Can God create a rock so big that He can’t pick it up? To understand the problem, we need to understand set theory. But I don’t really want to talk about Russell’s paradox quite yet – a big problem with set theory as it’s taught is that it doesn’t respond to a felt need, it’s just plopped down at the beginning of a calculus or logic textbook without explanation. Here’s a bunch of self-evident stuff! Go calculate what the union of the intersections is!

I’m not going to tell you how to do set theory here. You can look that up if you want. I’m just going to try to explain a little bit about why it matters, why you should be interested in it, and how to apply some set-theory-ish rules of thumb to your own thoughts.

Think about the difference between these two arguments:

The king of Freedonia is Phillip III.

The husband of Mary Teller is Phillip III.

Therefore, the king of Freedonia is the husband of Mary Teller.

Milk is white.

Snow is white.

Therefore, milk is snow.

The second argument looks just like the first one – but the first one works and the second one doesn’t. Why?

Well, I’ve deliberately made it tricky by using the verb “is” in each case. “Is” is one of those tricky verbs whose meaning is very context dependent. Here’s a more precise formulation of the arguments:

The king of Freedonia is the same as Phillip III.

The husband of Mary Teller is the same as Phillip III.

Therefore, the king of Freedonia is the same as the husband of Mary Teller.

Milk is one of the things that are always white.

Snow is one of the things that are always white.

Therefore, milk ??? snow.

it’s not even clear which* *spurious consequence is supposed to follow from the second argument anymore. Is this a specious proof that milk and snow are identical, or that all milk is snow, or that all snow is milk, or just that some things are both milk and snow?

Here’s another paired example:

A shark is an aquatic animal.

An aquatic animal is a living thing.

Therefore, a shark is a living thing.

A knife is an item in my silverware drawer.

An item in my silverware drawer is a spoon.

Therefore, a knife is a spoon.

And with more specific wording:

Every shark is an aquatic animal.

Every aquatic animal is a living thing.

Therefore, every shark is a living thing.

At least one knife is an item in my silverware drawer.

At least one item in my silverware drawer is a spoon.

Therefore, ???

Or better yet:

There exists at least one item that is both a knife and in my silverware drawer.

There exists at least one item that is both in my silverware drawer and a spoon.

Therefore, ???

Set theory is a way to force yourself to use statements more explicit than “X is Y”, to prevent you from accidentally equivocating and “proving” that knives are spoons. Since math is all about proving possibly counterintuitive things, this is kind of important in math. But it’s also important whenever you’re making explicit compounded arguments of the (A, B, THEREFORE C) style.

In set theory you never say “X is Y.” You instead are always talking about whether something is a member of a set. For now, think of a set as nothing more specific than a collection of things. There’s a problem with this, but I’ll get to it later.

You can say that something is a member of a set, or that if something is a member of one set, then it must be a member of another, or that there is at least one thing that is both a member of set A and a member of set B, etc. You can also negate these things – you can say that there are no things that are both members of set A and set B. Think about these sentences, and how to make them more precise:

- A mouse is in this cage.
- A mouse is an animal.
- This mouse is Pinky.
- Pinky is in this cage.
- Dallas’s football team is heavier than the people in China.
- A dragon is not real.
- WEF wrestling is fake.

Here are some formulations that are a little more set theory-ish:

- There exists at least one thing that is both a member of the set (is a mouse) and a member of the set (things in this cage)
- Every member of the set (is a mouse) is a member of the set (is an animal).
- Every member of the set (this mouse) is a member of the set (Pinky). Also, every member of the set (Pinky) is a member of the set (this mouse).

*(A pithier way to say that one is: Something is a member of the set (this mouse) if and only if it is a member of the set (Pinky). This is an “identity” relation.)*

- Every member of the set (Pinky) is a member of the set (in this cage).
- The average of the weights of all the members of the set (members of Dallas’s football team) is higher than the average of the weights of all the members of the set (the people in China).

*(This one is tricky – the original statement is ambiguous, because it’s worded as a statement about the set, but what exactly are we saying is heavier than what? Are we saying that each Dallas Cowboy is heavier than each person in China? Or that the Dallas Cowboys, weighed all together, are heavier than the people in China, weighed all together? Or that the average weight of a member of the first set is greater than that of a member of the second? It’s important to be specific about things like this when talking about group characteristics.)*

- There are no members of the set (dragons) that are members of the set (real things).
- Every member of the set (WEF wrestling matches) is a member of the set (fake things).

Do you get the pattern? You never simply talk about how something “is” or “is not” something else, only about whether a member of set A is never, sometimes, or always a member of set B, and whether an assertion is true or false.

This can be helpful in avoiding getting into stupid arguments. If someone says, “a mouse is an animal,” do they mean that there is at least one mouse that is an animal, or that every mouse is an animal, or that something is a mouse if and only if it’s an animal?

If they mean that there’s at least one mouse that’s an animal, then finding a mouse that’s not an animal (like a computer mouse, or a robotic mouse) is **not** evidence against their point – all they have to do to prove it’s true is find at least one mouse that is an animal. But if you phrase it explicitly like that, it’s harder for them to equivocate and “prove” that a computer mouse is an animal.

Or maybe more realistically, if I “prove” that wiggins are thieves by showing you one wiggin who steals something (which only proves that there is at least one wiggin who is a thief), I might then pretend that you should draw the inference that some other wiggin is also a thief (which would only be valid if I had proved that **every** member of set “is a wiggin” is a member of set “is a thief”).

