Okay. Two different things seem to be going on in this thread. One is the question of whether it is possible for an infinite number of
something to exist in reality. The other is the question of whether the universe has a beginning.
I read through most of Craig’s argument for the lack of existence of an actual infinity. I found it less than compelling. Plus, he got some of his facts on quantum physics wrong, which led him to wrong conclusions near the end. But quantum mechanics is hard, so I’ll ignore those for now.
Since the OP specifically asked about infinity, I’ll respond only to that part. The issue with the physical creation of the universe is a more difficult topic, and is still debated in scientific circles. I’m not sure that anyone here (myself in particular) is well-versed enough in the relevant mathematics and empirical results to comment on the origin of the universe in an educated fashion, so I will take the easy route and avoid the topic entirely.
I’m going to say that infinity
can exist. For instance, there could be an infinite quantity of empty space in the universe. This empty space could be quantified in some unit, such as hypercubes of some size, in which case there would be an infinite number of those hypercubes. (Note, that whether there really is unbounded empty space in the universe, or whether space is closed, is still not entirely clear.)
As lynx sort of suggested, we could also consider the infinitely small. Now this is more of a debate on whether you consider the universe to be continuous or discrete in nature. If the universe is discrete (and space is closed, as mentioned earlier), then there is almost certainly not an infinite quantity of anything. We know that matter and energy are discrete. That is to say, there is a “smallest” unit of energy, and it is impossible to find any energy in the universe that is in a smaller unit than that smallest unit, called a quantum. Thus, we cannot divide matter up into an infinitely small quantity. We are less certain about space and time. There are concepts called the Planck length and Planck time, but these are theoretical constructs, and are not necessarily physical boundaries on reality.
Assuming that space and time are continuous in nature (i.e. there is no “smallest” unit of space-time allowed by the universe), than infinity
does exist in the sense that within every finite unit of space there exists an infinite number of infinitesimal units of that space, each having size 0.
(Aside: In response to that, you might say, “Well, White_Tree, if I have a bunch of units all of size 0, and I add them up, then I still get 0, since 0 + 0 + 0 + … = 0. That defies logic, so your argument doesn’t hold.” However, that argument about an infinite sum of zeroes does not apply in this case, because such a series is only valid when considering
countable infinities (i.e. those infinite sets whose elements have a one-to-one mapping to the natural numbers). However, in dividing a finite unit of space into an infinite number of infinitesimal units, there would naturally need to be an
uncountably infinite quantity of such units, to which such an infinite series argument does not apply.)
And Coolduude, I think your counterexample about “counting to infinity” or back again is somewhat off-base, because it implicitly assumes that infinity is a number, which it isn’t. There is no number “infinity,” and thus you cannot count to it (or from it, for that matter).
So I’m going to say that an actual infinity can indeed exist, conditional on some quantity in the universe being continuous, or on space not looping back on itself. Either one of those assumptions holding would grant us the existence of an “actual” infinity.
Edit: Silly me. I almost forgot. Black holes, too. Black holes have positive mass and zero volume. Thus, they have infinite density. There’s an example for you of a physical object in the universe that has infinite something. So an actual infinity does exist, in terms of the density of black holes. That was easy.
That sounds like a whole other thread.
