[Zeeshan]

For the more mathematical minded here's a set-theoretical approach by Robert K. Meyer. I don't have access to the original article of God Exists! which appeared in Nous journal but here's the explanatory proof.

Link

Let S be the set of all entities that exist (or have existed). Define the relation <= on S by saying that x<=y if and only y is a cause of x. By verbal fiat we will define x to be a cause of x for all x in S (if we do not accept this definition, our assumptions will be slightly different; however, it is clear that the existence of x is necessary and sufficient for the existence of x, and that the existence of x is never strictly temporally posterior to that of x, so calling x a cause of itself is not such a bad idea.) Then, <= is transitive, and moreover if x<=y and y<=x, then x=y (i.e., there are no circles of causation). Hence, <= defines a partial ordering on S.

Premiss 1: The set S is a set in a universe of sets satisfying the Axiom of Choice.

Premiss 2: The set S is inductively ordered.

(Definitions will be given below.) Define a God to be an element G of S with the property that if X is another element of S such that G<=X, it must be the case that X=G. Define a divine creator of an element X in S to be a God G such that X<=G. (By definition, each God is a divine creator of itself.)

Theorem: If Premisses 1 and 2 hold, then a God exists and, moreover, any existent entity then has a divine creator.

The Theorem is a deductive consequence of the premisses together with some appropriate axioms of set theory (just use Zorn's Lemma for the proof). No claim is made in the Theorem about uniqueness (or about the divine creator of X being the same for every X other than him).

To specify the premisses further, for Premiss 1 we need an appropriate set theory in which we can embed S. Note that each element of S is to be an ens realis. This is important since if we allowed either an ens mentalis or a potential being to be in S, then it would not be clear that we can embed S in a set theoretic universe (since then the set S might contain all other sets, etc.) And we need that the Axiom of Choice should hold, which means simply that for every set T whose elements are a pairwise disjoint collection of non-empty sets, it is possible to choose a set V which contains precisely one element from each of the sets in T. (If T is finite, this is obviously true. The reason the Axiom of Choice is not completely obvious is that while for each of the sets in T we can choose an element of that set, it is not clear what is meant by making an infinite number of such arbitrary choices. Perhaps after all it might not be possible to specify a rule for making the choice, so that if sets are to be defined by rules deciding if a given element is in the set or not, then it may not be so clear that the Axiom holds.)

Premiss 2, however, is the one which is metaphysically loaded. This premiss is a very strong statement of the principle that each entity has a cause. More precisely, the assumption of inductive ordering says that given a chain T in S, there exists an entity X in S such that X is the cause of every entity in the chain T. By a chain T in S is meant a set of elements of S with the property that if x and y are distinct elements of T, then either x<=y or y<=x, so that T is totally ordered by causation. One might think intuitively that given Premiss 2, it is obvious that there is a first cause for every entity, and so the Theorem is essentially of no significance. However, in point of fact, from Premiss 2 it does not seem to immediately follow that the Theorem holds: one appears to need (unless one can instead make some simplifying assumption like that S is finite, or maybe that S is countable, or some other nice assumption that would have to be metaphysically justified) Premiss 1, and the non-trivial Zorn's Lemma. Given Premiss 1 and Zorn's Lemma, the Theorem follows.

If one is willing to accept S as a set and to accept the Axiom of Choice, then the proof of the existence of a God (and of the claim that each entity has a divine creator) needs only that Premiss 2 should hold. Now, Premiss 2 is stronger than the usual claim of a causal nexus, namely that every element has a cause (in our current setting, this usual claim would be trivial, since each element is a cause of itself). Meyer tries to argue that Premiss 2 is the right way of formulating the intuitive claim "Everything has a cause", because it is the one way of taking into account that explaining the position of a ball at time t in terms of the positions at times t_1>t_2>t_3>... (where t_1
To justify Premiss 2 in full generality would appear to require some metaphysical argument like the one that Aquinas or Aristotle tried to use in their versions of the Cosmological Argument. While one might think that because of this nothing is gained by Meyer over and beyond the position that Aquinas and Aristotle were in, this is not quite so. If Premiss 1 be accepted (and it is not very unreasonable to accept it), then the argument shows beyond Premiss 2 everything in the Cosmological Argument is correct. To prove or disprove Premiss 2, however, is non-trivial. Obviously, until a justification of Premiss 2 is given, the "proof" remains inconclusive.

Final remark: It is interesting to note that Premiss 1, namely the Axiom of Choice, is not needed in the Theorem if we make the additional assumption that there is no causal overdetermination, namely if we assume that if A causes X and B causes X, then it follows that either A causes B or B causes A. For, fix X. Let S be the set of all entities that cause X. Then, by the assumption that there is no causal overdetermination, it follows that S is a chain. But by Premiss 2, it follows that there is an entity G which causes every entity in S. Suppose now that H causes G. Then, because G causes X, it follows that H causes X, and hence H is in S, so that G causes H since G causes every entity in S. Therefore we see that H causes G and G causes H, so that G and H coincide. This proves that G is a God, and indeed a divine creator of X.