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For the Birds

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Achilles and the Tortoise have been admiring Escher's “Sky and Water I,” featuring birds metamorphosing into fish and vice versa. As they wander home, to their surprise, they encounter a flock of birds flying above them!

Achilles: What a pleasant coincidence!

Cardinal: Coincidence?

Tortoise: Er—never mind. I, as you can likely tell, am a Tortoise, and this is my friend Achilles. It's a pleasure to meet you.

Owl: Hello there. I am an Owl, and these are my friends the Cardinals. But I really must be going back to work.

Second Cardinal: Excuse me!

Owl: What?

Second Cardinal: How very dare you?

Owl: Er...pardon me.

Second Cardinal: Don't you even start lumping me in with her over there, for she is just a plain old Regular Cardinal, whereas I am a distinguished Singular Cardinal. Pfft!

The Singular Cardinal fluffs her wings and spirals above the regular cardinal with an obvious sense of self-importance.

Achilles: I'm sure she didn't mean any trouble. I'll just be going, then.

Tortoise: Easy for you to say, giant. Down here all you wingy sorts look the same. Why should I care whether you get all puffed-up?

Achilles: I don't think this is a good—

The Singular Cardinal swoops down, obviously affronted.

Singular Cardinal: Well, let me tell you...

Achilles decides to hustle off below the busy-looking owl rather than stick around to hear any more of this.

Singular Cardinal: Don't you know how to count to one hundred?

Tortoise: Um, is this a trick question?

Singular Cardinal: Maybe?

Regular Cardinal: One, two, three...

Tortoise: Every kid knows that you go “one, two, skip a few, ninety-nine, one hundred.”

Singular Cardinal: Exactly! You and I, such sophisticated, erudite mathematicians, know all about these tricks of the trade. To reach truly far-off, infinite objects, you can employ similar tricks without having to take as many infinite steps as you would expect. Whereas regular cardinals, like her, well...I'm afraid they'll just have to count the slow way, and they'll be counting a long, long, time.

Tortoise: But the slow way is the best way.

Singular Cardinal: Oh, be like that!

Meanwhile, Achilles is following the Owl towards her workplace.

Achilles: So where are you off to in such a rush?

Owl: Parliament, of course!

Achilles: What?

Owl: Haven't you ever heard of a parliament of owls?

Achilles: Well, yes, but...

Owl: We have a lot of things to legislate. Who has to prune the Gentzen trees, whether we can find an infinite path through the other forests, that kind of thing.

Achilles: I see. And what party do you belong to?

Owl: Party? I party whenever I wish, usually when the ostriches have us over. They are rather sadly ignorant of the political situation, you know, but they sure do have some great big nests!

Achilles: ...Sorry, I mean, how do you vote?

Owl: Oh, that. Well, I vote as I wish, of course, and if the chance comes to join a coalition to get an omnibus bill I believe in passed, then I take it.

Achilles: An omnibus?

Owl: The biggest bus of all! We owls are quite fervent believers in public transportation. Which is a bit ironic because we can fly, you see, but I suppose it's important to hover on principle.

Achilles: I'm not sure I understand.

Owl: Well, maybe I can explain it to you. We have just a few articles that make up our constitution, and these are the foundations on which our government is built.

Achilles: Castles in the air, it seems to me.

Owl: The First Article is that every time we vote on a bill, only the ayes and nays tell us whether it will pass or fail! If a coalition of owls is mighty and vocal enough to pass one bill, that same coalition can pass another the next day, even if they're singing a different song.

Achilles: I suppose that's democratic. You don't have a vice-president to break ties?

Owl: Of course not, we would never dabble in vice!

Achilles: But supposing there are as many ayes as nayes?

Owl: We're not fit to be tied, we have ways around that.

Achilles: I see.

Owl: The Second Article says that if a bill passes, and then is amended to include a rider, such that all the previous supporters still support it but some additional supporters do as well, the bill will still pass.

Achilles: Now, hang on, if a bill has already passed, why would you ever want to include additional riders?

Owl: It's our job to serve our constituents, you know. And omnibuses can seat lots and lots of riders!

Achilles: I can understand throwing in some pork to gain supporters when a bill would otherwise fail, but you wouldn't need to modify a bill that already had narrowly passed...

Owl: Throwing in some pork?

Achilles: You know, random amendments?

Owl: I'm afraid I don't know what you're talking about. Some of my favorite lobbyists are anthropomorphic pigs gifted with the power of speech, but I certainly wouldn't want to mash them up and throw them into a piece of legislature. That would be gross.

Achilles: But isn't that how the sausage gets made?

Owl: What?

Achilles: ...Never mind.

Owl: And the Third Article is that if we have a coalition big enough to pass one bill, and we also have a coalition to pass another bill, then the owls who support both of them will be able to pass the bill we get by stringing them both together.

Achilles: First of all, why would you pass a bill that duplicates the results of the first two?

Owl: Oh, sometimes we don't even bother having a vote on the first two if we know the third will also pass. These omnibuses, you see! So much more convenient.

Achilles: I'm not sure I'd say that.

Owl: We happen to like them.

Achilles: Secondly, this isn't how human parliaments work. If a slim majority of our members think we should have salad for dinner, and a slim majority think we should have soup for dinner, that doesn't mean I could get a majority to vote up a proposal for having soup and salad for dinner. They might be different people!

