Thomas Aquinas, Compendium of Theology

There cannot be more than three persons in God. For the divine persons cannot be multiplied by a division of their substance, but solely by the relation of some procession; and not by any sort of procession, but only by such as does not have its term in something outside of God. If the relation had something external as its term, this would not possess the divine nature, and so could not be a divine person or hypostasis. But procession in God that does not terminate outside of God, must be either according to the operation of the intellect, whereby the Word proceeds, or according to the operation of the will, whereby love proceeds, as is clear from our exposition. Therefore no divine person can proceed unless He proceeds as the Word, whom we call the Son, or as love, whom we call the Holy Ghost.

Moreover, since God comprehends everything in His intellect by a single act of intuition, and similarly loves everything by a single act of His will, there cannot be several words or several loves in God. If, then, the Son proceeds as Word, and if the Holy Ghost proceeds as love, there cannot be several Sons or several Holy Ghosts in God.

Again: the perfect is that beyond which there is nothing. Hence a being that would tolerate anything of its own class to be outside itself, would fall short of absolute perfection. This is why things that are simply perfect in their natures are not numerically multiplied; thus God, the sun, the moon, and so on. But both the Son and the Holy Ghost must be simply perfect, since each of them is God, as we have shown. Therefore several Sons or several Holy Ghosts are impossible.

Besides, that whereby a subsistent thing is this particular thing, distinct from other things, cannot be numerically multiplied, for the reason that an individual cannot be predicated of many. But the Son is this divine person, subsisting in Himself and distinct from the other divine persons by sonship; just as Socrates is constituted this human person by individuating principles. Accordingly, as the individuating principles whereby Socrates is this man cannot pertain to more than one man, so sonship in the Godhead cannot pertain to more than one divine person. Similar is the case with the relation of the Father and the Holy Ghost. Hence there cannot be several Fathers in God or several Sons or several Holy Ghosts.

Lastly, whatever is one by reason of its form, is not numerically multiplied except through matter; thus whiteness is multiplied by existing in many subjects. But there is no matter in God. Consequently whatever is one in species and form in the Godhead, cannot be multiplied numerically. Such are paternity and filiation and the procession of the Holy Ghost. And thus there cannot be several Fathers or Sons or Holy Ghosts in God.