God is busy setting up the universe. (If you’d prefer I not be flippant about god, please take it to be an alternate universe…)
God starts by creating angels. Since she has a potentially infinite set of tasks to perform, she creates infinitely many angels. She can do it, she’s god. And she is a “do it right the first time” kind of god: rather than create some angels now, and then realize she needs more later, she will create angels for every conceivable task. Hence the infinity of angels.
God foresees that the angels may want to form associations with each other. For example, all the angels responsible for the animals might want to form their own club; all those responsible for properties of numbers might want to form another; and so on.
Now, as I said, god does not like to do things twice. So, thinking ahead, she creates ALL possible clubs. For every possible collection of two or more angels that could exist, she enters it in a catalog, gives it a provisional name, prints up membership cards, builds a clubhouse. (She doesn’t give it a final name, because she doesn’t know what that particular collection of angels might want to form a club for. The angels do have some free will, after all.)
Being a fan of order, God decides to assign a leader to each potential club. For every potential club, there will be some angel that is its leader. Should that collection of angels choose to form a club in future, the clubhouse will already be built, and the leader will already be chosen. It will fall to the leader to collect membership dues, keep careful minutes, and fill out the necessary forms. (God likes order, as I said.) Since the duties can be somewhat onerous, god decides that each angel will lead only one potential club at the most. No angel is responsible for two or more clubs, potential or actual.
As she begins the assignments, she realizes that it will not always be possible for the leader of a club to be a member of that club!
Problem: Pick any three angels, let’s call them A, B, and C. (It wasn’t until later that angels got into fancy names.)
a) List all the possible clubs that A, B, and C could potentially form.
b) Explain why it is not possible that every potential club can have a leader who belongs to that club.
Well, no matter. God can start assigning leaders to some clubs who do not belong to the clubs they lead. There are infinitely many angels, after all, to assign to these infinitely many tasks. So the catalog is written, with all potential clubs of angels, and for each potential club, an angel leader, who will be called upon should that club decide to form. The leader may or may not be a member of the club that they are assigned to lead; no matter.
Time goes on, various clubs form, and leaders are called upon, referring to the infinite catalogue. All is well.
Then one day, some angels are talking on one of their AOL message boards (these were still the early days of the universe). The group who are talking consists of several angels, who each led groups to which they did not belong. They felt that it was difficult to lead a group, if you did not belong to it. The group felt that you did not understand their issues, and (the angels suspected) did not always invite you to their pizza parties.
These angels, (let us call them ostracized angels, in view of their social difficulties), decided to form a club. It would be a club for all angels who did not belong to the clubs (potential or actual) that they led: the club of ostracized angels. There would be much to discuss; no longer would they feel so left out.
Excited, the angels made a list of all angels who were ostracized in this sense. They took the list, and looked it up in god’s catalogue: recall, in god’s infinite foresight, she has provided ahead for ALL possible clubs that might form in future. Somewhere, waiting for them, was a clubhouse, all fitted out with chairs and a mediocre coffee maker.
And, most importantly, (for angels love hierarchy), they excitedly looked in god’s infinite catalog to find which angel she had chosen to the lead them.
There, they made a troubling discovery.
Problem: What was the troubling discovery? (Think about it for a bit, and then scroll down to the bottom of this page for a hint.)
Problem: What is the meaning of this disturbing outrage? What can we conclude from it?
Hints follow, don’t scroll down until you’re ready for the hint
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Are you sure you’re ready? I think you might still be able to figure it out on your own.
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Are you really sure? There’s time to turn back.
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OK, OK, I could do this all day.
Here’s the hint: it would be terribly ironic if the leader of the ostracized angels was herself ostracized, would it not? Is she? Or isn’t she?
