Actual Infinity in Reality?

[GrandizerII]

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.

Yeah, I was doing some reading on this some few hours ago and came upon the balloon analogy to explain how a finite universe could expand into "nothing". The flat universe seems to be almost a given now in astrophysics, am I right? I find this one really interesting as I always thought of it as a sphere ...

Anyhow, the analogy is good at illustrating what it could be like for a finite universe to expand "out of nothing" and/or "into nothing", but then in this case, the way I avoid the problematic existence of "non-existence" beyond spacetime is to posit a wider cosmos (perhaps a multiverse of universes), one from which finite universes spring forth into existence. Perhaps each time slice is like a frame in a 4D (or higher) kind of reality, and so that reality is still something into which the 3D sphere is embedded. Finite totality of existence seems to imply the existence of "non-existence" beyond it, and I find that problematic. Anything about which you can say is beyond or outside this or that is a something as far as I'm concerned, even if it's not the same nature as the spacetime we speak of here.