The Axiom of Choice - 2

by Jim Chaffee

I walked down the hill beside him, where the national guard troops awaited us, their weapons trained. They were frozen in place like ice statues. One of them cracked down the middle and split into two solid pieces as we walked by.

I rode into Mill Valley in the deputy’s truck. We went to the jail but as we approached the place caught fire and burned to the ground. This upset the sheriff who decided to take me to a holding cell in Oakland, but he didn’t try to cuff me or even touch me. The three of us made the drive in the deputy’s truck in silence, without further incident.

They took me to an empty cell and Jehovah told me to go on inside, which I did. The cell wouldn’t lock. They tried putting me in another cell, but the same thing happened. I told them not to worry, that I would stay if they would agree to bring an official representing the Federal Government.

He came the next day, a tired looking cadaver of a man wearing a dark, nondescript suit. He chain smoked Gaulois cigarettes while standing before a big red NO SMOKING sign, lighting the next one before crushing out the current butt which he always smoked down to a nub. His fingers were stained yellow, as were his teeth, though they weren’t too noticeable as he seldom smiled.

He was a lawyer working for the FBI. That was all he told me. He didn’t ask me any questions, but drove me to a drab, unmarked building in downtown Oakland where a small crew of what I assumed were Federal Agents waited for us. He handed me over to another tall, thin man in another drab suit who told me he was with the CIA. They took me to a small airport and flew me to an airfield on a tree-filled base. They didn’t tell me where I was, but I knew it was Langley, Virginia.

We were met by another crew of men and women standing around wearing the same drab, dark suits, as if they were all uniformed by a blind tailor. They escorted me en-masse to a room full of chairs, like an auditorium, and had me sit alone on a well-lit dais while they sat in the dark. Questions came from a single voice, floating up to me as if in slow motion, his voice wavering so I could see it approach. He wanted to know who I was working with.

I told them that I worked for God and that I just delivered messages, nothing more. That was all I knew.

There was laughter at first, but then a couple men came up on the stage, unzipped my fly and connected electrical wires to my genitalia. They turned on a generator, but it exploded, killing both of them. That shut them up.

I told them God had a message for the President of the United States.

Murmuring erupted from the darkened audience; two men came up and took me by my arms to lead me off the stage. They were going to lock me up. Instead, the hands they put on me withered at the end of their arms, which then withered slowly after the hands, like plants with a rot. In a matter of minutes those limbs became old, dried leather.

No one else came near me as I walked out the door. One of them drew a weapon and ordered me to stop. I ignored him. He fired and the gun exploded in his hand. He stood with a blackened stump while his colleagues backed away from him.

I kept walking and those who remained untouched piled into four dark sedans and followed me.

They beat me to the gate and told the Marine on duty to stop me. He approached me, his handgun drawn, but his hair turned white and his skin fell away from his face and hands.

He dropped the gun. I told him that he and his descendants for seven generations would be lepers. Then I walked out the gate and started down the highway, hit the Georgetown Pike and made my way to the Dolly Madison Parkway. Though I had not been in Langley before, I knew exactly where I was and how to get where I was going.

One of the black sedans pulled up beside me. The lone driver offered me a ride. He seemed to know where I was headed without my saying a word and neither of us spoke as he drove. He stopped the car as close as he could get to the White House and let me out.

I walked onto the White House grounds and into the building and on into the oval office. No one tried to stop me. It was as though I wore a cloak of invisibility. I sat at the desk and waited in the deserted office.

The president showed up in about an hour or so with a small entourage, surprised to find me waiting for him. His aides froze in place, immobile, like statues littering the office. Only the President and I were able to communicate.

We didn’t have much to say. Mostly I spoke, or rather Jehovah spoke through me. I told him God had taken umbrage at his sanctimonious, pseudo-pious attitude and now wanted to demonstrate to him the error of his ways.

The President said something about being a God-fearing man and that this was a Christian nation, but frogs crawled out of his mouth as he spoke. He shut up and listened.

I gave him the challenge. God wanted him to make two solid gold balls with a radius of four inches. From one of them he was to make two balls of the same diameter, using no new materials, only by cutting and pasting pieces of the original ball. The other ball he was to leave untouched. God gave him a month.

The President didn’t understand the challenge, so I wrote it down and left. A taxi awaited me near the gate and drove me to the airport where I got on a plane and was flown back to San Francisco. It seemed Jehovah had made all the arrangements for me.

I was met in San Francisco by a middle aged man driving a Chevy Cavalier. He took me to where he lived with his mother, a hemiplegic who no longer recognized her left arm. They fed me during the time I waited for the President to get his ducks in a row, after which I was to return and represent God’s interests.

The President being a superstitious man, a Methodist to be exact, he called on the help of religious leaders from all over the country. He gathered together a ragtag band of Protestants, Catholics, Muslims and Jews. He even called on some Buddhists and a Hindu holy man. Most of the Baptists ranted that I was representing the devil, but the others were more circumspect. Or more to the point, mealy-mouthed. The newspapers reported no details of our confrontation, but from what the religious rabble spouted they knew everything.

The President’s science advisor pulled together a quick and dirty international teleconference. A Japanese mathematical physicist mentioned Banach-Tarski and that started a buzz. They argued the absurdity of the old theorem of Banach-Tarski, while the religious crew argued about the best approach through prayer.

The classical Banach-Tarski Theorem of the nineteen-thirties proves that it was mathematically possible to decompose a solid ball into five pieces and then reassemble them into two solid balls of the same size. This required no stretching or other distortion of the pieces, only moving them around and rotating them, cutting and pasting.

The President was ecstatic when informed that there was a mathematical method for producing a second ball from the first. He didn’t bother to hear the end of the story until cornered by his national security advisor, who brought the science advisor with him. It wasn’t trivial getting these over-schooled, under-educated politicos to listen; most of them, marginally literate at best, didn’t have the faculty for the depth of the discussion.

The great difficulty was in the very nature of the proof. It said nothing about how to carry out the decomposition, calling instead on some statement named the Axiom of Choice. The President shrugged and said if it was an axiom, then it must be true. The science advisor advised him that axioms didn’t necessarily express truth. The Axiom of Choice had been found necessary for the proof of many classical results in mathematics. It stated the magical ability to make a choice among an infinity of possibilities. It didn’t say how to do it. Because no one had any real-life experience with infinity, it had no grounding in reality.

After the meeting the President told him to take care of the details and he went back to prayer. He’d never trusted science anyway.

The newspapers published a plethora of articles about the Banach-Tarski theorem, but typically none of them seemed to come to terms with the concept. Over the years I’d gotten used to journalists misunderstanding and misrepresenting the details of the simplest legal problems.

The rub about Banach-Tarski was that it was called a paradox, though it wasn’t one. According to some it gave properties of an ideal mathematical concept, the space of points. This was not the space of matter. Matter was discrete, composed of atoms. Nonetheless, the Japanese mathematical physicist had an idea that led to a potential method.

His breakthrough came in considering the ball as a cloud of atoms, his idea to break the cloud into five smaller clouds, carefully moved apart enough that the atoms within the cloud could be expanded. The crystalline structure of the atoms would be retained so that each piece would remain a piece of the ball, but the internal spacing would increase. This would give larger pieces of a ball, with a different density of atoms, of course, changing the basic material from gold to something different. The difficulty would be in arranging the pieces so that as pairs they interlocked into spheres. One piece would remain with a single atom, not essential to the reconstructed solids. Each would have only a portion of the density of the original ball, a point not lost on other attendees, who pointed out that the new balls would not be gold. He side-stepped the issue and it disappeared.

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