Gödel, and his Incompleteness Theorem
"Provability is a weaker notion than truthÖ"
- Douglas R. Hofstadter
Honorís Collegium 41
The concept of Completeness captivates mankind because of its infinite implications. Completeness bestows upon a body of knowledge a stigma of high aptitude, but more importantly illustrates a final state incapable of being improved upon. Completeness, in a conventional, non-technical sense, simply means: to make whole with all necessary elements or parts. The finality of any work that is "complete" should be the goal of every creative individual. In 1931, Kurt Gödel ís Incompleteness Theorem illustrated that in a mathematical system there are propositions that cannot be proved or disproved from axioms within the system. Moreover, the consistency of axioms cannot be proved. Such a shattering theorem wrought havoc within the mathematical community. Partially due to its disturbing consequences, Gödelís Incompleteness Theorem has remained one of the lesser known (though most profound) advancements of this century. With its 1931 publication, Uber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme showed that a sense of "completeness" for the mathematical community was out of reach in certain respects. That is to say, "It's not really math itself that is incomplete, but any formal system that attempts to capture all the truths of mathematics in its finite set of axioms and rules."
Mathematics in the late 19th and early 20th century was undergoing a renovation. Formal systems and works such as Bertand Russell and Alfred North Whitehead's Principia Mathematica were being devised following strict rules of inference. In a system where theorems came from axioms like so many branches from a tree, this new paradigm in mathematics was giving a new face to the typically haphazard mathematical muddle; where vague intuitions and exact logic had previously been on an equal plane. Lofty ideals of a clear-cut axiomatic mathematical system came to a screeching halt when Gödel emerged. Few breakthroughs have left such sour tastes in the mouths of a scientistís contemporaries.
Gödelís theorem not only brought the end of formalism (as formalism was known in 1931), but it also cast shadows of doubt on far reaching subjects. As Boyer stated in History of Mathematics, "one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions...It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced." Though Gödelís Theorem cast much into doubt, it nevertheless brought about the birth of recursive functions, not to mention the genesis of a new way of thinking. An apparent curse on all whom strove for absolute truth, Gödel's Incompleteness Theorem quickly had scientists rethinking the definition of provability.
Gödelís theorem is deeply associated with the concept of loops, and self-reference. "The proof of Gödelís Incompleteness Theorem hinges upon the writing of a self-referential mathematical statementÖ" Such statements have been seen throughout antiquity. The Epimenides paradox "All Cretians are liars" is a self-referential statement of language. Having language speak of language is not a large task; as Hofstadter goes on to say in Gödel, Escher, and Bach, "whereas it is very simple to talk about language in language, it is not at all easy to see how a statement about numbers can talk about itself." We may only measure the true genius of Gödel when we witness the delicate way in which he illustrated that math can speak of math.
Kurt Gödel was born on April 28, 1906 at Brünn in Moravia. Brünn (known today as Brno) was the most important city in Moravia because of its position as the center of the textile industry. Moravia, a predominantly Czechoslovakian region, also had a small German speaking population; it was to this minority that Gödelís parents belonged. At the age of 5, Gödel had what his brother, Rudolf, characterized as a "light anxiety neurosis, which later completely disappeared." When Gödel was still a young child he suffered from a severe bout of rheumatic fever. An apparently life changing event, Gödelís brother believes that it was this episode which led to Gödelís lifelong preoccupation with his health Ė a preoccupation which some may characterize as hypochondria. Rudolf Gödel on this subject says, "At about the age of eight my bother had a sever joint-rheumatism with high fever and thereafter he was somewhat hypochondriacal and fancied himself to have a heart problem, a claim that was, however, never established medically."
As a young boy, Gödel gave the impression of a troubled child. He often felt uneasy when his mother was absent, or when he lost a game (usually chess). The youngster earned the name "Mr. Why" (der Herr Warum) because of his overactive inquisitiveness. This natural curiosity must have served him well; as a student in secondary school (1916-1924) Gödel is remembered as never having made a mistake in Latin grammar (throughout all his schooling!). Gödelís geometry homework and report cards are still preserved, his geometry work showing a natural and keen draftsmanship. The young man only once received a grade that was not highest in his class, this was, ironically, in the subject of math.
