Nested Scrolls

by Rudy Rucker

  But after the older man left, Takeuti remarked, “He did some work thirty years ago that nobody cares about anymore. Now he should be quiet.”

  Takeuti had his own farfelu way of looking at set theory, and I think I may have been one of the very few people in the world who ever understood what he was talking about. The issue he was dealing with had to do with a problem in talking about the class of all sets.

  What is a set? You might call it the form of a possible thought—as long as you assume that thoughts can be infinitely large. Or you might say that a set is the abstract form of any conceivable entity. The class of all sets—sometimes called V for short—is viewed as not being a set itself. The reason is that if V were a set, then, as the class of all sets, it would have to be a member of itself, and this would eventually lead to a contradiction. But if V isn’t a set, then V isn’t conceivable, as, once again, a set is the form of any conceivable entity. Okay, fine, but if V isn’t conceivable, then how can it be that set theorists are always talking about it?

  Takeuti’s answer to this problem was something that he called Nodal Transfinite Type Theory. He argued that whenever we think we’re talking about the full set-theoretic universe V, we’re really just talking about an approximation to it. In Takeuti’s theory, saying something is true for V really means that it’s true for most of the approximations to V. But, of course, his theory gets a lot more complicated than this—as math always does. Eventually, in 1977, I’d publish a paper about Takeuti’s work called “The One/Many Problem in the Foundations of Set Theory.”

  At one of our seminar meetings I told Takeuti that I’d proved an important result about Nodal Transfinite Type Theory that he’d been inching towards. I’d worked the proof out in my mind during the drive from Highland Park to Princeton. I tended to see these complex set theory proofs not in terms of symbols but in terms of patterns—things like radiating fins or Gothic flying buttresses or cascades of collapsing concentric shells. But when I tried to explain my new proof to Takeuti, I realized that it had a hole in it.

  “I always like to quit for the day after I prove a very big theorem,” said Takeuti kindly. “Then at least I am happy for a little while.”

  He meant that, in the morning, you’d usually find a mistake in yesterday’s proof. Math was very harsh that way. Often there was no way to fix a broken proof. You’d find an unbridgeable gap and be left with nothing.

  Initially I’d been writing my thesis on a particular problem that Erik Ellentuck proposed in the subject of model theory. It had to do with enumerating how many different models a given mathematical theory might have. I’d gotten stuck on a proof I was trying to find, and then, in the kitchen with me at another math professor’s party, Ellentuck had offered a suggestion about a new approach I could use. I finished the proof, and I thought maybe I had enough for a thesis.

  In the summer of 1970, Sylvia and I left baby Georgia with my parents for a week, and went to a big mathematical logic conference at the University of California at Berkeley, in honor of Alfred Tarski. All the most famous logicians were there. I gave a short talk about the new result that I’d proved for my thesis, and it was well received. Before the talk, I was more nervous than I’d ever been in my life. Although I did mention that Ellentuck was my advisor, I didn’t say anything about the fact that he’d helped me with my proof. I was eager to take as much credit as I could for this, my first theorem.

  Sylvia and I were excited to be in California. There were a couple of Hungarian mathematicians that she thought were cute, Erdös and Rado, always talking together. Like real old-school Hungarians, they wore their white shirts with the big collars smoothed down on the outsides of their suit jacket lapels.

  We took a day off from the conference, and went to a free concert by hippies in Golden Gate Park. People were smoking pot and painting their faces, just the way they were supposed to. Walking around Berkeley, we could see clear out past the San Francisco Bay, and the fresh clean ocean air carried a whiff of menthol from the eucalyptus trees.

  We celebrated our third wedding anniversary in Berkeley, and went out to a fish restaurant recommended by our perennial seafood maven, Greg Gibson. Sylvia wore a beautiful red dress that night, with a frilly bib pattern on it. After all our baby care, it was a thrill for us be out on our own together again, seeing a bit more of the world.

