Shouting in the Darkness

I post to this blog for the same reasons that a castaway writes in the sand or that men on mountaintops talk to the stars. I have no hope that the words will  reach anyone else, but I derive some pleasure from translating my ideas into a physical form. It is a protest against the tragedy that so many worthwhile thoughts die without notice even from the thinker.

Return of the Jug

Has anyone else noticed that in "The Return of the King" (2003) the pitcher Pippin puts in Gandalf's arms when he takes the palantír looks just like the one he has in Minas Tirith a few minutes later? The first time is about 00:30:09 into the film, the second is 00:52:46. I'm not suggesting that the filmmakers made a mistake, only I was surprised to see that Pippin or Gandalf would bother to pack it. Or that such identical jugs were in mass production in a pre-industrial society.

Greed

Regardless of the system, the people at the top are capitalists.

The only question is whether they allow the other people to be.

Milton Friedman is unstoppable. He taught at Chicago University (one of the world's finest school for economics) where not even the professors could find room to disagree with him once he got started explaining things. He went on to win the Nobel Prize in economics, largely for his explanations of American monetary policy and explaining why the government should basically stay out of businesses' way.
Here's part of a chat he had with Phil Donahue:

The first video for this interview can be found below. The entire show was uploaded by the same user.

Math Majors

What does it mean to be a math major? Well...

• It means you make jokes that rely on the Banach-Tarski Paradox (and your "friends" get it).
• It means you're more comfortable finding numerical solutions to nonlinear differential equations than calculating a 15% tip.
• It means you enjoy pain.
• It means you keep track of strange things and post them on your Blog.
• It means you can't show pie charts to your peers because they'll criticize you for using a cute, unscientific presentations of data.
• It'll be even worse if they notice that you used Excel instead of Matlab.

Why a Franklin Planner is Better than a Girlfriend

This list is difficult to find, so I thought I'd help get it out there:

10. A Franklin will never make you late.
9. You don't have to pay attention to your Franklin if you don't want to.
8. Your Franklin will still be there when you get off your tour of duty.
7. If you lose your Franklin, it won't keep your tapes and sweaters.
6. You only have to pay for your Franklin once.
5. You never have to worry about someone else using your Franklin.
4. Your Franklin will wait in car while you play basketball.
3. Your Franklin doesn't care if you have another planner.
2. Your Franklin already comes with rings.
1. If your Franklin gets too big, you can just remove some pages.

1984 in the 23rd Century

As a fan of "Star Trek: The Next Generation," I have watched most episodes several times. "Chain of Command" (both parts) was one of my favorites since both Picard and Riker get to show their metal.
However, I was disappointed to realize that half of "Chain of Command"seems to be borrowed from George Orwell's "Nineteen Eighty-Four."
In the episode, Picard is captured by Cardassians and imprisoned for espionage. He was working with Worf and Crusher, who escaped without Picard's knowledge. Picard is then treated with, shall we say, less dignity than befits his rank (claiming POW rights would have required admitting he was acting under Starfleet's orders).
The specifics of Picard's predicament, in many important ways, resemble Winston Smith's re-indoctrination. Observe:

• They are punished for insisting that the four objects before them don't make five
• They are tortured first by humiliation and starvation, then by mysterious devices that inject pain directly
• They are fed lies about the wars outside
• They are concerned for the safety of the women they were working with
• They are given the opportunity to save themselves by betraying their respective women
• Their captors express a mental kinship to them, wishing to be able to explore their minds further

While Riker's exploits in "Chain of Command" would make the episode worth watching anyway, the apparent copying is disappointing.

Countability of Countable Union of Countable Sets Requires Countable Choice

The union of a bunch of countable sets is usually countable. This is very obvious if there are finitely many sets, say A, B, C. Pick an order for each set A={A1, A2, A3, ...} and then you can use the obvious ordering

1. A1
2. B1
3. C1
4. A2
5. B2
6. C2
7. A3
8. B3
9. C3
10. ...

So the finite union of countable sets is countable. So far so good.

But what if you have infinitely many countable sets to union together? If there are countably many sets, you might think of the elements of the union in a sort of lattice arrangement and trace the diagonals:

1. A1
2. A2
3. B1
4. A3
5. B2
6. C1
7. A4
8. B3
9. C2
10. D1
11. ...

It would take a little more work to write a closed form expression for what number goes with which element, but every element of every set gets covered eventually. It's obvious, right?

I sure thought so. I have used this principle fairly frequently throughout my schooling assuming it was a straightforward principle. Speaking with several other math folks, no one seemed to be aware of a subtle problem: each set is countable, meaning each set has a bijection with the natural numbers (there is at least one way to enumerate the elements of each set). We could also say that for each set, there is a non-empty set of bijections with the natural numbers.

The problem is that, picking one element (bijection/enumeration) from each of infinitely many sets (the sets of ways to enumerate the elements of our sets) is not always possible under the ZF axioms. In other words, since all you know is that it is possible to enumerate each of the sets, the normal assumptions about math don't let you assign an enumeration to each of infinitely many things. It requires a special assumption called the axiom of choice (or the weaker "countable choice" for the case of a countable collection of sets).

So be warned! You must not assume willy-nilly that the countable union of countable sets is countable. You must include countable choice with your other assumptions.

If you're still interested, I recently found this interesting discussion of the problem:
https://www.dpmms.cam.ac.uk/~tf/cupbook3AC.pdf