Claim: universe infinity and eternal infinity are the same size; or, \(|\mathbb{R}^4| = |\mathbb{R}|\).

For universe infinity, we suppose that the universe we live in is a continuous three-dimensional phenomenon. Suppose it stretches endlessly in every direction -- up, down, left, right, forward, backward.

It is like three copies of the continuous real line \(\mathbb{R}\), all perpendicular to each other. Once a center point and coordinates are chosen, every place in the universe can be described by three parameters, say \(x\), \(y\), and \(z\). These parameters correspond to distances in space.

In math, we would write this space as \(\mathbb{R}^3\), or \(\mathbb{R} \times\mathbb{R}\times\mathbb{R}\). It means the set of all triplets \([x,y,z]\), where each of \(x\), \(y\), and \(z\) is some real number. This is the space we live in.

But for universe infinity we want to imagine every event that has happened or can happen -- at any place, at any time. In other words, we add a fourth dimension: time. Everything that has ever happened or will happen, happens somewhere, at some time. Thus each event is described by four numbers, \([w,x,y,z]\), where \(w\) is the time coordinate. We will let our coordinates be arbitrarily large, positive or negative.

So we can equate universe infinity with the set \(\mathbb{R}^4\), i.e. \(\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\).

On the other hand, for eternal infinity we are only considering instants in time. There is only one coordinate, and it can be at the present, or arbitrarily far in the past or in the future. As in the third module, we can equate eternal infinity with the set \(\mathbb{R}\).

And so the claim that universe infinity is the same size as eternal infinity, is the same as the claim that \(|\mathbb{R}^4| = |\mathbb{R}|\).

(By the way, in the in-depth explanation, we'll also show that this statement is true with \(4\) replaced by any other positive integer. So \(|\mathbb{R}^{12}| = |\mathbb{R}|\), etc. So if you want to imagine our universe, as string theorists do, as having more than \(3\) spatial dimensions, our claim still holds true that universe infinity is the same size as eternal infinity.)