Actual Infinity in Reality?

[polymath257]

The *best* answer to the question 'what is the universe expanding into?' is THE FUTURE.

The universe is expanding into the future.

And, while that may sound like a trick of language, there is a strict sense in which it is completely correct in general relativity.

So, as discussed above, spacetime is a four dimensional construct: it consists of all of space and all of time as a single geometric entity. Spatial cross sections of this entity (i.e, space) are 'larger' for later times than they are for earlier times. That is what is means to say space is expanding. It just means that a later 3 dimensional cross section is larger (in some sense) that an earlier one.

A very basic analogy may make this easier to comprehend. Suppose that the surface of the earth represents spacetime. In this analogy, think of higher latitudes as being later in time. Different longitudes are different spatial locations.

So, a 'time slice' is a cross section of the surface of the earth that has the same latitude: a latitude line. This represents space at a specific time.
 

Now, in the southern hemisphere, as we move north (which is later in time), the latitude lines get larger: space is expanding. In the northern hemisphere, as we move north, the latitude lines get smaller: space is contracting.

Also, we have a 'beginning' at the south pole and an 'end' at the north pole. In this analogy, time is finite with 'singularities' at the poles.

Now, what are the latitude lines expanding into? Different latitude lines aren't even in the 'same space' because they are at different times. They expand or contract into the future!

Now, this is a *very* simplistic analogy, but the essences are there: spacetime is a single geometric construct, space corresponds to cross sections of that geometry. Different times correspond to different cross sections, and expansion/contraction are determined by comparing two different cross sections.

But then what about the earth itself and beyond? There is nevertheless still something beyond "finite space" in this analogy (which, as you say, is a simplistic one anyway). So even if we were like 2D beings who couldn't understand the notion of any sort of direction in the third dimension or whatever, and weren't aware that the "latitude timeslices" were placed in the context of a planet that is situated in an outer space, it does seem like even in our ignorance of what it is exactly, we still have to logically conclude there has to be something out there, as opposed to literally nothing.

Perhaps that's not what you were saying, but I'm just saying it here just to let you know where I'm coming from here.

In this analogy, we would be *one* dimensional beings.

You are, essentially, assuming that spacetime has to be embedded into something larger. But in this analogy, spacetime *is* just the surface of the Earth.

Perhaps a different analogy.

In this analogy, we use three dimensional space as our analogy for spacetime, but in this, the *radial* dimension from some central point is our time variable. So, later times are those farther away from the center.

What does a time cross section now look like? It is the *surface* of all points some fixed distance from the center: a sphere.

For later times (again, radius), the sphere is larger. So, space is 'expanding' into the future. Here, *space* is a sphere (not all of 3D, which is spacetime).

In this analogy, we (at a single time, that is) would be two dimensional beings on the surface of a sphere.

Now, this model has a 'beginning' (at the center) with no time prior to that. It is infinite into the future and space is finite at all times, but expanding.

One amusing aspect of this analogy is that spacetime is 'flat' while space is curved. In the previous analogy, both were curved. In cosmology, we talk about a flat space and curved spacetime.