# Bounded versions of Gödel's and Löb's theorems

While writing up decision theory results, I had to work out bounded versions of [Gödel's second incompleteness theorem](http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems#Second_incompleteness_theorem) and [Löb's theorem](http://en.wikipedia.org/wiki/L%C3%B6b's_theorem). Posting the tentative proofs here to let people find any major errors before adding formality. Any comments are welcome!

**Bounded Gödel's theorem**

Informal meaning: if a theory T has a short proof that any proof of T's inconsistency must be long, then T is inconsistent.

Formal statement: there exists a total computable function f such that for any effectively generated theory T that includes Peano arithmetic, and for any integer n, if T proves in n symbols that "T can't prove a falsehood in f(n) symbols", then T is inconsistent.

**Proof**

If the theory T takes more than n symbols to describe, the condition of the theorem can't hold, so let's assume T is smaller than that.

Let's take g(n)>>n and f(n)>>exp(g(n)). For any integer n, let's define a program R that enumerates all proofs in the theory T up to length g(n), looking for either a proof that R prints something (in which case R immediately halts without printing anything), or a proof that R doesn't print anything (in which case R prints something and halts). If no proof is found, R halts without printing anything.

Assume that the theory T is consistent. Note that if the program R finds some proof, then it makes the proved statement false. Also note that the theory T can completely simulate the execution of the program R in exp(g(n)) symbols. Therefore if the program R finds any proof, the resulting contradiction in the theory T can be "exposed" using exp(g(n)) symbols. Since f(n)>>exp(g(n)), and the theory T proves in n symbols that "T can't prove a falsehood in f(n) symbols", we see that T proves in slightly more than n symbols that the program R won't find any proofs. Therefore the theory T proves in slightly more than n symbols that the program R won't print anything. Since g(n)>>n, the program R will find that proof and print something, making the theory T inconsistent. QED.

(The main idea of the proof is taken from [Ron Maimon](http://mathoverflow.net/questions/72062/what-are-some-proofs-of-godels-theorem-which-are-essentially-different-from-th/72151#72151), originally due to Rosser, and adapted to the bounded case.)

**Bounded Löb's theorem**

Informal meaning: if having a bounded-size proof of statement S would imply the truth of S, and that fact has a short proof, then S is provable.

Formal statement: there exists a total computable function f such that for any effectively generated theory T that includes Peano arithmetic, any statement S in that theory, and any integer n, if T proves in n symbols that "if T proves S in f(n) symbols, then S is true", then T proves S.

**Proof**

Taking the contrapositive, the theory T proves in n symbols that "if the statement S is false, then T doesn't prove S in f(n) symbols". Therefore the theory T+¬S proves in n symbols that "T+¬S doesn't prove a falsehood in f(n) symbols" (I'm glossing over the tiny changes to n and f(n) here). Taking the function f from the bounded Gödel's theorem, we obtain that the theory T+¬S is inconsistent. That means T proves S. QED.

(The main idea of the proof is taken from [Scott Aaronson](http://www.scottaaronson.com/democritus/lec3.html), originally due to Kripke or Kreisel.)

As a sanity check, the regular versions of the theorems seem to be easy corollaries of the bounded versions. But of course there could still be errors...