The relentless champion was even smiling to himself as he turned for the final length, further proof that he senses almost anything is within his telescopic reach.
The case of the Speed-up Theorem where the length of proof is measured by the number of symbols was proved by Mostowski in 1952 (Mostowski 1982).
We turn our grudge into an object and hold it out at arm's length -- proof of what we have suffered, a badge of honor, a way to remind others and ourselves of our pain and deserving-ness.
This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers.
The computation time that was previously ignored now appears explicitly as the length of proofs [17] and theories can naturally evolve by the addition and transformation of proofs in the collection, with new concepts emerging by the introduction of cuts and definition inferences.
Gödel measures the length of proofs by the number of formulas; but there are also other possibilities, such as the number of symbols in the proof.
Other publications of the 1930s include those on the decision problem for the predicate calculus, on the length of proofs, and on differential and projective geometry.
It must be anticipated anyhow that, due to their length, all proofs in this section have been left in a working-paper.8.
We aim for an implementation enabling us to formalise, within the logical framework, proof-theoretic results such as the cut-elimination theorem for δRA, and any associated increase in proof length.
This method is proven to be asymptotically optimal among brute-force strategies in terms of proof length, space and time [4].
Though both measures may be equally natural candidates for measuring the length of a proof, proving the theorem for length measured by the number of symbols avoids a technical complication introduced by the other measure: there are only finitely many proofs with a given number of symbols, whereas there are infinitely many proofs with a given number of formulas.