Owl: Well, maybe you're just thinking too small.

They finally emerge into the Parliament, which stretches up into the rafters as far as the eye can see—maybe infinitely far! The Owl takes her place in one of the lower offices, and Achilles stares up at the dizzying spiral of ascending desks. Seeing her page, an Eagle, soar nearby, the Owl waves a friendly wing.

Owl: Can you get me my committee notes?

Eagle: Yes, of course!

The Eagle takes off to go about the busywork of Parliament.

Owl: You see, back in the day there were not so many of us, and it was just a finite bunch that gathered to make the laws here, under the same constitution there is now. But the day did come when they did not vote unanimously, and we had to decide which bloc's votes were numerous enough to pass and which would fail.

Achilles: I suppose you chose the more numerous bloc to pass?

Owl: We did—for a time. But there was internal dissension, and a separate faction voted to pass a bill on a different subject. By the Third Article, you see, the people in the smaller group that had voted for both passing bills were able to pass a third successful bill the next day. But once that group started disagreeing, and even within their own faction they could not come to a consensus, they split once again, and we had to decide who would carry the day.

Achilles: Sounds rather arbitrary.

Owl: On and on it went, and our finite sets were winnowed ever narrower. Finally, we were left with only two people within the dominant clique voting at odds, and we finally realized that all the bills passed exactly when, and only when, Oliver Owl supported them!

Achilles: But that's no democracy at all!

Owl: It was by our standards! If Oliver and some large group liked a bill, it would pass just as surely as if Oliver and some small group did—or if Oliver alone did. And if Oliver was part of the coalition declaring war on starlings and he was also part of the coalition making peace with kestrels, why, the coalition that wanted both war on starlings and peace with kestrels found itself very happy.

Achilles: I should hope you overthrew that regime.

Owl: It turned out the only way to dilute the power of a dictatorship was to expand the Parliament rather drastically—to a bureaucracy of infinite size!

Achilles: That seems a bit unwieldy.

Owl: It sure did decentralize power!

Achilles: But isn't that just as vulnerable to the same problem? “Anything that Oliver likes passes” is just as stable a government under your constitution no matter how many of you there are.

Owl: You're right! Which is why our first order of business was to show our constituents that there was a new and better way, in the hopes that we could win our trust back.

Achilles: That couldn't have been easy.

Owl: It wasn't. So we decided to go about things from another direction. Instead of thinking about legislation that we wanted to pass, we tried to figure out what bills should be doomed to failure from the start.

A mockingbird zooms by out of the blue, pointing and laughing at the owl.

Mockingbird: Doomed to failure from the start!

Owl: Go away, Smullyan, you're in the wrong story.

Mockingbird: You're in the wrong story!

The mockingbird glides away again.

Owl: As I was saying, clearly a bill that could muster no support would be vetoed. And it soon became clear that in an infinite Parliament, any bill that only had a finite number of proponents was a non-starter; there was a supermajority against it. It was not the will of a single dictator telling us that those were going nowhere, so much as the collective.

Achilles: That sounds good.

Owl: On the other hand, those bills with a supermajority in favor, and only a finite number opposed, would surely pass with flying colors! Well, all of our colors are flying, but you know what I mean.

Achilles: And if a few more stragglers were to join after seeing a rider they liked thrown in, that would leave only a smaller finite number holding out—your Second Article at work!

Owl: The Third Article works just as well, for say that everyone opposed to Athena's bill is below level three, and everyone opposed to Mercury's bill is below level five.

Achilles: Then everyone opposed to the unified bill is still below level five, and there are still infinitely many owls beyond that to pass the combined bill. So you're all set!

Owl: Of course, that still left a bunch of bills that didn't leave us with any consensus. Sometimes everyone on this side of the aisle would vote aye, and everyone on that side of the aisle would vote nay.

Achilles: An infinite tie!

Owl: So it took some very delicate work from the first generation of true democrats, our founding mothers and fathers, to convince our voters that no matter what, there would still be a non-dictatorial resolution to every bill. There are still a few skeptics who threaten to secede every now and then because they don't like our underlying laws, but on the whole, people are convinced, and we leave the details to our interns now. Speaking of which, here he comes.

Eagle: There's a tortoise hanging out outside.

Achilles: That's probably my friend.

Eagle: Can I abduct him?

Owl: No.

Eagle: Please?

Owl: No.

Eagle: And go dive-bombing playwrights?

Owl: No—what?

Achilles: Where I come from it's a thing.

Owl: Where did you come from and why would you ever come here instead?

Achilles: I didn't have much choice in the matter, but I kind of like it.

Owl: Come on, let's go say hi.

They reunite with the Tortoise outside, who is still being talked at by the Cardinals.

Singular Cardinal: ...and so, if the continuum hypothesis is true for all these lowly birds below me, then it can be true for me, too!

Tortoise: What.

Singular Cardinal: It's called Silver's Theorem! You can check it out in my new book, Copper, Silver, Gold; A Do-It-Yourself Handbook In Proof Mining.

Achilles: Come on, buddy, let's go home.

Regular Cardinal: ...ninety-nine...one hundred!

Singular Cardinal: Don't sound too excited, or they'll think you're factorializing.