A friend recently posed this question:

Anyone willing to help me answer a Fedora Tipper on this? I’m not an expert on Set Theory.

“Consider the set of all of the things God knows; we’ll call it A. The power set, P(A), of A, the set of all subsets of A, has a higher cardinality than A [this is a theorem of set theory]. Now, consider the set B = {“x is a subset of A”|x∈P(A)}. There exists an obvious [bi]jection between the set of propositions in B and the elements of P(A). Thus, P(A) and B have the same cardinality. So, there exists a set B of truths containing infinitely more truths than the set of all truths known by God”

They assert over and over that this is a “proof against God”.

This argument deserves more to be laughed at than answered. That said,, here is an answer.

The powerset of a set is merely the set of all subsets of that set. So for the set {a} the powerset is {{}, {a}}, i.e. the set containing both the empty set and the set containing a. (Set theory is really hard to say. This helps it to sound more impressive than it is.) You can also write this as {∅, {a}}. It means the same thing.

Now, to see why the Fedora Tipper’s argument is ridiculous (apart from his defining B to be identical to the powerset of A and pretending he did something), let’s consider an example of a finite set of knowledge. Finite sets are easier to work with than infinite sets and the principles are the same. We’ll keep it simple and use a set containing 3 pieces of knowledge: {“11 > 4”, “Beavers mate for life”, “pie is round” }. Now let’s look at its powerset. For clarity (hah!) I’ll offset it:

{∅, {“11 > 4”}, {“Beavers mate for life”}, {“pie is round” }, {“11 > 4”, “Beavers mate for life”} , {“Beavers mate for life”, “pie is round”}, {“11 > 4”, “pie is round”}, {“11 > 4”, “Beavers mate for life”, “pie is round” }}

Now, obviously there are more elements in the powerset than in the original set. For finite sets, it’s actually 2^n elements, in this case 2^3=8. And it’s true that for infinite sets you get a different cardinality than the original set (“a higher order of infinity”). But this is completely irrelevant to the question of knowledge. Let’s go back to our example.

Suppose that we said that some man, call him George, knows the set above mentioned, i.e. he knows that 11 > 4, beavers mate for life, and pie is round. It would be ridiculous to say:

“Ha ha! But George doesn’t know that 11 > 4

andthat beavers mate for life! He only knows that 11 > 4, beavers mate for life, and pie is round! The ignorant fool!”

Which is what the Fedora Tipper above is trying to say. This disproves God about as much as it proves that George knows nothing about beavers. I.e. not in the slightest.

The moral of the story is that powersets are mathematical constructs, not real things. They don’t change reality. If you take a balding man and consider the powerset of his remaining hair, he doesn’t gain a full head of hair. There’s no more hair in the powerset of his hair then in the set of his hair; there are just a lot more ways to consider the little hair he has left.

Or, to invoke an old trope, you can’t use powersets to make a rock so big that God can’t lift it.

This is definitely a strange line of argument. I’m certainly not a theist, but even if there was no other flaw in the logic, it seems easy enough to defeat the whole thing simply by denying that there IS a set of all things which God knows.

This is precisely why we cannot consistently discuss “the set of all sets,” for example. If we have some set, A, which we believe to be the set of all sets, we can take its powerset, P(A). Since P(A) has a greater cardinality than A, it cannot be true that A was the set of all sets.

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