Mathematical Proof that God Does Not Exist?

I was recently scanning through my copy of The Cambridge Companion to Atheism and came across what purports to be a proof that an omniscient being cannot possibly exist. Since God is traditionally considered to be an omniscient being, this essentially amounts to a proof that God does not exist.

Following along with the text, I will proceed with a reductio proof that omniscient beings cannot logically exist. Assume that there is an omniscient being. Such a being would, by definition, know the set of all truths. Therefore, there must exist a set of all truths; call this T. Now, consider the power set of T (the set of all subsets of T); call this PT. By Cantor's theorem, PT will always contain "more" members than are in T (technically, the cardinality of PT will always be greater than T.) But for each member x of PT, there is at least one propositional truth; namely, that x is in PT. So, there must be some truths which are not in T. Therefore, T cannot be the set of all truths. This is a contradiction -- and we must conclude that the set of all truths does not exist. This implies that omniscient beings cannot logically exist.

Note that this also rules out a certain kind of omnipotent being. If by "omnipotent being", one means to refer to a being that can do absolutely everything, then surely this rules out that sort of being as well. Why? An omnipotent being of that kind would  be able to know everything, since knowing any given thing is something to do. But if the proof just given is correct, then surely it is impossible to know everything. Ergo, an omnipotent being of that kind cannot exist.

However, the theist might be able to save omnipotence. Since Aquinas, the word "omnipotent" has been commonly taken to refer to a being that could perform any logically possible task and not one that could do absolutely anything. I wouldn't be surprised if this were even written in the Catholic catechism somewhere.

This is because, standardly, logically contradictory tasks are not understood to be tasks at all. As it is usually construed, propositions which may be taken to refer to logically contradictory tasks are actually understood to be contentless (they do not refer to any tasks at all, even ones performable only by God.)

There are some alternative thoughts, but most theologians would go at least this far. Thus, contrary to the thoughts of some lay believers, most theologians understand God to be bounded by logic. You could certainly take an alternative route, but that would be to discuss a different sort of thought than the one we are broaching here.

Besides, if such a being were truly beyond any kind of human reasoning, it's existence would be presumably beyond our ability to know, and therefore we could never be justified to think that just a being exists. Arguably, to be maximally charitable to the theist, we should be willing to say that an omnipotent being would still be bound by logic.

However, a powerful response to this objection can be made. Knowing a given proposition isn't logically impossible, even for human beings. Thus, knowing any given true proposition would surely not be a burden on an omnipotent being.

But since we have already shown that no being could know all true propositions (since there is no such set), there seems to be a contradiction here which escapes Aquinas' re-definition of "omnipotence". We have constructed an impossible task, but which is constructed from an infinite series of possible tasks.

Thus, no omnipotent beings can exist -- even under Aquinas' re-definition -- because that would contradict our previous result on the non-existence of omniscient beings.
















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References

Grim, P. (2007). Impossibility arguments. In Cambridge Companion to Atheism (pp. 199-214). New York, NY: Cambridge University Press.