This essay originally appeared in the Ellsworth American, August 27, 1992, Section I, p. 2.Copyright © 1992, Peter Suber.
This is a brief account of Kurt Gödel's trip to Blue Hill, Maine, in the summer of 1942, written for the audience that knows Blue Hill better than Gödel. (The Ellsworth American serves the Blue Hill area.)
For this HTML version I restore the footnotes, which I did not submit to the newspaper, and a sidebar, which the newspaper omitted perhaps for being too technical.
50 Years Later, The Questions Remain
Kurt Gödel in Blue Hill
Peter Suber, Philosophy Department, Earlham College
Who was Kurt Gödel and what was he doing in Blue Hill in 1942? It turns out that both questions are difficult to answer.
Kurt Gödel (1906-1978) has been called the greatest logician since Aristotle. He was unquestionably the greatest logician of the 20th century, which has been the greatest century for logic since Aristotle's. Despite this stature, his name is little known outside professional circles of logic and mathematics, and astonishingly little is known about his life. His low profile cannot be due to the fact that his major achievements are complex and demanding, unintelligible to the uninitiated, for that is also true of his best friend, Albert Einstein. Nor can it be that Gödel was more conventional than Einstein. He did comb his hair and wear socks, but Gödel's reclusive and reserved personality, thin owlish appearance, forbidding glances, hypochondria, and paranoia are at least as easy to romanticize as Einstein's eccentricities. Part of the cause of his narrower fame must lie in the fact that Einstein's theories were acclaimed as triumphs, while the initial judgment on Gödel's theorems was that they constituted a first-rate calamity for logic and mathematics.
Among physicists Gödel is known as the man who proved that time travel to the past was possible under Einstein's equations. But this was just dabbling outside his field. His main field was logic and the foundations of mathematics, and in this field he is best known for his two incompleteness theorems. The first of them showed that axiomatic systems, like Euclid's exemplary system for geometry, could never capture all the truths of arithmetic. The second showed that the consistency of such a system could never be proved by reasoning inside the system.
The first incompleteness theorem showed that some perfectly well-formed arithmetical statements could never be proved true or false. Worse, it showed that some arithmetical truths could never be proved true. More precisely, for every axiomatic system designed to capture arithmetic, there will be arithmetic truths which cannot be derived from its axioms, even if we supplement the original set of axioms with an infinity of additional axioms. This shattered the assumption that every mathematical truth could eventually be proved true, and every falsehood disproved, if only enough time and ingenuity were spent on them.
The second incompleteness theorem showed that axiomatic systems of arithmetic could only be proved consistent by other systems. This made the proof conditional on the consistency of the second system, which in turn could only be validated by a third, and so on. No consistency proof for arithmetic could be final, which meant that our confidence in arithmetic could never be perfect.
In quick succession, Gödel deprived arithmetic of its hope of completeness and its certainty of consistency. These were devastating blows to the concepts of logic and mathematics that prevailed in 1931 when Gödel published his proofs at age 25. But the conviction that genius and time could conquer every conjecture and hypothesis, not only in arithmetic, but in all of mathematics generally, had prevailed for the two or three millenia of mathematical history before Gödel's theorems. For this reason Gödel truly seemed to overturn the glory of this glorious subject and bring an epoch to a close. With time, however, mathematicians have adapted to the post-Gödelian world, and many now find mathematics to be more beautiful incomplete than it could ever have been when considered completeable.
Kurt Gödel and his wife Adele spent the summer of 1942 in Blue Hill, in the Blue Hill House at the top of the bay. Gödel was taking a vacation from the Institute for Advanced Study in Princeton, where he and Einstein were the most celebrated members of the faculty --Einstein celebrated by the world, Gödel by the considerably smaller cadre of logicians and mathematicians who followed his work.
Gödel was not merely vacationing, and had a very productive summer of work. Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson, Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942.[Note 1] Gödel's close friend Hao Wang supports this conjecture,[Note 2] noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.
If true, this would be important. Gödel had published a related result in 1938 for a stronger form of set theory. The 1938 proof established a weaker proposition in a stronger theory; the reputed 1942 proof established a stronger proposition in a weaker theory. Together they show him zeroing in on the elusive quarry --a full proof of the independence of the axiom of choice for the stronger form of set theory, a result first obtained by Paul Cohen in 1963.
