From proof theory to theories theory

What is a theory? An algorithm. The logic and the theories When constructing a proof, we can use rules that are specific to the objects the proof speaks about and others that are ontologically neutral. We call the rules of the first kind theoretical rules and those of the second logical rules. The position of the border delimiting the logic from the theories is not always clear and, for instance, it can be discussed if the notion of set belongs to the realm of logic or to that of a specific theory. It seems that the dominant point of view at the end of the XIXth and at the beginning of the XXth century — Frege, Whitehead and Russell, . — was that this notion of set, or of concept, was logical. But Russell’s paradox seems to have finally ruined this point of view: as we must give up full comprehension, and either restrict comprehension to specific propositions, as in Zermelo set theory, or add a type system as in the Principia Mathematica, there are too many ways to do so — with or without the replacement axiom, with or without transfinite types — hence the notion of set is theoretical. We may speculate that, even if full comprehension had not been contradictory, this idea that the notion of set is ontologically neutral would have been finally given up, as even with ∗École polytechnique and INRIA. LIX, École polytechnique, 91128 Palaiseau Cedex, France. gilles.dowek@polytechnique.edu, http://www.lix. polytechnique.fr/~dowek full comprehension, there still would have been choices to be made for the set existence axioms: first because comprehension depends on the choice of a language, then because other set existence axioms, such as the axiom of choice may be added. This understanding of the theoretical nature of the notion of set has lead, at the end of the twenties, to a separation of set theory from predicate logic and to the constitution of predicate logic as an autonomous object. As a corollary, it has been assumed that any theory should be expressed as a set of axioms in predicate logic. As we shall see, and this will be the main thesis of this paper, this corollary is far from being necessary. From a historical point of view The constitution of predicate logic as an autonomous object, independent of any particular theory, and the simplicity of this formalism, compared to any particular theory such as geometry, arithmetic, or set theory, has lead to the development of a branch of proof theory that focuses on predicate logic. A central result in this branch of proof theory is the cut elimination theorem: if a sequent Γ ` A has a proof, it also has a cut-free proof, both in natural deduction and in sequent calculus. Once we have a notion of proof of a sequent in predicate logic, we may define a notion of proof of a sequent in a theory T , defined as a set of axioms, and say that a proof of the sequent Γ ` A in the theory T is a proof in predicate logic of a sequent Γ, T ′ ` A where T ′ is a finite subset of T . And from results about proofs in predicate logic, we may deduce results about proofs in an arbitrary theories. For instance, a cut elimination theorem: if a sequent Γ ` A has a proof in a theory T , it has a cut free proof in this same theory. Yet, although such results do have some content, they often are disappointing: they do not say as much as we could expect. For instance the cut free proofs of a sequent ` A in predicate logic always end with an introduction rule, but this in not the case for the cut free proofs in a theory T . Thus, while a corollary of the cut elimination theorem for predicate logic is that there is no proof of the sequent ` ⊥ in predicate logic, this corollary does not extend to arbitrary theories. And, indeed, there are contradictory theories. Thus, this definition of a cut free proof of a sequent Γ ` A in a theory T as a cut free proof of a sequent Γ, T ′ ` A for some finite subset T ′ of T is