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The third way is taken from possibility and necessity, and runs thus. We find in nature things that are possible to be and not to be, since they are found to be generated, and to corrupt, and consequently, they are possible to be and not to be. But it is impossible for these always to exist, for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence. Now if this were true, even now there would be nothing in existence, because that which does not exist only begins to exist by something already existing. Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence — which is absurd. Therefore, not all beings are merely possible, but there must exist something the existence of which is necessary. But every necessary thing either has its necessity caused by another, or not. Now it is impossible to go on to infinity in necessary things which have their necessity caused by another, as has been already proved in regard to efficient causes. Therefore we cannot but postulate the existence of some being having of itself its own necessity, and not receiving it from another, but rather causing in others their necessity. This all men speak of as God.
ST 1. Q. 2, A 3.
The 13 step argument given by Robert in the interview:
1. We find in reality some things that are capable of existing and not existing, because they are found to be generated and to corrupt.
2. We call such beings possible beings.
3. A possible being cannot be the cause of its own existence.
4. This is so for two reasons: (1) It would already have to exist in order to cause its own existence, but if it already exists then it does not need to cause its own existence, and (2) If it caused its own existence then it would be both prior to itself and also not prior to itself, which is a contradiction and thus impossible.
5. Therefore a possible being must get its existence from a cause that exists external to it.
6. It is impossible that everything which exists is a possible being.
7. The reason is that nothing could have begun to exist in reality if everything were a possible being because a possible being only comes to exist through an already existing cause external to it, which would not exist if everything were a possible being.
8. But if nothing could have begun to exist in reality then nothing would have existed in the past and nothing would exist now, because “from nothing, nothing comes.”
9. But this is absurd because things exist now.
10. Thus not all things are possible beings—at least one necessary being must exist.
11. There are 2 ways for a being to be necessary: (1) it can get its necessity from another; (2) it can get its necessity from itself (per se).
12. If it has its necessity from another, then it requires a cause external to it.
13. An infinity of beings that get their necessity from another would not explain how anything came to exist, just as it is clear from the above that an infinity of possible beings would not explain how anything came to be.
Conclusion: There must be a cause that has of itself (per se) its own necessity (i.e., it does not receive necessity from another), and on which all other beings are, ultimately, dependent for their existence. And this we call God.
Robert mentioned a paper, "There must be a First: Why Thomas Aquinas Rejects Infinite, Essentially Ordered, Causal Series." Read it here: http://www.tandfonline.com/doi/abs/10.1080/09608788.2013.816934