#incompleteness theorem

So although Gödel published his results eighty-five years ago, the theorem endures in the popular imagination. This month, a group of latter-day Gödelians convened on Thursday nights for a crash course in incompleteness, on the roster at the Brooklyn Institute for Social Research. One syllabus text promised “Gödel Without (Too Many) Tears,” while supplementary reading unpacked his work “in Words of One Syllable.” According to “Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse,” the chameleon interpretations emerge in discussions of math, philosophy, computer science, and artificial intelligence, as one might expect, but also in ruminations on physics, evolution, religion, atheism, poetry, hip-hop, dating, politics, and even the Constitution. (Famously, Gödel informed the judge at his U.S. citizenship hearing about an inconsistency that he had discovered in the Constitution, which would allow a dictator to rise in America.)

The students in the incompleteness class, held at the Brooklyn Commons, included a computer scientist obsessed with recursion (that is, self-referential things, like Russian nesting dolls or Escher’s drawing of a hand drawing a hand); a public-health nutritionist with a fondness for her “Philo_sloth_ical” T-shirt; a philosopher in the tradition of American pragmatism; an ad man versed in the classics; and a private-school teacher who’d spent a lonely, life-changing winter reading Douglas Hofstadter’s “Gödel, Escher, Bach,” which won the Pulitzer in 1980. Even Hofstadter, a professor of cognitive science at Indiana University, still has Gödel on the brain. He delivered two talks on the logician recently, billing him as “The True Inventor of Programming Languages and Data Structures.” When I told Hofstadter about the course—not everyone’s ideal Thursday night out on the town—he said, “It ought to be quite fun.”

In total, eight people enrolled for the class, a fitting number, since “8” rotated by ninety degrees is “∞,” and infinity is, in a sense, where the trouble began with this trippy episode in the history of mathematics. In the nineteenth century, the German mathematician Georg Cantor launched an investigation into various sizes of infinities, thereby inventing set theory, which became an overarching paradigm for mathematics. “A set is a Many that allows itself to be thought of as a One,” Cantor said. But, from nearby realms of logic, paradoxes emerged, such as the Russell set, the set of all sets that do not contain themselves. In the early twentieth century, Bertrand Russell and Alfred North Whitehead, with “Principia Mathematica,” and David Hilbert, with “Hilbert’s Program,” attempted to construct a solid foundation for mathematics, creating a formal system based on axioms and rules. Gödel, with his incompleteness theorem (two theorems, actually), put an end to these dreams. He proved, to borrow from the course outline:

For any consistent axiomatic formal system that can express facts about basic arithmetic:

There are true statements that are unprovable within the system.

The system’s consistency cannot be proven within the system.

Puzzled? Russell admitted the same. He wondered, “Are we to think that 2+2 is not 4, but 4.001?”

A mathematician is said to be a machine for turning coffee into theorems, and at that Gödel excelled, although he said that the coffee in Vienna was wretched. For Peter O’Hearn, an engineering manager at Facebook and professor at University College London, the incompleteness “wow moment” was fuelled by visits to the brewpub during graduate school. O’Hearn is the co-recipient of this year’s Gödel Prize—he and a colleague, Stephen Brookes, invented concurrent separation logic, a revolutionary proof system for computer software. “Gödel’s theorem has a major impact on what all computer scientists do,” he told me. “It puts a fundamental limit on questions we can answer with computers. It tells us to go for approximation—more approximate solutions, which find many right answers, but not all right answers. That’s a positive, because it constrains me from trying to do stupid things, trying to do impossible things.”

At the Brooklyn Commons, a keg of Turmeric Sunrise kombucha was the popular drink during class breaks, which people spent pondering the pronunciation of “Gödel.” (“Goy-del?” “Good-ul?” “Girdle?”) I asked my classmates whether they had heard of the sci-fi writer Rudy Rucker’s book “Spacetime Donuts,” which includes a minor character named Professor G. Kurtowski. The answer was no, but as the American pragmatist noted, “Kurtowski is significantly easier to pronounce than ‘Guuuudel.’ ” In Gödel’s archives, there is a full file of letters from Rucker. “I was looking for a guru,” Rucker told me. He first visited with the great man in 1972, as a graduate student, and thereupon Gödel joined his other mentors—Frank Zappa, Robert Crumb, William Burroughs, and Thomas Pynchon. Pynchon, in his novel “Gravity’s Rainbow,” paraphrased incompleteness by mapping it onto Murphy’s Law, “that brash Irish proletarian restatement of Gödel’s Theorem—when everything has been taken care of, when nothing can go wrong, or even surprise us … something will.” “Yeah,” our instructor said. “Metaphorically, yeah, I can see it.”