Russell's Paradox and Pascal's Wager

In the foundations of mathematics you have to be careful. You need to set things up to be completely self-consistent, for if a singe paradox occurs, it can be used to prove anything. (This is because "false implies true" is a true statement.) Bertrand Russell noticed that a certain way of constructing sets led to a contradiction, now known as Russell's Paradox. Let X be the set of all sets which are not elements of themselves. Now the question is whether X is an element of itself. If it is, since X is defined to consist of those elements which are not elements of themselves, then X must have this property. In other words X is not an element of itself. Oops, that's a contradiction. On the other hand if X is not an element of itself, that means that X does not satisfy the condition for inclusion in X, which is that X is not an element of itself. That means X is an element of itself, again a contradiction. So the set X leads to logical inconsistency. The way this is resolved in modern mathematics is to limit the sorts of sets that can be formed.

There is a similar question in theology: can God create a rock so heavy that She can't lift it? If She is omnipotent, then She can do anything, so She can make a rock She can't lift. But that rock demonstrates Her lack of omnipotence. This is Russell's paradox in a different guise. It really demonstrates that the concept of omnipotence is logically inconsistent. I used to think this argument conclusively demonstrated that God doesn't exist, but in actual fact it shows that human language is an imprecise tool. This argument rules out an omnipotent God in this technical sense of the word omnipotent, which is concept that is not well-defined. It doesn't rule out an extremely powerful being, force or entity.

On the other side of the coin we have Pascal's wager. What Pascal basically said was that you might as well believe in God, because of the following four possibilities:

Believe? | God exists? | Payoff
-------------------------
Yes | Yes | Infinite
Yes | No | None
No | Yes | Eternal damnation
No | No | None

So if you want to maximize your payoff, you should believe in God. (Basically you have nothing to lose and everything to gain.)

However this argument is complete hogwash. The above chart does not give a complete list of all logical possibilities, and it assumes a great deal about the properties of a God that exists. It is logically possible that God exists, but that He only rewards those who don't believe in Him. It also give no hint about the "right" religion. What if you choose the wrong one?

So the Russell-type rock paradox is not a good argument against God's existence, and Pascal's wager is not a good argument for God's existence. The glib lesson to be drawn is that it's hard to get anywhere using logic alone. Additional input is needed.

And on that ambiguous note, I bid you adieu.