  When I got back to Rutgers, I found that Ellentuck was envious about my talk, and angry with me. Someone had told him that I hadn’t emphasized his role in my proof. And now he said that, in order for him to get his fair share of the credit, I had drop the proof from my thesis and write it up as a joint paper to be published in a mathematics journal under our two names. I’d have to find something else for my thesis—and it was up to me to figure that part out on my own. He wasn’t going to help me any further.

  He was in an extremely agitated state. He kept looking out his living room window and asking me if his mother was hiding inside my car. He thought he could see her out there. He was nuts.

  What to do? I wrote up most of our paper, and he added a final section on his own. It was a nice little article, “Martin’s Axiom and Saturated Models,” and in 1972 we’d publish it in a journal called the Proceedings of the American Mathematical Society. As for my thesis—well, I was just as happy to proceed on my own and pursue some ideas that I had. I wasn’t really very interested in saturated models.

  For a little while in 1970, I hoped to unearth some kind of logical contradiction in the foundations of mathematics—briefly Ellentuck became concerned that I might succeed, and said my efforts were “immensely destructive,” which was music to my ears.

  But in the end no contradictions were to be found. Mathematics is like some tough old goddess, powerful enough to withstand the onslaughts of generation after generation of young ruffians.

  For my first few years at Rutgers, the math department was housed in a historic-landmark-type Victorian house right beside the main library. But in 1971, the department moved into a huge, brand-new concrete building that also housed those lower forms of life known as computer scientists.

  The movers cleaned all the furniture and boxes out of the old math building, the janitors removed all the grime and dust, and the building stood completely empty for about a year. I still had the key to the building, and I took to doing my thesis work in there, working on some ideas about definability and set theory that stemmed from the great logician Kurt Gödel’s celebrated Incompleteness Theorem of 1931.

  In mathematical logic, it’s possible to define a coding system whereby each possible statement about mathematics corresponds to its own code number. It would be convenient if there were some simple and mechanistic procedure that we could apply to any candidate code number so as to decide whether this number represents a true sentence or not. But—and this is the interesting thing—there can never be such a formula. In other words, no mathematically precise formula can define mathematical truth.

  Why not? Well, if you could precisely define mathematical truth, then you could craft a mathematical Liar Sentence that in effect says, “This sentence isn’t true.” But the Liar Sentence is true if and only it’s not true—which is a contradiction. Therefore you shouldn’t have been able to construct the Liar Sentence. And therefore truth must not have been definable after all. Mathematical logicians love thinking along these lines—it’s what we call a proof by contradiction, or a reductio ad absurdum.

  The logician Alfred Tarski gets some credit for the undefinability of truth result, as he wrote about it in 1936. But the bulk of the credit lies with Kurt Gödel and his Incompleteness Theorem of 1931, whose proof develops the machinery for constructing paradoxical mathematical expressions like the Liar Sentence.

  So okay, fine, truth is undefinable. Could I prove that undefinability is truth?

  That was to be the main topic of my thesis. I wanted to show that the undefinability of truth might be the only reason why any other sets were undefinable. To be more precise, I thought
in terms of a set called Truth that holds the code numbers of all the true sentences. We knew that Truth was undefinable. I was wondering if, given any undefinable set X at all, I could always find a way to define the Truth set in terms of X. In these cases, I could say that X was undefinable because truth is undefinable. And if all undefinable sets X have this quality, then it would indeed make sense to say that undefinability is truth.

  On a typical morning in 1971, I’d let myself into the empty old math building and go upstairs, lying on a pleasantly soft parquet floor, writing with a pencil on sheets of paper that I was assembling into a three-ring binder. I erased and revised a lot. I loved those days alone with mathematics.

  I’d bring along a kalimba, or African thumb piano. It was like a miniature cigar box with curved strips of metal. I’d gotten it in a toy store. When I wanted a break, I’d play little songs, enjoying the echoes. Or maybe I’d just lie on my wood floor, pondering the beautifully three-dimensional shape of the room, thinking about the spacetime trails everything was leaving as we moved forward in time.

