Gödel's speed-up theorem


In mathematics, Gödel's speed-up theorem, proved by, shows that there are theorems whose proofs can be drastically shortened by working in more powerful axiomatic systems.
Kurt Gödel showed how to find explicit examples of statements in formal systems that are provable in that system but whose shortest proof is unimaginably long. For example, the statement:
is provable in Peano arithmetic but the shortest proof has at least a googolplex symbols, by an argument similar to the proof of Gödel's first incompleteness theorem: If PA is consistent, then it cannot prove the statement in fewer than a googolplex symbols, because the existence of such a proof would itself be a theorem of PA, a contradiction. But simply enumerating all strings of length up to a googolplex and checking that each such string is not a proof of the statement, yields a proof of the statement.
The statement has a short proof in a more powerful system: in fact the proof given in the previous paragraph is a proof in the system of Peano arithmetic plus the statement "Peano arithmetic is consistent".
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. For example, the statement
is provable in Peano arithmetic, but the shortest proof has length at least A, where A=1 and A=2A. The statement is a special case of Kruskal's theorem and has a short proof in second order arithmetic.
If one takes Peano arithmetic together with the negation of the statement above, one obtains an inconsistent theory whose shortest contradiction is unimaginably long.