A semantic approach to independence
Giorgio Venturi (Universidade Estadual de Campinas)
A practical response to the philosophical issues that followed the development of mathematics at the end of the XIX century consisted in a revival of the axiomatic method. Through the innovative work and the authority of David Hilbert, the simultaneous development of logic and mathematics lead to an attempt to discharge the ontological and epistemological controversies onto the axiomatic presentation of a theory. Even set theory underwent the same transformation and eventually ZFC was accepted as a foundation for the whole mathematics. Nonetheless the faith in the axiomatic solution to every set-theoretical problem was proved unfounded by the plethora of independent results produced by forcing. In order to over- come the problem that the independence phenomenon posed to mathematical truth, Gödel  proposed to extend ZFC with new principles able to solve questions like CH. Contrary to Gödel’s hope in an uncontroversial axiomatic solution, his program opened once again the philosophical debates that the axiomatic method was meant to solve. Indeed, the so-called Gödel’s program has shown its limits in deciding between competing, incompatible extensions of ZFC , . But, then, how to overcome the limits of independence by purely mathematical tools?
In order to approach this problem we suggest to turn upside down the traditional perspective on independence and instead of completing the syntactic side, we propose to complete set theory semantically by applying to set theory model theoretic techniques meant to produce complete models. The study of model-theoretics tools to complete theories goes back to the work of Robinson and to the concept of model completeness from the period 1950–1957. This notion, together with those of existentially complete theories and model companion- ships, aimed at generalizing to other algebraic contexts, the peculiar role that algebraically closed fields play with respect to the class of fields.
We will discuss the philosophical significance of this new approach together with some preliminary results obtained by the application of Robinson infinite forcing to the collection of set-forcing extensions of models of ZFC.
In  it is shown that Robinson infinitely generic structures exist and that they validate many generic absoluteness principles like Maximality Principles, Resurrection Axioms, and Bounded Forcing Axioms. This unified perspective, thus, shows an interesting connection be- tween a semantic perspective and generic absoluteness results. Moreover, in , this new model theoretic approach and Woodin’s classical results on the absoluteness of second order arithmetic are combined to show the existence of the model companion of ZFC plus large cardinals.
We believe that we are only scratching the surface of a new promising interaction be- tween model theory and set theory, able to produce interesting new results and a deeper understanding of set theory.