  Before too long, I was able to prove my result about undefinability and truth, although only for the somewhat limited universe of the so-called constructible sets. But it was a nice theorem in any case, and in 1974 I’d be able to publish it under the title “Undefinable Sets,” in the Annals of Mathematical Logic. But in 1972, I still hoped to find some bigger and better theorems for my thesis, and I worked on into the spring.

  Takeuti gave me some help, as he was still visiting at the Institute for Advanced Study. And eventually Ellentuck started helping me again too. He didn’t act crazy all the time—on a good day, he could be perceptive, helpful, and even genial. But I don’t think he actually read much of what I wrote.

  As I mentioned earlier, an exciting thing about the Institute for Advanced Study was that the reclusive genius logician Kurt Gödel was still in residence there. As a student of mathematical logic, I thought of Gödel as a supreme guru.

  Not only had he proved the fabulous Incompleteness Theorem, he’d formulated the notion of constructible sets as a way to prove something about the legendary Cantor’s Continuum Problem.

  This problem—which would loom large in my life for many years—goes back to 1874 when the German mathematician Georg Cantor proved a startling fact: there are different levels of infinity. In particular, there are more points in continuous space than there are integers. This theorem leads to a very interesting problem, the Continuum Problem, where “continuum” is a scholarly word for a continuous stretch of space. Which level of infinity best characterizes the continuum, that is, how many points are there in continuous space?

  Cantor coined some esoteric-sounding names for the first few levels of infinity: alef-null, alef-one, alef-two, and so on. The set of integers has the size alef-null, but we don’t know which alef matches the size of the continuum. Is it alef-one, alef-two…or something else? Cantor thought the continuum’s size might be alef-one, and this guess is known as Cantor’s Continuum Hypothesis.

  In 1940 Gödel proved that the Continuum Hypothesis is indeed consistent with the known facts about set theory. But in 1963, the mathematician Paul Cohen proved that the negation of the Continuum Hypothesis is consistent with set theory as well! In other words, on the basis of what we currently know about sets, the Continuum Hypothesis is undecidable. You can’t prove it and you can’t disprove it on the basis of any currently accepted axioms for set theory. But Gödel had remarked several times that, by pondering the nature of infinity, he hoped to unearth some new axioms that would solve the Continuum Problem for once and for all.

  One day in 1970, a Xerox of a hand-written manuscript appeared in my mailbox in the Rutgers math department office. It was four or five pages covered with spidery writing, replete with corrections and footnotes, entitled, “Some considerations leading to the probable conclusion that the true power of the continuum is alef-two.” The author was Kurt Gödel.

  I never did find out who put that paper in my mailbox—probably it was Ellentuck, who liked being secretive. In any case, I flipped out with joy. I dropped everything and spent the next week poring over the little manuscript, struggling to decipher it.

  I recall that I scoured the depths of the Rutgers library, emerging with an obscure tome in French that elucidated some of the turn-of-the-century concepts that Gödel had used. He’d unearthed an arcane way of drawing pictures of higher infinities in terms of slanting curves, where a first curve is viewed as “bigger-by-end-pieces” than a second curve if, as you move out towards the right, the first curve eventually crosses the second, and remains above it. Gödel used the word “scales” for his sequences of steeper and steeper curves. I thought about them a lot, and eventually, I’d put pictures of them into my popular book on the philosophy of mathematics, Infinity and the Mind, 1982.

  Everyone at Rutgers was curious about Gödel’s new sketch of a proof. I gave a seminar talk on it, and discussed it with Takeuti. Gödel’s new argument was intriguing, but it seemed to have some—logical gaps.

  Takeuti knew Gödel fairly well, and perhaps due to Takeuti’s influence, the great man invited me to come to his office for a conversation early in 1972. It’s hard to express just how big a deal this was for me. It was as if I’d been at art school learning to paint—and suddenly Picasso had asked me to come to his studio. As I wrote in Infinity and the Mind:

  I didn’t know where his real office door was, so I went around to knock on the outside door instead. This was a glass patio door, looking out on a little pond and the peaceful woods beyond the Institute for Advanced Study. It was a sunny March day, but the office was quite dark. I couldn’t see in. Did Kurt Gödel really want to see me?

