"The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt." -- Bertrand Russell

Wednesday, July 11, 2012

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.



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


  1. Yeah, I'm going to need a minute to look up a few things regarding this before I respond with some snarky comment about how there is something wrong with your logic.

  2. I must admit the mathematics behind Cantor's Theorem seem to be beyond what I can figure out at the moment, but it seems to me that the argument you are trying to make goes something like this.

    1. Assume an Omnipotent Being exists.
    2. Omnipotent Being knows not only all truths, but all possible truths.
    3. Given 2, Omnipotent being knows all truths including those greater than the truth.
    4. For each possible truth, there msut be some truths which are not in the actual Truth.

    Therefore, Truth cannot be the set of all truths... And therefore any set of all truths does not exist.

    This has certainly been a case of Reducto ad Absurdum.

    Firstly: Omnipotence, Omniscience, etc, are all relative. Your argument purports that G-d would be a part of the existence over which he has omnipotence, which is not necessarily the case. For parallel: As a Scientist, I have the power to do a great many things to a petri dish. Using advanced (beyond current human capability) computers, I could determine every possible quantum state for the petri dish. No such quantum states includes me, as it is physically impossible for the entirety of my body to exist within a 25 square centimeter petri dish.

    What your argument does suggest, however, is that is impossible for reality as we know it to contain any such deity. But then there are many more arguments that support that argument.

  3. "In reply, Plantinga (Grim and Plantinga, 1993) holds that knowledge of all truths does not require the existence of a set of all truths. He notes that a parallel argument shows that there is no set of all propositions, yet it is intelligible to say, for example, that every proposition is either true or false. A more technical reply in terms of levels of sets has been given by Simmons (1993), but it goes beyond the scope of this entry. See also (Wainwright 2010, 50–51)."