There are good reasons to think that actual infinities are possible in the real world. In the first place, while the size and shape of the universe are not settled issues, the generally accepted theory fits better with the idea that the universe is physically infinite than with the idea that it is finite.

Likewise, the universe is certainly larger than the size of the observable universe, namely about 93 billion light years in diameter. Supposing you have a probability distribution which assigns a finite probability to the claim that the universe is physically infinite, there is no consistent probability distribution which will not cause the probability of an infinite universe to go to 100% at the limit, as you exclude smaller finite sizes. But if someone had assigned a reasonable probability distribution before modern physical science existed, it would very likely have been one that make the probability of an infinite universe go very high by the time the universe was confirmed to be its present size. Therefore we too should think that the universe is very probably infinite. In principle, this argument is capable of refuting even purported demonstrations of the impossibility of an actual infinite, since there is at least some small chance that these purported demonstrations are all wrong.

Likewise, almost everyone accepts the possibility of an infinite future. Even the heat death of the universe would not prevent the passage of infinite time, and a religious view of the future also generally implies the passage of infinite future time. Even if heaven is supposed to be outside time in principle, in practice there would still be an infinite number of future human acts. If eternalism or something similar is true, then an infinite future in itself implies an actual infinite. And even if such a theory is not true, it is likely that a potentially infinite future implies the possibility of an actual infinite, because any problematic or paradoxical results from an actual infinite can likely be imitated in some way in the case of an infinite future.

On the other hand, there are good reasons to think that actual infinities are not possible in the real world. Positing infinities results in paradoxical or contradictory results in very many cases, and the simplest and therefore most likely way to explain this is to admit that infinities are simply impossible in general, even in the cases where we have not yet verified this fact.

An actual infinite also seems to imply an infinite regress in causality, and such a regress is impossible. We can see this by considering the material cause. Suppose the universe is physically infinite, and contains an infinite number of stars and planets. Then the universe is composed of the solar system together with the rest of the universe. But the rest of the universe will be composed of another stellar system together with the remainder, and so on. So there will be an infinite regress of material causality, which is just as impossible with material causality as with any other kind of causality.

Something similar is implied by St. Thomas’s argument against an infinite multitude:

This, however, is impossible; since every kind of multitude must belong to a species of multitude. Now the species of multitude are to be reckoned by the species of numbers. But no species of number is infinite; for every number is multitude measured by one. Hence it is impossible for there to be an actually infinite multitude, either absolute or accidental.

We can look at this in terms of our explanation of defining numbers. This explanation works only for finite numbers, and an infinite number could not be defined in such a way, precisely because it would result in an infinite regress. This leads us back to the first argument above against infinities: an infinity is intrinsically undefined and unintelligible, and for that reason leads to paradoxes. Someone might say that something unintelligible cannot be understood but is not impossible; but this is no different from Bertrand Russell saying that there is no reason for things not to come into being from nothing, without a cause. Such a position is unreasonable and untrue.