If they mean that every mouse is an animal, then finding an example of a mouse that is not an animal is a counterexample, but finding an example of an animal that is not a mouse, like a dog, is a **not** counterexample. If they’ve shown to your satisfaction that **all** members of set “mouse” are members of set “animal”, then you can go on and assume that’s true for each new mouse you encounter – but it doesn’t imply that all members of set “animal” are members of set “mouse”.

Finally, if they show “if and only if,” then you would have been able to prove them wrong just by showing them a dog. But if they convince you of this, then – and only then – you should accept the inference both ways.

It’s easy to lose track of this when you say things like “mice are animals” or “wiggins are thieves”, so it can be helpful to use set-theoretic language (which is almost as compact), like “MICE is a subset of ANIMALS.”

OK, so what does this have to do with God’s rocks? Well, sets are important, right? And we want to be correct when talking about important things – and sets help us be correct. So we want to describe sets using other sets. And talk about sets of sets!

Like you might want to talk about the properties of “sets that have no members.” Or “sets that have a finite number of members.” This is fine. But there are limits.

Let’s walk through one of them – the rock paradox. It’s usually stated as:

God is omnipotent. That means God can do any thing.

Making a rock so big that God can’t pick it up is a thing.

Therefore, God can make a rock so big that God can’t pick it up.

But picking up an arbitrary object that exists is also a thing.

Therefore God can pick up an arbitrary object that exists.

Now, let that arbitrary object be “a rock so big that God can’t pick it up.”

Then, God can pick up a rock so big that God can’t pick it up.

Now, if the existence of such a rock were impossible, then this wouldn’t be a problem. But we just said that God can make one.

But it’s not really a rock so big that God can’t pick it up, if God can pick it up.

Thus, the omnipotence of God implies a contradiction.

Therefore, there can be no omnipotent God.

The problem here seems to be using omnipotence in the definition of one of the powers. If you don’t allow that, then there’s no way to get the contradiction.

This brings up another set-theoretic principle: the “things” a set can be a collection of have to be well-defined, before we define any of the sets. So if we’re talking about puppies, and we already know what puppies are, without using sets of puppies in the definition, then we can talk about sets of puppies. But we can’t just define a collection of “puppies and sets of puppies,” before we know what the sets of puppies are. And the sets of puppies can’t themselves be defined until the puppies are defined.

So does the rock paradox follow this rule? No.

“God is omnipotent” can be rephrased as:

For every ability X, let there be a set (entities that have ability X).

Every omnipotent being is a member of every such set.

God is an omnipotent being.

Therefore, for every ability X, God is a member of the set (entities that have ability X.)

Now, this works for abilities like “walk on water” or “use set-theoretic notation” or “make ten commands”. Because those things are well-defined even if we don’t know about God.

How about “make a rock so big that God can’t pick it up.” Is this well-defined before we start talking about sets of abilities? No, because the ability is defined by a reference to what God can do, and what God can do is defined by a particular set of abilities. So a collection of abilities that includes “make a rock so big that God can’t pick it up” is simply not a well-defined collection that we can take sets of.

In fact, “make a rock so big that [someone] is not a member of set (entities that have the ability to pick up a rock of that size)” is never a first-order ability.

A set-theoretically valid definition of omnipotence would be something more like this:

Define some collection of “abilities,” none of which reference other powers or omnipotence directly.

Define omnipotence as the set of all these abilities.

Now, maybe “make an arbitrarily large rock” is one of the powers. And maybe “pick up an arbitrarily large rock” is a power. But none of the powers refer to each other, or to sets of powers, no matter how indirectly. So “make a rock so big that God can’t pick it up” isn’t an ability.

We can then think of sets of abilities, like the set of rock-making and rock-picking-up. Omnipotence is the ability-set that is contains all abilities.

Now we need to use a concept called a “subset.” A X is a subset of Y if every member of set X is also a member of set Y. For example, “Puppies” is a subset of “Animals,” and “Animals” is also a subset of “Animals, but “Animals is not a subset of “Puppies.”

So every ability-set is a subset of omnipotence.

Of course, that doesn’t mean that no one can make a rock so big that someone else can’t pick it up. Or even a rock so big that they themselves can’t pick it up. But that’s a statement about *combinations* of abilities and inabilities.

So what if you wanted to describe all the collections of abilities that don’t include** **certain abilities? Well, that’s a second-order** **set. Call it a schmet. So you might have a schmet of ability-sets that include walking on water, but not swimming. Or making a 32kg rock, but not picking it up.

Now let’s get back to that paradox. Can God make a rock so big that He can’t pick it up? How does that cash out when thinking about sets of abilities?

If someone can make a rock so big they can’t pick it up, that means that their ability-set is a member of a certain schmet. In particular, it’s the schmet that includes ability-sets where for some size X, they include the ability “can make a rock of size X”, and also do not include any ability “can pick up a rock of up to size Y”, for any Y>=X.

So the question is, is God’s ability set (omnipotence) a member of that schmet? The answer is no: omnipotence is not a member of the schmet “can make a rock so big you can’t pick it up.”

There’s no paradox, because a schmet is not an ability. Remember, we had to define all the abilities before defining any of the ability-sets, and we had to define the ability-sets before defining the schmets. So there can’t be an ability that refers to a schmet! And omnipotence is an ability-set, so its definition can’t refer to schmets either – it’s just the ability set that includes all abilities.

If you look up Russell’s paradox explained, you will find a similar exposition, except it’s less fun because it isn’t about God and rocks.