Gödel reports that his interest in math started around the age of 14. Like Einstein, this genius was also largely self-taught. Rudolf Gödel states that Kurt Gödel learned a good deal of math before entering university. In secondary school, Gödelís ability was well above the maximum required. Gödel and Einstein have many interesting similarities; indeed, it is no surprise that the two intellects became good friends at Princeton. Though the two are alike in the their exceptional minds, self-taught background, and breakthrough theories, the two men did share their differences. Gödel, unlike Einstein, concentrated on philosophy for great lengths of time. Hao Wang, in Reflections on Kurt Gödel, attributes this to the fact that, "Einstein saw a sharper distinction between science and philosophy, and was less sanguine of achieving sufficiently definite results in philosophy." Maurice Solovine once wrote to Carl Seelig about Einstein, "when he was younger, [he] had a strong taste for philosophy, but the vagueness and arbitrariness which reigned there had turned him against it, and that he was now concerned solely with physics."
Historians and Mathematicians agree, 1930 was Gödelís most profound year Ė if one was to include the latter part of 1929 as well. It is in this year that Gödel states he first heard of Hilbertís proposed outline of a proof of the continuum hypotheses. In the summer, Gödel began work on trying to prove the relative consistency of analysis. Gödel soon discovered that truth in number theory is undefinable Ė he later went on to prove a combinational form of the Incompleteness Theorem.
In 1930, Gödel traveled several days to attend the Second Conference on Epistemology of the Exact Sciences (September 5-7). Towards the end of the Conference on the last day, Gödel spoke for the first time and, "criticized the formalist assumption that consistency of Ďtransfiniteí axioms assures the nonderivability of any consequence that is Ďcontentually false.í He concluded, ĎFor of no formal system can one affirm with certainty that all contentual considerations are representable in it.í And then v. Neumann interjected, ĎIt is not a foregone conclusion whether all rules of inference that are intuitionistically permissible may be formally reproduced.í" It was after this statement, that Gödel made the announcement of his incompleteness result, "Under the assumption of the consistency of classical mathematics, one can give examples of propositionsÖthat are contentually true, but are unprovable in the formal system of classical mathematics." It was these events which preceded the formal 1931 publishing of Gödelís article Uber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme.
In 1931, by the age of twenty-five, Gödel had published two very important papers. For the next brief period in his life, the logician would concentrate his work in mathematics and physics, lectures on his 1931 paper, and research on special topics in logic and geometry; which was published unofficially in association with Mengerís colloqium. In this time, Gödel had only one full length, formal publication: a ten-page paper on the Entscheidungsproblem (published in 1933, in the Montashefte). Throughout all this prolific work, Gödel was very reclusive about his achievements to his family; he would "hide his light under a bushel," as Rudolf Gödel put it.
As with other European scientists, the Nazi threat led to an immigration to the United States for Kurt Gödel, and his wife Adele. Before leaving, Hans Thirring warned Gödel to tell Einstein, "of the possibility of a German application of atomic fission to making bombs." Thirring, a mutual friend of Einstein and Gödel, foreshadowed what would ultimately become the Manhattan Project, and the turning point in World War II.
After having arrived to Princeton in 1940, Gödel achieved a great level of peace in his life. Both his marriage and professional carrier were more settled. As Hao Wang states, "There was no immediate external or internal pressure to complete an unfinished project or attain new heights quickly. He was in relatively good health, and could look forward to a long period of peace and quiet, devoid of the need to travel or change employment." During such a time, Gödel did not come out of his reclusive shell. Indeed it was the scientistís method to work out all problems in solitude. Both Einstein and Gödel agreed, "that thinking, the quest for truth, is an entirely solitary occupation." This however is not entirely true (at least on the part of Einstein); in the spring of 1905 Einstein went to talk to his friend M. A. Besso because, "Trying a lot of discussions with him, I could suddenly comprehend the matter [special theory]." Many believe that if Gödel had the benefit of such an able mathematician as a friend, he too would have been able to make more definite advances.
1949 was another excellent year for Gödel. In July he published solutions for Einsteinís field equations which were entirely novel. Einstein himself gave much credit to Gödel for making important advancements in the field. Gödel also published a discussion on the relationship between relativity theory and idealistic philosophy in the Einstein volume. This more technical paper was later followed by a more philosophical piece on the same subject. Also in this year, Gödel bought a house with his wife Adele on Linden Lane.
Gödel would spend his remaining years surrounded by his friends, though by and large living a most reclusive lifestyle. In his later years Gödel grew paranoid about the spread of germs, and he became notorious for compulsively cleaning his eating utensils and wearing ski masks with eyeholes wherever he went. He died at age 72 in a Princeton hospital, essentially because he refused to eat. Gödel left behind him a massive legacy of work and philosophy; his most notorious work being, of course, his Incompleteness Theorem. By showing us how "mathematics twists back on itself, like a self-eating snake," Gödel ushered in a new era of mathematics, and philosophy. Like most geniuses, he is remembered equally for his eccentricities as his mastery of subjects.