The manager of the Blue Hill House in 1942 was Louise Frederick, whose family came to Maine in 1764 and still owns the summit of Blue Hill itself. She remembered many details of Gödel's stay in the Blue Hill House, and shared them with me in an interview in July 1989.
Louise Frederick recalls no change of mood or expression that might have marked Gödel's moment of discovery of the independence proof. He was unremittingly taciturn and dour. Gödel's face struck most people as a scowl, she says. To her, however, it was the face of a man lost in thought.
Gödel did most of his thinking on long walks. To avoid disturbances and to aid concentration, he walked at night, leaving the inn at sunset and returning after midnight. He walked with his hands behind his back, leaning forward, looking down. In 1942 there were very few cars in Blue Hill, and his night-time perambulations would have been much more peaceful than the same walks today. (However, even today the summer population of Blue Hill is no more than 3,000.)
Gödel usually walked along Parker Point Road, a narrow road through the pine forest along the coast, abutting some of the wealthiest homes in the area. Because the shoreline was settled before the areas further inland, Parker Point Road was about as fully populated in 1942 as it is today. But even today, the road is barely wide enough for two cars to pass each other going in opposite directions. It is difficult to see the houses on the shore through the thick shield of trees, but at many spots the trees thin out to reveal a magnificent view of Blue Hill Bay and its many islands.
Throughout the summer Louise Frederick received agitated telephone calls from people of the town. Who was this scowling man with a thick German accent walking alone at night along the shore? Many thought Gödel was a German spy, trying to signal ships and submarines in the bay.
During the daylight hours, Gödel spent most of his time in his room. Adele also stayed in the room during the day, probably on the bed she lined with an entire trunkful of pillows. The Gödels' room was the only one in the inn at the time with a private bath, permitting them to remain out of sight all day long. Adele did the beds herself, and did not even allow the staff to enter the room. Louise Frederick thinks this was to protect Gödel's table of notes and books.
The inn had a common room for reading and conversation, but the Gödels were never seen in it. The residents ate in a common dining room as well. Although the Gödels came to the dining room for meals, they virtually never ate. (When I told Louise Frederick that Gödel died childless, she exclaimed, "Of course! He didn't have the strength to make a baby!")
When Gödel left at the end of the summer, taking the train from Ellsworth back to Princeton, he never returned. He wrote to Louise Frederick twice, accusing her of stealing his trunk key.
The Blue Hill House still stands, and has been renamed the Blue Hill Inn. Although the building has been given a new kitchen and dining room, Gödel's room is undisturbed on the second floor, at the southeast corner, at the tip of the tail of the "L".
We do not know whether Gödel really discovered a full indepence proof for finite type theory in 1942 and simply chose not to publish it.[Note 3] The answer may lie in Gödel's Blue Hill notebooks, which are written in the outmoded Gabelsberger shorthand also used by Edmund Husserl. The notebooks are now being transcribed by the editorial staff assembling his collected works for the Association for Symbolic Logic and the Oxford University Press. Time will tell what Gödel discovered in Blue Hill.
Despite its obscurity, the axiom of choice has been the subject of more written controversy than any other axiom in mathematics, including the Euclid's notorious parallel postulate. Essentially the axiom of choice says that if we have a set of nonempty sets, then we can construct a new set by taking a member from each of the existing sets. For example, we can make a new class of students by taking one student from each of many other classes. More precisely, the axiom holds that there will always be a definite rule (function) for making the selections, even if we have no idea what such a rule would look like.
The axiom of choice appears harmless, but if we accept it we are led to some very unusual results. For example, it helps us prove the Banach-Tarski theorem, which says that if we cut up a sphere into a large enough number of small scraps, then we can reassemble the scraps into a sphere with twice the surface area of the original. The axiom of choice also helps us prove that the real numbers can be well-ordered, that is, that they can be rearranged so that every subset of them has a smallest member. Some sets of real numbers, such as all those greater than 1 and less than 2, have no least member and no greatest member. We have no idea how to rearrange them so that they become well-ordered, but the axiom of choice assures us that there is such an arrangement.
The axiom of choice is rejected by many kinds of mathematicians, although it is probably accepted by a majority today. In Blue Hill Gödel worked single-mindedly on one aspect of its acceptability: its consistency with the axioms of finite type theory. In 1938 Gödel had proved that the axiom of choice could not be disproved from the axioms of stronger, standard set theory. This means that it can be added to standard set theory without introducing any inconsistencies that were not already there. Hence, we can have as much confidence in it as we have in the standard axioms. This is why most mathematicians accept the axiom today.
