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The Incomplete Gödel

Gregory Moore

Incompleteness: The Proof and Paradox of Kurt Gödel. Rebecca Goldstein. 296 pp. W. W. Norton, 2005. $22.95.

A World Without Time: The Forgotten Legacy of Gödel and Einstein. Palle Yourgrau. x + 210 pp. Basic Books, 2005. $24.

Such eminent 20th-century physicists as Albert Einstein, Niels Bohr and Werner Heisenberg are well known to almost all scientists, whether or not they happen to be physicists. Yet most scientists are unfamiliar with eminent mathematicians from the same period, such as David Hilbert (Germany) and Oswald Veblen (United States). A rare exception is John von Neumann (Hungary and the United States), a mathematician whose contributions to quantum mechanics, the stored-program concept for computers, and the atomic bomb resonate with many physical scientists.

One mathematician who deserves to be better known, and who was highly esteemed by von Neumann, is Kurt Gödel (1906-1978). In 1951 Gödel shared the first Einstein Award with physicist Julian Schwinger (who later won the Nobel Prize). At the award ceremony, von Neumann gave a speech calling Gödel's work "a landmark which will remain visible far in space and time."

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Who was Gödel, and why was his work of fundamental significance? These questions are the subject of two recent books, both popularizations: Incompleteness: The Proof and Paradox of Kurt Gödel, by Rebecca Goldstein, and A World Without Time: The Forgotten Legacy of Gödel and Einstein, by Palle Yourgrau. The association between Einstein and Gödel began when the latter was a visiting member of the renowned Institute for Advanced Study in Princeton for several years in the 1930s. Later they became close friends. Veblen had brought Gödel to the institute after hearing him speak at Karl Menger's colloquium at the University of Vienna. And it was Veblen who saved the physically frail Gödel from being drafted into the Nazi army by getting him a non-quota immigrant visa so that he could leave Austria permanently for the United States.

Gödel, who had originally intended to become a physicist, was turned to mathematics by Philip Furtwängler's course on number theory at the University of Vienna. Paralyzed from the neck down, Furtwängler lectured from his wheelchair without notes while his assistant wrote equations on the blackboard. Then Gödel, seeking results of more fundamental significance than he could find in number theory, moved to the new field of mathematical logic, which Hilbert—the most influential mathematician of the early 20th century—had made mathematically respectable during the 1920s. (Earlier work on mathematical logic, by Bertrand Russell among others, had reinforced the traditional tendency to regard logic as part of philosophy—and hence subject to all of philosophy's ambiguity—rather than as part of mathematics, where exactness is the watchword.)

Gödel's doctoral thesis, which was vastly more important than the average dissertation in mathematics, established the "semantic completeness" of first-order logic—a different matter from the "incompleteness" to be discussed below. The thesis settled a major problem, which Hilbert had proposed in 1928. What Gödel showed was that any sentence S can be proved in a formal system (that is, in a formal language, like a computer language, with a computable list of sentences and of axioms) if and only if S is "valid"—that is, true in all "models" (all sets with appropriate functions and relations on the set). Thus he established a connection between what is true in a formal system and what can be proved in it.

Certainly Hilbert expected that the semantic completeness of first-order logic could be established. However, neither he nor anyone else expected that attempts to establish the semantic completeness of the next more complicated kind of logic (second-order logic) had to fail! Yet this conclusion followed from Gödel's next results, his incompleteness theorems, which had consequences for both first-order and second-order logic. (Roughly speaking, first-order logic is adequate only for the simplest parts of mathematics, whereas second-order logic is needed to express adequately the parts of mathematics dealing with the infinite. Technically, first-order logic allows the quantifiers "for all x" and "there is some x" to apply only when x is an object, whereas second-order logic allows x also to be any set of objects.)

Gödel's two incompleteness theorems were his crowning achievement. They have often been misunderstood in their popular presentations. We must first be clear about what the theorems do not say. They do not state that there are absolutely unprovable truths. What the first theorem asserts is that if a system of axioms can express the arithmetic of both addition and multiplication of counting numbers (1, 2, 3 and so on), and if the system is consistent (that is, if it does not imply a contradiction), then there will be a sentence G of the system that is unprovable in the system but is nevertheless true. If this new sentence G is now added to the axioms of the system, then applying Gödel's procedure to this enlarged system will generate a new sentence H, which will be true but unprovable in the system with the added axiom G. So it would be more accurate to describe Gödel's theorems as having to do with "incompletability" rather than "incompleteness." The second incompleteness theorem states that, given the same hypothesis as the first, one such true but unprovable sentence expresses the consistency of the system.

Hilbert insisted that it is possible to prove the consistency of all of classical mathematics (and, in particular, of both the counting numbers and the real numbers) by using only the truths of the elementary arithmetic of the counting numbers. The attempt to do so became known as "Hilbert's program." Gödel's incompleteness theorems showed that Hilbert was wrong; the arithmetic of the counting numbers could not prove even its own consistency, much less that of the real numbers!

For Gödel's proof to work, however, it is essential that arithmetic have both addition and multiplication available. In fact, if we consider the counting numbers with just addition (and not multiplication) defined on them, then we can find a set of axioms that prove all true first-order sentences in the language of this system. So Hilbert was right—for this weaker system.

