See the first post in this series for an explanation and guide of our tour of
This may be proved in three ways. The first…
Summa Contra Gentiles. All posts are under the category SAMT
There are different kinds of infinity, a non-simple concept. But infinity is nothing in St Thomas’s hands. Those who know something of the history of mathematics will enjoy his arguments. See  &  in particular.
Chapter 43 That God is Infinite
 Since, as the philosophers teach, “the infinite accompanies quantity,” infinity cannot be attributed to God on the ground of multitude. For we have shown that there is only one God and that no composition of parts or accidents is found in Him. Nor, again, according to continuous quantity can God be called infinite, since we have shown that He is incorporeal. It remains, then, to investigate whether according to spiritual magnitude it befits God to be infinite.
Notes God is not a body, is not material; these things were proved. God is no infinite in a countable way, as, say, the integers are.
 We speak of spiritual magnitude with reference to two points: namely, power and the goodness or completeness of one’s own nature. For something is said to be more or less white according to the mode in which its whiteness is completed. The magnitude of its power likewise is measured from the magnitude of its action or its works. Of these magnitudes one follows the other. For, from the fact that something is in act it is active, and hence the mode of the magnitude of its power is according to the mode in which it is completed in its act. Thus, it remains that spiritual beings are called great according to the mode of their completion. Augustine himself says that “in beings that are great but not in bulk, to be greater is the same as to be better.”
Notes St Thomas is prepping us to recall God is pure act, i.e. He has no potentiality. Recall that having potentiality, which is in a way the ability to change, accounts or the lack of perfection. Only that which has no potentiality is perfect.
 We must therefore show that God is infinite according to the mode of this sort of magnitude. The infinite here will not be taken in the sense of privation, as in the case of dimensive or numerical quantity. For this quantity is of a nature to have a limit, so that such things are called infinites according as there is removed from them the limits they have by nature; which means that in their case the infinite designates an imperfection. But in God the infinite is understood only in a negative way, because there is no terminus or limit to His perfection: He is supremely perfect. It is thus that the infinite ought to be attributed to God.
 For everything that according to its nature is finite is determined to the nature of some genus. God, however, is not in any genus; His perfection, as was shown above, rather contains the perfections of all the genera. God is, therefore, infinite.
Notes God Is Not In A Genus: “Whatever is in a genus differs as to being from the other things contained in the same genus: otherwise a genus would not be predicated of several things.” Etc.
 …Furthermore, in reality we find something that is potency alone, namely, prime matter, something that is act alone, namely, God, as was shown above, and something that is act and potency, namely, the rest of things. But, since potency is said relatively to act, it cannot exceed act either in a particular case or absolutely. Hence, since prime matter is infinite in its potentiality, it remains that God, Who is pure act, is infinite in His actuality.
Notes Prime matter is a sort of goo from which all form has been removed. Formless building (shapeless, as it were) blocks of the universe; prime matter can become, i.e. be formed, into anything—though there is much more to it (see here for example).
 Moreover, an act is all the more perfect by as much as it has less of potency mixed with it. Hence, every act with which potency is mixed is terminated in its perfection. But, as was shown above, God is pure act without any potency. He is, therefore, infinite.
Notes An analogy. Perhaps it helps to think of perfection as a ratio of act to potential, and as potential progresses to the limit of 0, the ratio jumps to infinity. Again, an analogy.
 …Then, too, what has a certain perfection is the more perfect as it participates in that perfection more fully. But there cannot be a mode of perfection, nor is one thinkable, by which a given perfection is possessed more fully than it is possessed by the being that is perfect through its essence and whose being is its goodness. In no way, therefore, is it possible to think of anything better or more perfect than God. Hence, God is infinite in goodness.
Notes Recall that God’s essence and existence are one. The limit analogy is useful here, too.
 Our intellect, furthermore, extends to the infinite in understanding; and a sign of this is that, given any finite quantity, our intellect can think of a greater one. But this ordination of the intellect would be in vain unless an infinite intelligible reality existed. There must, therefore, be some infinite intelligible reality, which must be the greatest of beings. This we call God. God is, therefore, infinite.
Notes All will accept the first premise, but the second? This states a moderate realist and not a nominalist or Platonic view, a view which is being taken up (again) in mathematics, notably by Jim Franklin and others. And by you, dear reader, if you’ve ever used mathematics to understand the real world. Best short introduction here (pdf): “The neglect of epistemology accounts for two strange absences in the philosophy of mathematics: understanding (and mathematics is where one first goes to experience pure understanding) and measurement (the primary way in which mathematics joins to the world).”
 Again, an effect cannot transcend its cause. But our intellect can be only from God, Who is the first cause of all things. Our intellect, therefore, cannot think of anything greater than God. If, then, it can think of something greater than every finite thing, it remains that God is not finite.
Notes Don’t forget our intellect is not material. We are not our brains. And if not material, it has to come from something. St Thomas tackles this argument in the next chapter (next week for us, so belay questions for now).
 There is also the argument that an infinite power cannot reside in a finite essence. For each thing acts through its form, which is either its essence or a part of the essence, whereas power is the name of a principle of action. But God does not have a finite active power. For He moves in an infinite time, which can be done only by an infinite power, as we have proved above. It remains, then, that God’s essence is infinite…
 Each thing, moreover, is more enduring according as its cause is more efficacious. Hence, that being whose duration is infinite must have been from a cause of infinite efficaciousness. But the duration of God is infinite, for we have shown above that He is eternal. Since, then, He has no other cause of His being than Himself, He must be infinite…
Note A sweet little argument. Don’t forget eternal means, essentially, outside time.
 The sayings of the most ancient philosophers are likewise a witness to this truth. They all posited an infinite first principle of things, as though compelled by truth itself. Yet they did not recognize their own voice. They judged the infinity of the first principle in terms of discrete quantity, following Democritus, who posited infinite atoms as the principles of things, and also Anaxagoras, who posited infinite similar parts as the principles of things. Or they judged infinity in terms of continuous quantity, following those who posited that the first principle of all things was some element or a confused infinite body. But, since it was shown by the effort of later philosophers that there is no infinite body, given that there must be a first principle that is in some way infinite, we conclude that the infinite which is the first principle is neither a body nor a power in a body.
Notes St Thomas means to quote Aristotle (here, Section IV):
Belief in the existence of the infinite comes mainly from five considerations:
(1) From the nature of time-for it is infinite.
(2) From the division of magnitudes-for the mathematicians also use the notion of the infinite.
(3) If coming to be and passing away do not give out, it is only because that from which things come to be is infinite.
(4) Because the limited always finds its limit in something, so that there must be no limit, if everything is always limited by something different from itself.
(5) Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by everybody-not only number but also mathematical magnitudes and what is outside the heaven are supposed to be infinite because they never give out in our thought.
Aristotle also says, “Also, if void and place are infinite, there must be infinite body too, for in the case of eternal things what may be must be.” If the universe was already infinitely old, we wouldn’t be here.