By delaying this issue until Question 11 in the First Part of Summa Theologiae (and chapter 42 in the SCG, I), Thomas is indicating that it is easier to prove that there is at least one God than it is to prove that there is no more than one. It is also important to realize that the ‘one’ that we appeal to in proving God’s unity is not the number 1, but the concept of oneness that is convertible with being. God is supremely one precisely because He is supremely a being. We don’t count gods in the same way that we can count apples or doors. As we shall see, the nature of God excludes the very possibility of there being two or more gods. At the same time, since the oneness involved is not that of the numerical one, Thomas is leaving open the door for God’s being, in some sense, a plurality or multitude. This will help in working through both the problem of God’s ideas  and the Trinity.
 
In Summa Theologiae I, Q11, article 1, Thomas explains that the focal meaning of ‘one’ in ‘there is one God’ is that of being undivided. God is supremely indivisible, and this follows from His simplicity. Thomas argues that if oneness were not convertible with being, an infinite regress would result. If oneness is not equivalent to being, then it would have to be something that is added to being. But then we can ask what makes this addition one addition, and an infinite regress follows.
 
But what about multitudes, like crowds of people? If they exist, they must also be one. And, indeed, a crowd is one crowd. But how can something be both one and many? Isn’t that inconsistent? Thomas anticipates here an answer given much later by the German logician Gottlob Frege. A multitude is one in one way, and many in another. It is, for example, one crowd but many people. Thomas elaborates this point in article 2, again distinguishing between one as the principle of number and one as convertible with being. We can ask for the number of a crowd, and this question must appeal to some way of dividing the crowd—into families, or individuals, or human cells.  At the same time, a crowd must be, like anything that exists, in some more basic sense one thing. Some more recent metaphysicians, following the work of American logician George Boolos, have suggested that multitudes can exist without being one thing at all. Very large proper classes, like the class of all sets, for example, seems to be a real multitude that is not in any sense one thing. If this is right, it could create some difficulty for some of Thomas’s proofs (in article 3) for God’s oneness. Or, it might simply point to the fact that the oneness of God is consistent with His comprising a kind of multiplicity, so long as these things are not parts or attributes of God.
 
In objection 4, Thomas addresses the problem of the definition of one as undivided. Being undivided is a negative notion, signifying the absence of division. But being is perfectly positive. So, how can being be convertible with oneness? Thomas replies that division is prior to one only in the order of our understanding. Ontologically, being undivided is prior to being divided. It’s just that we are first aware of composite things before we are aware of their simple parts.
 
In article 3, Thomas offers three arguments for the oneness of God. The first argument appeals to God’s simplicity. God is made to be God by His divine nature, and that divine nature also makes Him exist as a particular being. For there to be two gods, there would have to be two divine natures, each of the same species. But for two natures to exist with the same species, there would have to be something responsible for making each distinct from the other. So, for example, two men can be two by virtue of being combined with two packets of prime matter. Two packets of prime matter have no actual nature of their own, and so they can be fundamentally or primitively distinct. The divine nature is an actual nature (it is maximally actual), and so two divine natures cannot be fundamentally distinct. Since God is identical to His own nature, there cannot be two instances of the divine nature, just as there cannot be two instances of a single angelic species.
 
In the second argument, Thomas appeals to God’s infinity. (This is a new argument, not present in the SCG.) In fact, he appeals to the infinity of God’s perfection, by which he means that nothing can be superior to God in perfection. Suppose that there were two such maximally perfect beings. In this argument, Thomas concedes (for the sake of argument) that there could be two distinct species of god. If there were two such species, something would have to differentiate them. One would have to have something that the other did not have. But this means that one would have to have some form of perfection that was lacking in the other. But God has all perfections. So, in fact, this argument actually appeals to God’s perfection, rather than His infinity.
 
Third, Thomas appeals to the apparent unity of the world. This is one of the relatively few cases in which Thomas appeals to some form of the Fifth Way—pointing to God as the cause of the world’s systematic harmony, the fact that the active and passive powers of the world’s created substances fit together in order to make a stable, scientifically intelligible universe. Thomas gives a more detailed version of this argument in Summa Contra Gentiles 1.42, paragraph 7.
 
In Summa Contra Gentiles 1.42, paragraph 5, Thomas also appeals to some details of Aristotle’s natural philosophy, especially the assumption that the movement of the heavenly spheres are regular and continuous. It’s not clear to me whether any of this can be salvaged, given the falsity of Aristotle’s astronomy.
 
