of the creatures. For instance, we probably demand some serious stability of these infinitely aged creatures if we want to prevent the argument that the creatures in one epoch were swapped out with a different set of infinitely aged creatures in another epoch.

As to how to treat causality of these infinitely aged creatures on their presumably less permanent environment, it’s hard to say. If the infinitely old creatures are the only fixed points of the world, so that everything about them is subject to change, then it is only meaningful if we suppose that there are meaningful features of the world as a whole in one epoch which arose as a result of the behaviour of the old ones in a previous epoch.

  • Did one of the old ones create a unique artifact in one epoch, which still exists albeit in a seemingly unrelated location in another epoch? The mere continued existence of the artifact is a causal link.

But these examples are still of infinitely enduring features of the world: the existence of an enduring phenomenon/species in one, and an enduring object in another. Even if we supposed that these infinitely aged creatures could create or destroy energy, the level of energy would be in a sense an enduring excitation of the matter/light fields caused by these creatures akin on a subtler level to making an enduring artefact (which itself is also just a very stable excitation of matter fields). Perhaps the only possible sense in which you can define continuity is in such infinitely enduring things, including the creatures themselves.

Supertask << Hypertask
If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which includes one task done for each ordinal number — thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. -Source

Surcomplex Spacetime


Infinitely many infinite beings contained w/in an unwritable ineffable !?


The Epoch …

(epoch-ellipsis; plural: epoch-ellipses)
& oddly after all of my logic & my theory,
i add a bit of t.swift, just so millennialz can hear me … *
0 = { | } = Endgame