# Reality is a countably infinite Sierpiński cube

I was thinking out loud on Twitter about what weird beliefs I hold, after realising (more or less as I was writing it) that my philosophical positions are practically banal (at least to anyone who has thought about these issues a bit, whether or not they agree with me).

I came up with a couple, but probably the most interesting (if very very niche) that I thought of is that one true and accurate mathematical model of reality is  time cube a closed, connected, subset of the countably infinite Sierpinski cube.

I consider this opinion to be not that weird and more importantly obviously correct, but I’m aware that this is a niche opinion, but hear me out.

Before we start, a quick note on the nature of reality. I am being deliberately imprecise about what I mean by “reality” here, and basically mean “any physical system”. This could be “life the universe and everything” and we are attempting to solve physics, or it could be some toy restricted physical system of interest and we are trying to nail down its behaviour. This post applies equally well to any physical system we want to be able to model.

Consider an experiment. Let’s pretend we can have deterministic experiments for convenience – you can easily work around the impossibility by making countably infinitely many copies of the experiment and considering each of them to be the answer you got the nth time you ran the experiment.

Also for simplicity we’ll assume that experiments can only have one of two outcomes (this is no loss of generality as long as experiments can only have finitely many outcomes – you just consider the finitely many experiments of the form “Was the outcome X?” – and if they have infinitely many outcomes you still need to ultimately make a finite classification of the result and so can consider the experiment composed with that classification).

There are three sensible possible outcomes you could have here:

• Yes
• No
• I don’t know, maybe?

Physical experiments are inherently imprecise – things go wrong in your experiment, in your setup, in just about every bloody thing, so set of experiments whose outcome will give you total certainty is implausible and we can ignore it.

Which leaves us with experiments where one of the answer is maybe. It doesn’t matter which answer the other one is (we can always just invert the question).

So we’ve run an experiment and got an answer. What does that tell us about the true state of reality?

Well whatever reality is we must have some notion of “an approximate region” – all of our observation of reality is imprecise, so there must be some notion of precision to make sense of that.

What does the result of a physical experiment tell us about the state of reality?

Well if the answer is “maybe” it doesn’t tell us anything. Literally any point in reality could be mapped to “maybe”.

But if the answer is yes then this should tell us only imprecisely where we are in reality. i.e. the set of points that map to yes must be an open set.

So an experiment is a function from reality to {yes, maybe}. The set of points mapping to yes must be an open set.

And what this means is that experiments are continuous functions to the set {yes, maybe} endowed with the Sierpiński topology. The set {yes} is open, and the whole set and the empty set are open, but nothing else is.

Now let’s postulate that if two states of reality give exactly the same answer on every single experiment, they’re the same state of reality. This is true in the same sense that existing is the thing that reality does – a difference that makes no difference might as well be treated as if it is no difference.

So what we have is the following:

1. Any state of reality is a point in the cube $$S^E$$ where $$E$$ is the set of available experiments and $$S = \{\mathrm{yes}, \mathrm{maybe}\}$$.
2. All of the coordinate functions are continuous functions when $$S$$ is endowed with the Sierpinski topology.

This is almost enough to show that reality can be modelled as a subset of the Sierpinski cube, not quite: There are many topologies compatible with this – reality could have the discrete topology.

But we are finite beings. And what that means is that any given point in time we can have observed the outcome of at most finitely many experiments.

Each of these experiments determine where we are only in the open set of some coordinate in our cube, thus the set that the experiments have determined us to be in is an intersection of finitely many open sets in the product topology on that cube, and thus is open in that topology.

Therefore the set of states of reality that we know we are in is always an open set in the product topology. So this is the “natural” topology on reality.

So reality is a subset of a Sierpiński cube. We now have to answer two questions to get the rest of the way:

• How many dimensions does the cube have?
• What sort of subset is it?

The first one is easy: The set of experiments we can perform is definitely infinite (we can repeat a single experiment arbitrarily many times). It’s also definitely countable, because any experiment we can perform is one we can describe (and two experiments are distinct only up to our ability to describe that distinction), and there are only countably many sentences.

So reality is a subset of the countably infinite dimensional Sierpiński cube.

What sort of subset?

Well that’s harder, and my arguments for it are less convincing.

It’s probably not usually the whole set. It’s unlikely that reality contains a state that is just constantly maybe.

It might as well be a closed set, because if it’s not we can’t tell – there is no physical experiment we can perform that will determine that a point in the closure of reality is not in reality, and it would be aesthetically and philosophically displeasing to have aphysical states of reality that are approximated arbitrarily well.

In most cases it’s usually going to be a connected set. Why? Well, because you’re “in” some state of reality, and you might as well restrict yourself to the path component of that state – if you can’t continuously deform from where you are to another state, that state probably isn’t interesting to you even if it in some sense exists.

Is it an uncountable subset of the Sierpinski cube? I don’t know, maybe. Depends on what you’re modelling.

Anyway, so there you have it. Reality is a closed, connected, subset of the countably infinite dimensional Sierpiński cube.

What are the philosophical implications?

Well, one obvious philosophical implication is that reality is compact, path connected, and second countable, but may not be Hausdorff.

(You should imagine a very very straight face as I delivered that line)

More seriously, the big implication for me is on how we model physical systems. We don’t have to model physical systems as the Sierpiński cube. Indeed we usually won’t want to – it’s not very friendly to work with – but whatever model we choose for our physical systems should have a continuous function (or, really, a family of continuous functions to take into account the fact that we fudged the non-determinism of our experiments) from it to the relevant Sierpiński cube for the physical system under question.

Another thing worth noting is that the argument is more interesting than the conclusion, and in particular the specific embedding is more important that the embedding exists. In fact every second countable T0 topological space embeds in the Sierpinski cube, so the conclusion boils down to the fact that reality is a T0, second countable, compact, and connected (path connected really) topological space (which are more or less the assumptions we used!).

But I think the specific choice of embedding matters more than that, and the fact that we the coordinates correspond to natural experiments we can run.

And, importantly, any decision we make based on that model needs to factor through that function. Decisions are based on a finite set of experiments, and anything that requires us to be able to know our model to more precision than the topology of reality allows us to is aphysical, and should be avoided.