OK, I think we've probably run into the same ambiguity that I mentioned in a different comment in this thread. There are basically two definitions that we might have of this "infinite" business. I'm working from the viewpoint of general relativity -- or some other metric-based theory of gravity -- since I'm wanting to be work with a theory capable of quantifying statements. Given this, I have (at least) two definitions of "infinite" here:
1) The model we are driven to employ, the Friedman-Lemaitre-Robertson-Walker model (or some mildly inhomogeneous or mildly anisotropic generalisation of this) says that the universe extends spatially to an extent that is infinite (in two of the three subcases) and is finite (in the other), and our data is only good enough to state that none of these is preferred over the other (but that given the level of non-flatness, the flat case is *theoretically* preferred, as a theoretical, non-observational, bias.)
2) The past light cone is finite-volumed, as it obviously is, pending some revolution in our understanding.
So far as I understand your point, you're stating that the past light cone is finite-volumed. If that's what you're saying, excellent, we're in agreement. The problem is you used the word "space", which I interpret to mean "space", and the spatial extent of the universe is basically untestable except by reference to the density of the universe; if it is at the critical density, then it is flat (and infinite), and if it is at less than the critical density then it is hyperboloidal (and infinite). If it is greater than the critical density, then it is spherical (and finite).
If we're not talking about spatial extent but instead talking about the 4-volume of the past light cone then we agree.
(I might also add that in GR the entire manifold is basically set -- the "future" of an event (mapped out by the future light cone), the "present" which are regions connected by spacelike geodesics, and then the "past" which is mapped out by the past light cone. From this point of view, which is one that GR forces us to (although it may very well be in contradiction of quantum mechanics), we can't tell if the universe is finite or not temporally. We do know that the past light-cone appears to be finite, but we cannot know what the future light-cone is since we do not have backwards-propagating photons to bring us information along them. The future light-cone could be finite, which would imply that the universe recollapses in the future. The likeliehood right now is that the future lightcone is infinite. That would mean that if we're going to generalise what we mean by "infinite" to four dimensions, we still end up with an infinite 4-volume. However, that's ultimately supposition and extrapolation.
(It can also be commented that actually we don't know that the past light cone itself is finite, since the theory cannot be used to propagate the light cone to, let alone beyond, the singularity. This doesn't actually mean that the universe formed at the singularity; it means that our theory cannot be used to propagate light that near to it - or anything else, for that matter. It may very well be that in a quantum theory of cosmology, there is no singularity, and that we have a bouncing universe. In that event, the past light cone is *not* finite and may very well, in fact, be infinite.
(This part of it though is academic in many ways too since as I commented in another post, we can't see back anything like to the singularity anyway. Our only probe is light, and the universe goes inconveniently opaque at the CMB, which is effectively a photo of the universe when it was a bouncing baby of 300,000 years or so. We simply cannot see beyond this, other than indirectly, unless we manage to observe the gravitational wave background or, even less plausibly, the neutrino background.)