Objection 1. It seems that something else besides God can be essentially infinite. For the power of anything is proportioned to its essence. Now if the essence of God is infinite, His power must also be infinite. Therefore He can produce an infinite effect, since the extent of a power is known by its effect.
Objection 2. Further, whatever has infinite power, has an infinite essence. Now the created intellect has an infinite power; for it apprehends the universal, which can extend itself to an infinitude of singular things. Therefore every created intellectual substance is infinite.
Objection 3. Further, primary matter is something other than God, as was shown above (Question 3, Article 8). But primary matter is infinite. Therefore something besides God can be infinite.
On the contrary, The infinite cannot have a beginning, as said in Phys. iii. But everything outside God is from God as from its first principle. Therefore besides God nothing can be infinite.
I answer that, Things other than God can be relatively infinite, but not absolutely infinite. For with regard to infinite as applied to matter, it is manifest that everything actually existing possesses a form; and thus its matter is determined by form. But because matter, considered as existing under some substantial form, remains in potentiality to many accidental forms, which is absolutely finite can be relatively infinite; as, for example, wood is finite according to its own form, but still it is relatively infinite, inasmuch as it is in potentiality to an infinite number of shapes. But if we speak of the infinite in reference to form, it is manifest that those things, the forms of which are in matter, are absolutely finite, and in no way infinite. If, however, any created forms are not received into matter, but are self-subsisting, as some think is the case with angels, these will be relatively infinite, inasmuch as such kinds of forms are not terminated, nor contracted by any matter. But because a created form thus subsisting has being, and yet is not its own being, it follows that its being is received and contracted to a determinate nature. Hence it cannot be absolutely infinite.
Reply to Objection 1. It is against the nature of a made thing for its essence to be its existence; because subsisting being is not a created being; hence it is against the nature of a made thing to be absolutely infinite. Therefore, as God, although He has infinite power, cannot make a thing to be not made (for this would imply that two contradictories are true at the same time), so likewise He cannot make anything to be absolutely infinite.
Reply to Objection 2. The fact that the power of the intellect extends itself in a way to infinite things, is because the intellect is a form not in matter, but either wholly separated from matter, as is the angelic substance, or at least an intellectual power, which is not the act of any organ, in the intellectual soul joined to a body.
Reply to Objection 3. Primary matter does not exist by itself in nature, since it is not actually being, but potentially only; hence it is something concreated rather than created. Nevertheless, primary matter even as a potentiality is not absolutely infinite, but relatively, because its potentiality extends only to natural forms.
Source: Summa Theologica (New Advent)