In paragraph 8, Thomas argues that if there were two gods, at least one would have to be composite. But no composite being could be necessary through itself (as established in the Third Way). It’s not obvious here why one of the gods would have to be composite, but the two arguments that Thomas gives in the Summa Theologiae support this premise: either the two gods would belong to the same species, in which case each would have to have a part that individuates it from the other, or they would belong to two species, in which case each would have to add some differentia to their common genus.
 
Paragraphs 9-11 contain another interesting argument that appeals to God’s necessity of being. This argument involves a complex dilemma: if there were two necessary beings (each necessary per se), then either (1) the two differ by something required for the completion of the necessity of being, or (2) not.
 

• Assuming (2). (Paragraph 9) If not, then they are different by something accidental. This accident must have a cause (since all accidental additions must be caused), which is either (a) the essence of the necessary thing, or (b) some external thing. We can rule out (a), since both necessary beings would have the same accident, if it is a proper accident determined by their common essence. If (b), then the existence of two necessary beings depends on the action of this third thing. But then the existence of each thing would also depend on the third thing, and so neither could be necessary in itself.
•  
• Assuming (1). (Paragraph 10) That which differentiates the two necessary beings would be either (a) something included in the common nature of necessary being, or (b) something that differentiates the two natures into two species. Case (a) is impossible because in that case, the factor will be present in both necessary beings. In case (b), the common nature must be made actual by each of the two differentiae, in which case, each necessary being would derive its existence from something else. Hence, the two things cannot be necessary in themselves, nor could each be simple and identical to its own act of existence. This complex argument seems to be mirrored in the first two arguments in the Summa Theologiae. The SCG helps us to see that these are really two parts of a single argument, based on a dilemma (either one species of god or two).

 
Paragraph 12 involves a very simple argument, based on the thesis that God is identical to His nature and to His act of existence. If there were two gods, then each god would have the same divine nature. But this divine nature would then have to be identical to two distinct acts of existence. But nothing can be identical to two distinct things.
 
Paragraphs 13 and 14 contain another complex argument.
 

1. If x is a per se necessary being, then x’s necessary being belongs to x insofar as it is x. That is, the necessity of x is grounded by x’s being x.
2. Suppose there were two per se necessary beings, x and y. Then x’s being x would also consist in x’s not being y, since if x and y are distinct, x’s being x depends on its not being y.
3. Consequently, x’s necessity would belong to it insofar as x is distinct from y.
4. This means that x’s necessity depends on the existence of y.
5. Consequently, x’s necessity would not be per se. Contradiction
6. So, there cannot be two per se necessary beings.

 
Thomas defends premise 1 in paragraph 14. If x’s necessary being depends only on x, then x’s necessary being must belong to x insofar as it is x.
 
In paragraph 18, Thomas appeals to the fact that God is supreme being. Since being is convertible with oneness, God must be supremely one, and so undivided. This applies equally well to the divine nature.
 
Thomas also appeals in paragraph 20 to the superiority of monarchy as the form of government. Since God is the perfect governor of the universe, God must be one.

As I have argued, Thomas proposes that every actual being has its own act of existence (actus essendi). An act of existence is a real being, although it is not a substance. In this sense, the act of existence is something like an accident, since accidents are also real but non-substantial beings. Nonetheless, an act of existence is not an accident of its substance, since it is causally prior to that substance. The act of existence gives the substance actual existence. Substances do have (by virtue of their essences) a kind of being (potential being) that is independent of the act of existence. It is potential beings that make up the structure of mere possibilities (including Leibniz's 'possible worlds').
 
Do acts of existence themselves exist, and, if so, do we face an infinite regress, with each act of existence requiring a further act to give it existence? I think acts of existence do exist, but they exist in a way that is different from the way that substances and accidents exist. They exist simply by being acts of existence, while substances and accidents require distinct acts to give them actual existence. Substances and accidents can have potential existence, which is why they can be contained in mere possibilities. Acts of existence cannot have potential existence: they are intrinsically actual. Hence, no mere possibility contains any act of existence.
 
An act of existence must give whatever actuality a thing has. Hence, acts of existence, as such, are infinite, while all substances and accidents are finite. If acts of existence as such were finite in any way, then there would be possible entities that could not be actualized. But that is a contradiction in terms: to be a possible entity is to be possibly actualized. Hence, acts of existence must have the power to actualize everything, to the outermost limits of possibility. They must be intrinsically infinite.
 
Furthermore, if any act of existence were finite as an individual, then this finitude would define a particular kind of being. In other words, the act of existence would have a kind of essence built into it. But if the act includes such an essence, then it would make sense for the act to exist only in potentiality. But, as we've seen, this is impossible. Every act of existence must be actual.
 
