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Extra info for 1998 European Summer Meeting of the Association for Symbolic Logic. Logic Colloquium '98
Meanings are given in term of actions and are adapted to states using the view of state formulae given above. B , actions: to do A or to do B but one does not know which one (outer non-determinism), states: one of A and B is true but one does not know which one; (Dependent sequentiality) A > B, actions: to do B by beginning to do A , states: B is true but one must first prove A in order to prove B: (Independent sequentiality) A >> B , actions: to do A and then to do B, states: A and B are true but A is proved before B; (Logical implication) A + B, actions: if you know how to do A , you can do B, states: B can be logically deduced from A: (Causality) A --+ B, action: doing A causes doing B, states: nothing; (Universality)V x .
E-mail: daquinomunina. it. ) Let M be a model of IAo + and K be the residue field of M for a nonstandard prime pin M. e.. it satisfies the following axioms of Ax (1) there exists a unique extension of each degree n ; (2) every absolutely irreducible curve has a point in the field. In the case of Open Induction. Macintyre and Marker proved in  that for any field L of characteristic 0 there is a model M of Open Induction and a prime p E M such that the residue field K is elementary equivalent to L.
Prove() does A. Axiomatisation. Our logic is a sequent calculus. The axiomatisation is made difficult because of the need of linearity. Moreover, realizations are imperative programs which cannot be combined in any way. In the sequent, the 8 connective intuitively links formulae on both sides of the entailment sign t-. Therefore, our structural rules are unusual: These rules make the proofs much more natural than classical sequent calculus rules by allowing a linear structure of proofs. They also allow a more human-like automatic research of proofs.