Objection 1. It seems that God cannot know infinite things. For the infinite, as such, is unknown; since the infinite is that which, "to those who measure it, leaves always something more to be measured," as the Philosopher says (Phys. iii). Moreover, Augustine says (De Civ. Dei xii) that "whatever is comprehended by knowledge, is bounded by the comprehension of the knower." Now infinite things have no boundary. Therefore they cannot be comprehended by the knowledge of God.
Objection 2. Further, if we say that things infinite in themselves are finite in God's knowledge, against this it may be urged that the essence of the infinite is that it is untraversable, and the finite that it is traversable, as said in Phys. iii. But the infinite is not traversable either by the finite or by the infinite, as is proved in Phys. vi. Therefore the infinite cannot be bounded by the finite, nor even by the infinite; and so the infinite cannot be finite in God's knowledge, which is infinite.
Objection 3. Further, the knowledge of God is the measure of what is known. But it is contrary to the essence of the infinite that it be measured. Therefore infinite things cannot be known by God.
On the contrary, Augustine says (De Civ. Dei xii), "Although we cannot number the infinite, nevertheless it can be comprehended by Him whose knowledge has no bounds."
I answer that, Since God knows not only things actual but also things possible to Himself or to created things, as shown above (Article 9), and as these must be infinite, it must be held that He knows infinite things. Although the knowledge of vision which has relation only to things that are, or will be, or were, is not of infinite things, as some say, for we do not say that the world is eternal, nor that generation and movement will go on for ever, so that individuals be infinitely multiplied; yet, if we consider more attentively, we must hold that God knows infinite things even by the knowledge of vision. For God knows even the thoughts and affections of hearts, which will be multiplied to infinity as rational creatures go on for ever.
The reason of this is to be found in the fact that the knowledge of every knower is measured by the mode of the form which is the principle of knowledge. For the sensible image in sense is the likeness of only one individual thing, and can give the knowledge of only one individual. But the intelligible species of our intellect is the likeness of the thing as regards its specific nature, which is participable by infinite particulars; hence our intellect by the intelligible species of man in a certain way knows infinite men; not however as distinguished from each other, but as communicating in the nature of the species; and the reason is because the intelligible species of our intellect is the likeness of man not as to the individual principles, but as to the principles of the species. On the other hand, the divine essence, whereby the divine intellect understands, is a sufficing likeness of all things that are, or can be, not only as regards the universal principles, but also as regards the principles proper to each one, as shown above. Hence it follows that the knowledge of God extends to infinite things, even as distinct from each other.
Reply to Objection 1. The idea of the infinite pertains to quantity, as the Philosopher says (Phys. i). But the idea of quantity implies the order of parts. Therefore to know the infinite according to the mode of the infinite is to know part after part; and in this way the infinite cannot be known; for whatever quantity of parts be taken, there will always remain something else outside. But God does not know the infinite or infinite things, as if He enumerated part after part; since He knows all things simultaneously, and not successively, as said above (Article 7). Hence there is nothing to prevent Him from knowing infinite things.
Reply to Objection 2. Transition imports a certain succession of parts; and hence it is that the infinite cannot be traversed by the finite, nor by the infinite. But equality suffices for comprehension, because that is said to be comprehended which has nothing outside the comprehender. Hence it is not against the idea of the infinite to be comprehended by the infinite. And so, what is infinite in itself can be called finite to the knowledge of God as comprehended; but not as if it were traversable.
Reply to Objection 3. The knowledge of God is the measure of things, not quantitatively, for the infinite is not subject to this kind of measure; but it is the measure of the essence and truth of things. For everything has truth of nature according to the degree in which it imitates the knowledge of God, as the thing made by art agrees with the art. Granted, however, an actually infinite number of things, for instance, an infinitude of men, or an infinitude in continuous quantity, as an infinitude of air, as some of the ancients held; yet it is manifest that these would have a determinate and finite being, because their being would be limited to some determinate nature. Hence they would be measurable as regards the knowledge of God.
Source: Summa Theologica (New Advent)