Gödel on the net

Every day, Gödel's incompleteness theorem is invoked on the net to support some claim or other, or just to whack people over the head with it in a general way. In news, we find such invocations not only in sci.logic, sci.math, comp.ai.philosophy, sci.philosophy.tech and other such places where one might expect them, but with equal frequency in groups dealing with politics or religion, and indeed in alt.cuddle, soc.culture.malaysia, rec.music.hip-hop, and what have you. In short, whenever a bunch of people get together on the net, sooner or later somebody will invoke Gödel's incompleteness theorem.

Unsurprisingly, the bulk of these invocations covers a range from the nonsensical to the merely technically inaccurate, and they often give rise to a flurry of corrections and more or less extended technical or philosophical disputes.

My purpose in these pages is to provide a set of responses to many such invocations, couched in non-confrontational and hopefully helpful and intelligible terms. There are few technicalities, except in connection with a couple of technical (and less frequently raised) issues. All of my comments and explanations are intended to be non-controversial, in the sense that people who are familiar with the incompleteness theorem can be expected to agree with them. (Thus, for example, I don't present any criticism of so-called Gödelian arguments in the philosophy of mind, but only a couple of technical observations relevant for the discussion of such arguments.)

I have included a sketch of how the incompleteness theorems can be proved using the so-called Gödel sentence for a theory. This, Gödel's original proof, is the proof that most often prompts the various ideas and arguments commented on on these pages. I've added a few biographical facts, with references to the literature.

A rather full presentation of Gödel's theorem, which includes expositions of set theory and formalized arithmetic, is given in Around Gödel's Theorem, a "Hyper-textbook for students in mathematical logic" made available on the web by Karlis Podnieks. (The author also has a philosophical axe to grind, as quickly becomes apparent, but this does not detract from the value of the exposition.)

The list below contains links to comments on the claims or questions listed. I hope the list will be expanded and the comments improved as time goes on.

Comments that are quoted without attribution (and shown in green on color screens) are taken from Usenet postings.

Torkel Franzén's webpage (archived)