An axiom is "independent" if it cannot be derived from the other axioms, and they cannot be derived from it. In general, axioms ought to be independent of one another. If they are not, then at least one of them could be removed from the set of axioms without reducing its strength. To leave a superfluous principle in the set of axioms is simply inelegant.
But proofs of independence are important for more than elegance. If an axiom is independent, then it can be replaced by its negation without creating inconsistency in the system. Euclid's parallel postulate is again the most famous example. For centuries it did not seem independent, yet attempt after attempt to derive it from Euclid's other axioms ended in failure. On the assumption that the parallel postulate was independent, Gauss in the 18th century, and Bolyai and Lobachevski early in the 19th, replaced the postulate with its negation and produced consistent Non-Euclidean geometries. The parallel postulate was finally proved independent by Hilbert in 1899.
If the axiom of choice is independent of the standard axioms of set theory, then the controversy over its acceptability quiets down a few notches, for then both those who accept it and those who reject it are using consistent set theories (or are not using inconsistently theories simply on account of their decision on the axiom of choice). Moreover, if it is independent, then we have discovered a "doorway" through which to create non-standard set theories.
Set theory in which the axiom of choice is replaced by its negation is called Non-Cantorian set theory for a somewhat involuted reason: the negation of the axiom of choice implies the negation of the generalized continuum hypothesis (GCH), and the GCH is intimately associated with Georg Cantor. The GCH holds that infinite numbers only come in certain kinds. If we let a stand for the number of natural numbers, let b stand for 2a, let c stand for 2b, and so on, then the generalized continuum hypothesis asserts that the only infinite quantities are a, b, c, .... For a time Gödel disputed this, and tried to prove that there was an infinite quantity in between a and b. He agreed with Cantor that the number of real numbers was b, but he thought this was the second, not the first, infinite quantity greater than a or the number of natural numbers. Eventually he changed his mind about this and spent the rest of his career trying without success to prove either the continuum hypothesis or its generalized variant.
Proving the independence of the axiom of choice would not falsify the generalized continuum hypothesis, but open up the world of non-standard set theories in which it was replaced by its negation. Gödel's 1938 proof provided the related result that the negation of the axiom of choice could not be derived from the standard axioms. In Blue Hill in 1942 he tried to work out a complete independence proof, though from the weaker system of finite type theory; and according to Dawson and Wang, he succeeded.
1. See Dawson's chronology of Gödel's life in Solomon Feferman (ed.), Kurt Gödel, Collected Works, Vol. I, Publications 1929-1936, Oxford University Press, 1986, at p. 41. [Resume]
2. Hao Wang, Reflections on Kurt Gödel, MIT Press, 1987, p. 108. [Resume]
3. There is evidence on both sides. Hao Wang says that Gödel had discovered partial proofs in 1941 and 1943, and never wrote them up. Gödel told Wang in 1976 or 1977 that he was ready to reconstruct his proofs when his health improved; but he died first. Hao Wang, "Some Facts About Kurt Gödel," Journal of Symbolic Logic, 46 (1981) 653-59, at p. 657. The man who catalogued Gödel's papers after his death, John Dawson, believes that nothing of large mathematical importance is likely to be found in Gödel's untranscribed notebooks with the single possible exception of the reputed 1942 independence proof of the axiom of choice. On the other hand, Dawson writes, while Gödel was "fastidious" in polishing his work before releasing it for publication, "there is no evidence that he actively withheld important mathematical discoveries". John W. Dawson, Jr., "Kurt Gödel in Sharper Focus," in S.G. Shanker (ed.), Gödel's Theorem in Focus, London: Croom Helm, 1988, pp. 1-16, at p. 13. Gödel was also generous in his praise of Paul Cohen's 1963 proof, describing it as "the greatest advance in abstract set theory since its foundation by Georg Cantor." Dawson, ibid., p. 15, note 8. [Resume]
I would like to thank F. E. Robertson for helping me remove an error from the online version of this article.
Earlham College, Richmond, Indiana, 47374,
firstname.lastname@example.org. Copyright © 1992, Peter Suber.