The popular impact of Gödel's incompleteness theorem
Torkel Franzén · Epubl LTU · 2006
The author, whose untimely passing in April 2006 was a great loss to the logic community, used this short paper primarily to dispel a few popular and not so popular misinterpretations of Gödel's incompleteness theorems. The most obvious misconceptions arise in areas having no direct connection with mathematics. But even within scientific circles, it is useful for the author to have pointed out that no unsolved problem in "traditional mathematics" has been shown to be undecidable via Gödel's first incompleteness theorem. (The Paris-Harrington undecidable problem does have to do with standard mathematical concepts, but it was not obtained from Gödel's result.) The author also makes illuminating remarks about Gödel's second incompleteness theorem concerning unprovability of the consistency of sufficiently strong mathematical theories. For example, the role of consistency proofs in justifying mathematical reasoning has been overemphasized. Moreover, there are informal `proofs' on the same level as ordinary mathematical argumentation that may convince most mathematicians of the consistency of, say, Peano arithmetic. For a more extensive treatment of all of these matters, the author refers the reader to his recent book Gödel's theorem, A K Peters, Wellesley, MA, 2005.