His first proof is that whatever is moved is moved by something else. The chain of movement must lead back to a first mover, and the first mover is God.
His second proof is similar to the first. Nothing can be the cause of itself. Every cause may be caused by something else, so there must be a first cause of things, and the first cause is God.
His third proof is that not all being can be merely possible, some being must be actual. But there could not have been any actual being without some other being necessary for it to become actual. The chain of necessary being must lead back to some being necessary through itself, and this necessary being is God.
His fourth proof is that among all things, there can always be found one thing that is more true, more good, more perfect, etc than some other thing, but there must also be one thing that is the most true, most good, most perfect, etc. of all things, and which is the greatest in being. This being, which no other being can be greater than, is what we call God.
His fifth proof is that everything is governed by something else and acts for the sake of some end. The end toward which things move is governed by something that is intelligent, and so there must be some intelligent being by whom all things are governed or directed toward an end, and this intelligent being is God.
The first proof is called the argument from motion. The second proof is called the argument from causation. The third proof is called the argument from contingency. The fourth proof is called the argument from degree or gradation. The fifth proof is called the argument from design (or the teleological argument).
Aquinas's first three proofs are cosmological arguments (arguments regarding the nature of the universe). He is critical of the ontological argument (proposed by Anselm, whom he doesn't mention) that God is that than which nothing greater than can be thought, and that therefore God exists because if God existed only as a thought in the mind and not in actual reality, then something else could be greater than God, which would be impossible. Aquinas says it isn't self-evident that God is that than which nothing greater can be thought, and that some people may not necessarily think of God in this way. Furthermore, even if God were that than which nothing greater than can be thought, this would only confirm God's existence in the world of thought and not in the world of actual reality.
All of Aquinas's proofs are based on the premise that the chain of movement, causality, necessary being, comparative quality, or intelligence cannot be infinite and must have an end, and that when we arrive at the end of the chain we arrive at God. But if the universe is infinite, then there may be an infinite number of movers, causes, necessary beings, comparative qualities, and intelligences, and by definition there isn't any place where infinity ends. Thus, to say that there is God, under the premises adopted by Aquinas, is to say that the universe ends somewhere, and that it is not infinite. And indeed, Aquinas says in Question 7 of the Treatise, on "Divine Infinity," that God is absolutely infinite, but no created thing is absolutely infinite.
Whether the universe is actually finite or infinite may be both a metaphysical and a scientific question, and one for which we're still seeking an answer.
RESOURCES
Anselm, Proslogion, translated by M.J. Charlesworth (Notre Dame: University of Notre Dame Press, 1979.
Thomas Aquinas, "The Treatise on the Divine Nature," in Thomas Aquinas: Basic Works, edited by Jeffrey Hause and Robert Pasnau (Indianapolis: Hackett, 2014), pp. 36-49.
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