Returning to the arithmetic of counting numbers with addition and multiplication: Why not simply take all the true sentences of this system as axioms? What Gödel's theorems show, in this case, is that there is no algorithm (or equivalently, no computer program) that will decide whether any given sentence of arithmetic is true or not. Although Hilbert had carefully distinguished the "true" and the "provable" in a formal system, he believed that these two concepts were in fact identical. He was wrong. Yet he was never able to accept that Gödel had mathematically proved him to be wrong.

The high points of Goldstein's book, Incompleteness, are the first and last chapters, in which she gives some of the flavor of Gödel's personality. The book starts with an imagined description of Einstein walking home from the institute with Gödel. Here she quotes the recollections of Ernst Gabor Straus about Einstein's friendship with Gödel: Einstein was "gregarious, happy, full of laughter and common sense." Gödel, on the contrary, was "extremely solemn, very serious, quite solitary and distrustful of common sense as a means of arriving at the truth."

They were very, very dissimilar people, but for some reason they understood each other well and appreciated each other enormously. Einstein often mentioned that he felt that he should not become a mathematician because the wealth of interesting and attractive problems was so great that you could get lost in it without ever coming up with anything of genuine importance. In physics, he could see what the important problems were and could, by strength of character and stubbornness, pursue them. But he told me once, "Now that I've met Gödel, I know that the same thing does exist in mathematics."

At the end of her book, Goldstein gives an absorbing account of the last two decades of Gödel's life, after Einstein died. Here she recounts the intellectual and personal isolation of Gödel in his final years. His incipient paranoia finally overcame him, and he starved himself to death for fear of being poisoned.

What is of lasting importance, however, is not Gödel's psychological idiosyncrasies but the great conceptual depth of his mathematical ideas. And unfortunately, between its first and last chapters Goldstein's book is the "death of a thousand cuts" to anyone who knows the history of mathematical logic and of Gödel's work in particular. Like hydrofluoric acid, the sheer accumulation of errors, both minor and major, erodes all the trust that the informed reader might have had in her book. She even gets the year in which Gödel first proved his incompleteness theorem wrong—it was 1930, not 1929 as she suggests on page 156. She does not begin to understand the basis for Gödel's mathematics, as she makes plain by stating (on page 139) that the set of real numbers is of "higher ordinality" than the set of natural numbers. What she meant was a completely different concept, "higher cardinality." She even misunderstands what the incompleteness theorems state, because she writes (on page 191) that "For Gödel, for each formal system there will be truths expressible in that system that will not be provable." But this is false, because there are infinitely many different formal systems in which every truth expressible in the system is provable.

Unfortunately, Goldstein's errors are precisely the sort that many copy editors would not catch: factual and conceptual errors. Such an error occurs when Goldstein (on page 82) says that Olga Neurath was the sister of Otto Neurath and the wife of Hans Hahn, when it was actually the other way around: She was Neurath's wife and Hahn's sister. Goldstein's book needed, prior to publication, a reader who knew mathematical logic and its history and who could correct both conceptual and factual flaws (such as the repeated misspelling of the Austrian logician Georg Kreisel's name as "Kreisl"). The mathematical logician Simon Kochen, whom Goldstein thanks in her acknowledgments, did not serve as such a reader; he has informed me that he did not read the entire manuscript but merely pointed out certain errors in her outline of the incompleteness theorems.

In sum, Goldstein does not understand mathematical logic and set theory, the subjects of Gödel's mathematical work. Her book would have benefited if she had just left them out and not pretended to explain them. In any case, she should have avoided such sophomoric touches as renaming first-order logic "limpid logic." This strikes a mathematician as condescending. A physicist might have the same reaction if Goldstein decided to rename quantum mechanics "limpid physics."

Yourgrau's book, A World Without Time, although by no means perfect, is a vast improvement on Goldstein's. He too tells the story of Gödel's life, but with particular emphasis on the friendship with Einstein. From that friendship emerged Gödel's totally unexpected models of the general theory of relativity: rotating universes in which it is possible to travel into the future or the past if a spaceship can go at least two-thirds the speed of light.

Yourgrau discusses Gödel's incompleteness theorems more adequately than Goldstein does. He understands that these theorems originated from Gödel's attempt to carry out part of Hilbert's program (in particular, to prove the consistency of the axioms for the real numbers) and not from any attempt to refute that program. And Yourgrau's prose is at least as compelling as Goldstein's.

In the end, Gödel's search for the deep truths of philosophy and metaphysics—undertaken in the spirit of the great philosopher/mathematician Gottfried Leibniz—was only partly successful, even in his own opinion. Yet Gödel shared with Einstein a deep-seated belief that the truth about physics and mathematics was objective. This truth was not merely the creation of the human mind but was there to be discovered.

In conclusion, the best biography of Gödel remains Logical Dilemmas: The Life and Work of Kurt Gödel (A K Peters, 1997), by John W. Dawson, Jr., which recently became available in paperback. Both Goldstein and Yourgrau rely heavily on Dawson's book but do not always make it clear how much they do so.

If a reader wants to look at just one book about Gödel, it should be Dawson's. Those who want a clear, simplified, but fundamentally correct explanation of the incompleteness theorems should read Ernst Nagel and James R. Newman's short book Gödel's Proof. And those seeking a popular introduction to Gödel's unusual rotating models of the laws of general relativity should read Yourgrau's A World Without Time. Goldstein's book, with its many errors, is best ignored.

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