MATHEMATICAL PLURALISM AND SOME GENERIC ABSOLUTENESS RESULTS IN SET THEORY: A PHILOSOPHICAL INQUIRY

The forcing technique was discovered by Paul Cohen in the early sixties. Since then forcing has appeared to be a very powerful tool to provide independence results in Set theory. Actually, because of the foundational role played by Set theory with regard to the rest of classical mathematics, and because of the possibility to mimic from the standard axiomatic basis of Set theory, ZFC, the proof of the existence of almost any mathematical object, forcing has been applied to different areas of mathematics revealing to us the undecidability of many different important questions connected with different branches of mathematics. Given the pervasive presence of the independence phenomenon in Set theory determined by forcing, a natural philosophical question arises: is forcing the ultimate horizon of Set theory, or is it (as a source of undecidability) to be considered as a pathology that needs to be neutralised? A special kind of results in Set theory, known in literature as generic absoluteness results, give mathematical substance to the perspective that the real challenge that the discovery of the forcing technique places to the set theorist, as well as to the philosopher of mathematics, goes beyond the idea that the right answer to questions such as the Continuum Hypothesis is given by computing the precise extent of their undecidability. In fact, when it is possible to relieve generic absoluteness for a certain mathematical structure, a different framework appears where forcing can be exploited and, so we may say, integrated into the practice of the mathematician as a strong tool for proving theorems. In chapter 1 of my dissertation I recall some main aspects of the forcing technique developed following the Boolean valued-models approach introduced by Scott, Solovay, and Vopenka starting from 1965. In chapters 2 and 3 I analyze some main motivations behind Viale's and Woodin's alternative strategies for producing generic absoluteness for the structure at the level of the Continuum problem. I try to stress, in particular, the essential use of the so-called forcing axioms that is inherent Viale's generic absoluteness results and that, to some extent, conflicts with Woodin's choice to introduce a strong logic as the appropriate setting for studying the possibility of generic absoluteness at the level of the Continuum problem. In chapter 4 I open the philosophical discussion and I try to correlate the pure mathematical phenomenon of generic absoluteness described in chapters 2 and 3 to the more general philosophical debate concerning the question of Pluralism in Set theory and the search for new axioms. Insofar as we are interested in spell out Viale's and Woodin's absoluteness results in terms of the right axiomatisation for the structure theory where the Continuum problem is expressible, I try to sketch an argument according to which the possibility to unify the two distinct theories offered by Viale and Woodin emerges as one of notable philosophical importance.