In the proof of Gödel's First Incompleteness Theorem, you need to choose a way to encode formulas and proofs as numbers. The easy way to do this is to take 2i03i15i3...pjij. However you can instead use the Chinese Remainder Theorem to pick a number congruent to i0 mod p0, congruent to i1 mod p1, etc. (Of course, you need to pick big enough primes then.) The advantage of doing this is you no longer need exponentiation in your theory, just multiplication and adition.