Argument from Contingency

The argument from necessity/contingency does not require the world to have had a beginning in time, which is what makes it an especially  powerful proof. It has two parts. First, it is shown that it's impossible for absolutely everything that exists to be contingent. For something to be contingent means that its existence depends on the existence of  something else.

        A exists only if B exists.
        B exists only if C exists.
        C exists only if D exists., etc.

If this regression were to go on infinitely, neither A nor B nor C  etc. can exist, since there will always be lacking some ultimately  necessary existent upon which all the others depend. This is the  metaphysical analog to the impossibility of infinite regression in a  logical proof.

        P is true only if Q is true.
        Q is true only if R is true.
        R is true only is S is true. etc.

If this goes on infinitely, you cannot establish that P is true. In  fact, in modern logic, if it can be shown that the proof of a theorem  requires infinite steps, that is the same as showing it is unprovable.

Getting back to the existential regression, note that this is not necessarily a regression in time. A cause can be temporally simultaneous with its effect. This is forgotten by modern physicists,  who are so accustomed to using spacetime diagrams to determine the  range of possible causal links between events (i.e., event A cannot  cause B if it is in the future of B), that they tend to confine  causality to spatiotemporal relation, though this is just a shorthand.  In the 3rd proof, we don't even need to insist that B is the cause of  A, only that the existence of A depends on the (simultaneous)  existence of B.

Aristotle's cosmos could be eternal with an only finite regression,  since there were only finitely many eternally moving spheres, the  outermost being the first mover from which all others received their  motion. This is compatible with St. Thomas' argument, except now we  are speaking of existence rather than motion. You might say that the  existence of an animal depends on the existence of its constituent 
chemicals, which in turn depend on some other contingency, etc.  Ultimately, however, for existence to be grounded, there must be some  necessary existent. Even atheists effectively acknowledge as much when  they posit spacetime or the multiverse or some other substratum as  some ultimate existent that must be, so that there cannot be nothing.

Where St. Thomas goes beyond Aristotle and to a truly theistic  argument is that he does not stop at a necessary existent and leave it  at that. He then asks whether that existent's necessity comes from  itself or some other necessary being. For example, in a fatalistic  universe, all beings would be necessary, yet some necessities would be  determined by others. Ultimately, there must be some being that is not  only necessary, but derives its necessity from itself and nowhere else.

To St. Thomas and other Scholastics, it would be too obvious for  comment that even eternally persisting matter could not derive  necessity from itself. Matter cannot exist by itself, but needs form,  and needs to receive its form or determination from somewhere. Nor can  a material being with some eternally fixed form be self-necessary,  since its materiality implies the possibility of change, so there must  be something preventing it from changing. Then there should be some  immaterial being that is self-necessary. In this regard St. Thomas has  only to show, as he does elsewhere, that angels or other purely  spiritual beings are not self-necessary.

Whatever remains, though we can't describe or define it, is something  that does not receive form or any determination of being or even  simple existence from anywhere but itself. Not only must it exist, but  this necessity derives from itself. Such a being deserves the name of  God if anything does.