What do we know when we know that a theory is consistent?

Given a first-order theory and a proof that it is consistent, can we design a proof-search method for this theory that fails in finite time when it attempts to prove the formula False

From proof theory to theories theory

In the last decades, several objects such as grammars, economical agents, laws of physics... have been defined as algorithms. In particular, after Brouwer, Heyting, and Kolomogorov, mathematical proofs have been defined as algorithms. In this paper, we show that mathematical theories can be also be defined as algorithms and that this definition has some advantages over the usual definition of theories as sets of axioms