Gödel's speed-up theorem
Kurt Gödel showed how to find explicit examples of statements in formal systems that are provable in that system but whose shortest proof is absurdly long. For example, the statement:
is provable in Peano arithmetic (PA) but the shortest proof has at least a googolplex symbols, by an argument similar to the proof of Gödel's first incompleteness theorem: PA (if consistent) cannot prove the statement in fewer than a googolplex symbols, because the existence of such a proof would itself be a theorem of PA, that would contradict the statement which PA supposedly proved. But simply enumerating all strings of length up to a googolplex and checking that each such string is not a proof (in PA) of the statement, yields a proof of the statement that is necessarily longer than a googolplex symbols.
The statement has a short proof in a more powerful system: in fact it is easily provable in Peano arithmetic together with the statement that Peano arithmetic is consistent (which, per the incompleteness theorem, cannot be proved in Peano arithmetic).
In this argument, Peano arithmetic can be replaced by any more powerful consistent system, and a googolplex can be replaced by any number that can be described concisely in the system.
Harvey Friedman found some explicit natural examples of this phenomenon, giving some explicit statements in Peano arithmetic and other formal systems whose shortest proofs are ridiculously long (Smoryński 1982).
If one takes Peano arithmetic together with the negation of the statement above, one obtains an inconsistent theory whose shortest contradiction is ridiculously long.
- Buss, Samuel R. (1994), "On Gödel's theorems on lengths of proofs. I. Number of lines and speedup for arithmetics", The Journal of Symbolic Logic 59 (3): 737–756, doi:10.2307/2275906, ISSN 0022-4812, MR 1295967
- Buss, Samuel R. (1995), "On Gödel's theorems on lengths of proofs. II. Lower bounds for recognizing k symbol provability", in Clote, Peter; Remmel, Jeffrey, Feasible mathematics, II (Ithaca, NY, 1992), Progr. Comput. Sci. Appl. Logic 13, Boston, MA: Birkhäuser Boston, pp. 57–90, ISBN 978-0-8176-3675-3, MR 1322274
- Gödel, Kurt (1936), "Über die Länge von Beweisen", Ergebinisse eines mathematischen Kolloquiums (in German) 7: 23–24, Reprinted with English translation in volume 1 of his collected works.
- Smoryński, C. (1982), "The varieties of arboreal experience", Math. Intelligencer 4 (4): 182–189, doi:10.1007/bf03023553, MR 0685558