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.

- Can Gödel's theorem really be proved, like ordinary mathematical theorems?
- From Gödel's theorem, we know that the Bible is necessarily incomplete or inconsistent.
- Gödel showed that we can't really prove anything in mathematics!
- By Gödel's second incompleteness theorem, we can't know that mathematics is consistent.
- Gödel's theorem shows that there can't be any complete and consistent theories in mathematics.
- Gödel showed that we can "step outside the system" and see the truth of propositions that can't be proved within the system.
- Gödel's theorem is all about the power of self-reference!
- By Gödel's theorem, there are truths that can't be proved.
- Nah, Gödel's theorem has nothing to do with truth!
- Did Gödel really prove that the constitution of the US is inconsistent?
- Did Gödel really have a proof of the existence of God?
- Is there really such a thing as Gödel's completeness theorem?
- Were Gödel and Einstein really buddies?

Torkel
Franzén's webpage (archived)

[Torkel
Franzén: 1 April 1950 – 19 April 2006]