Kurt Gödel

Happy 100th birthday, Kurt Gödel!

Gödel proved a series of fundamental results in logic, starting with the completeness theorem for predicate logic, which he proved in his doctoral dissertation in 1929. The incompleteness theorems (1931) are what he is famous for, but his work in set theory and constructivism has also been of great importance. In the incompleteness theorems Gödel proved fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system.

This theorem has been used over and over again in all kinds of fields — especially in the human sciences. Even if over-deployed, it is a useful way of thinking about other than mathemetical systems. I especially like what Régis Debray did with Gödel’s Incompleness theorem in his (badly translated, heavily abridged and unhappily much maligned) Critique of Political Reason. Here’s one quote from that book:

The statement of the “secret” of collective misfortunes, that is to say, of the a priori condition of all political history, past, present and to come, is to be found in a few words, simple and childish…This secret has the form of a logical law, a generalization of Gödel’s theorem: there is no organized system without closure, and no system can be closed with the help only of the elements belonging to the system…

Re which Michel Serres commented:

Régis Debray applies to social groups or finds in them the theorem of incompleteness, which holds for formal systems, and shows that societies only organize themselves on the express condition that they are founded on something other than themselves, external to their definition or boundary. They cannot be self-sufficient. He calls this foundation religious. By way of Gödel he completes Bergson.

Yet, as Kevin Mulligan notes in a review of the infamous Sokal book (from which review the above quotes are also taken), Debray, in a text subsequent to that quoted above, claims that “Gödelity is an illness that has become widespread.”

Information about Gödel’s life and work can be found in the Collected Works (Oxford University Press, ed. Solomon Feferman et al.) and in the book by Hao Wang, Reflections on Kurt Gödel. There is also a biography by John Dawson, Logical Dilemma: The Life and Work of Kurt Gödel.