Finally, existence itself can have no limit, since a limit implies some possible thing beyond the limit, but nothing can exist “outside” of existence. A limit is something that receives existence, and that limits the existence it receives. Existence itself cannot be or have a limit.
 
God is a pure act of existence, without any associated essence. Hence, God is absolutely infinite. He must possess every possible perfection without any limit whatsoever.
 
Now, infinity seems to be something negative—the lack of finite boundaries or limitations. However, when infinity is combined with God’s perfection, we do get something positive. God must have each perfection to a greater degree than any possible creature.
 
Infinity is, for Thomas, a complicated matter. For Aristotle, infinity is generally a bad thing. Being not-finite means lacking sharp boundaries. A non-finite thing is somewhat amorphous and shapeless, lacking any sharp definition. Shadows, clouds, or crowds would all be examples of in-finite, somewhat vague entities. Clearly none of this applies to God. God is definitely what He is, with no vagueness or amorphousness.
 
In addition, Aristotle generally associates infinity with potentiality. Nothing in the material world is ever actually infinite. A real line segment (say, one on the surface of a box), for example, does not contain an actual infinity of points. Instead, each of its internal points exists only potentially, since the line segment can be divided in any of an infinite number of quantitatively different ways. Similarly, the future is potentially but not actually infinite. We will never reach a day that is infinitely many days after this one. God, in contrast, is pure actuality. So, it seems that God’s essence and infinity should be incompatible.
 
But, of course, God’s infinity is not a quantitative infinity. He is not infinite by having an infinite number of parts, or by filling (in a physical way) an infinite volume of space or an infinite duration of time. The kind of finitude that material substances have is itself foreign to God. My form is made finite by my matter, since my matter is quantitatively limited prior to my generation. At the same time, my matter is also made finite by my form, since it is form that gives a definite size and shape to my matter. Thus, finitude emerges in the cooperation of matter and form.
 
There is nothing corresponding to matter in God. Hence, there is nothing that can put a limit to His “form”. And, conversely, there is nothing that could be limited by His form. Hence, God is in-finite.
 
When we turn to article 2 of Question 7, we see that creatures can be “relatively” infinite, but only God can be absolutely infinite. Matter, for example, is relatively infinite, in the sense that it is capable of taking on any substantial form for a material substance. A material substance can be relatively finite with respect to its possible accidents, like shape. Angels have no matter, and yet they too can be only relatively infinite, since their being (esse) is “received and contracted into a determinate nature.” Only God is pure, absolute being, and so only He can be absolutely infinite.
 
God’s infinity is an infinity of active power and maximum nobility. He and He alone has the full power of being itself.
 
In the Summa Contra Gentiles, I I.43, Thomas talks of God having an infinite “spiritual” magnitude. We shouldn’t read too much into the word ‘spiritual’ (spiritualis). The term is used in two quite different ways in scholastic philosophy. It can mean purely intellectual substances, like God, the angels, and the separated soul. But it can also mean the more ethereal and subtle aspects of the physical world. Light, for example, is sometimes classified as spiritual. So too are the ‘animal spirits’ which scholastic philosophers took to be circulating through the body (like our modern electrical impulses of the nervous system).
 
Thomas identifies two dimensions of spiritual magnitude: active power, and the goodness or completeness (perfection) of one’s nature. Clearly, God is infinite in both these respects.
 
In natural magnitudes, as we’ve seen, infinity is a kind of privation, a lack of definition and form. In God’s case, it is a pure, non-primitive negation, a simple absence of boundaries and limits, without any implication that those boundaries and limits ought to be there.
 
Things that have limits have those limits either due to the definition of their nature (reflected in their genus and species) or from something into which they are received. So, anything whose existence and essence are distinct must have a limit: the essence will have limits due to its definition, and the act of existence will be limited by being received by something limited (the essence). God has no definition or genus, and His existence is not received into any distinct nature. Hence, God’s existence is infinite.
 
Thomas also argues that God is infinite in the sense that He cannot be exceeded by anything else. Since He has no passive potentiality, nothing can be more actual than He is. (SCG 1.43, paragraphs 6-7, 9-11) He argues, in somewhat Platonic fashion, that nothing could be better than a being that is its own goodness (par. 9). He appeals to something like Anselm’s conception of God as the greatest conceivable thing in par. 10, arguing that there must be a greatest of all intelligible beings. These are interesting, but they fall short of showing that God is infinite, as opposed to simply the greatest of all finite beings.
 
In par. 11, Thomas argues that we can think of something greater than any finite being, and yet our intellect cannot think of something greater than its own cause. So, God (our intellect’s cause) must be infinite. (This is close to one of Descartes’ arguments for God in the Meditations, a sort of hybrid of the cosmological and ontological arguments.)