  Suddenly he was there, floating up before the long glass door like some fantastic deep-sea fish in a pressurized aquarium. He let me in, and I took a seat by his desk.

  I felt comfortable with Gödel right away. For some reason he reminded me my German grandmother—partly it was his accent, and partly it was his kindly, inquisitive demeanor.

  Gödel had a way of instantly understanding anything that I said. When I described my thesis work to him, it took him about ten seconds to figure out my proof that undefinability is truth.

  We spent an hour talking together—about set theory, about the nature of time, and about the teachings of mysticism. He laughed a lot, and even the laughter seemed like speech. It was wonderful to be in contact with so powerful a mind. It wasn’t at all like being with a regular person. Compared to Gödel, the rest of us are like talking dogs.

  I managed to visit Gödel two more times over the next couple of years. Perhaps he had some hope that I might be the one to fix up the logic of his manuscript on the continuum problem. And in any case, he seemed to enjoy my enthusiasm and my odd way of looking at things.

  I remember, for instance, a discussion about time travel paradoxes. If you can travel to the past, then you might conceivably kill your father as a boy, so you wouldn’t be born, but then you wouldn’t have traveled to the past…and so on. One escape route used by science fiction writers is to suppose that you don’t actually travel to your own past, but rather to the past timeline of some parallel world. I thought about this for awhile and drew up some intricate spacetime diagram, referring to the dimension between the parallel timelines as “eternity.”

  Gödel got a kick out of my drawing. “This is a very strange idea,” he said. “A bizarre idea.” I was proud.

  Years later I’d meet the famous mathematician Paul Cohen, and he expressed his own sense of wonder and good fortune at having met Gödel. “I was walking across the lawn to Gödel’s office,” said Cohen. “And I’m thinking, how does a kid from New Jersey end up sitting down with the world’s greatest mathematician?”

  When I met Gödel, I felt the same way. Just a few years ago, I’d been a feckless college boy, and now I was discussing philosophy and logic with the smartest man I’d ever meet. It was as if I’d broken out of a chrysalis at Rutgers,
or as if I’d emerged from a long series of dreams.

  I’d been hoping that I might get a post-doctorate position at the Institute for the fall of 1972, but that didn’t pan out. My paper with Ellentuck and my thesis work, although publishable, weren’t compelling enough to land me a position as a high-powered logician. I began to sense that the highest echelons of mathematical achievement would remain inaccessible to me. I’d had my chance with this thesis—and I hadn’t scored.

  I tried to tell myself that I was okay with being, as it were, a low-level stone carver among the artisans working to perfect the great cathedral of set theory. But honestly I wasn’t content with this. I wanted to be a star at something—and if I wasn’t going to win at technical academic mathematics, I was going to have to find another game to play.

  Even finding a “low-level stone carver” type of job turned out to be hard. During the spring of 1972, I mailed out an exceedingly large number of job application letters, and I attended a couple of dreary math-job fairs. Finally I got a tip from the Rutgers math chairman. There was a position open at a state college in Geneseo, New York, up near the Great Lakes. And another Rutgers graduate was on the recruitment committee there. I phoned him up, and he helped me get the job.

  My friends and I loaded all our crap into a U-Haul truck, and in August, 1972, a math pal named Nort Fowler helped me drive the stuff to Geneseo and unload it. Nort and I drove the truck back to Highland Park, and then I ferried my little family to Geneseo.

  A few month later, in January, 1973, I had to return to Rutgers one last time for my formal thesis defense. Anyone at all can show up for these seminar-like events, possibly to pose challenges to the work that the candidate has done. A Princeton mathematician turned up and asked me a series of juicy questions, suggesting some interesting further work that I might do. Fearing a rewrite request, Ellentuck intervened and shut down the discussion. Enough said. I’d passed. I had my Ph. D.