Forcing in Contemporary Philosophy of Set Theory

Beschreibung

In the early 1960s Paul Cohen developed the forcing technique as a method to show that the Continuum Hypothesis is independent from the standard axiomatization of set theory ZFC. On the mathematical side, the impact of forcing on set theory has been enormous for it allows to build models of ZFC+A, for (reasonable) mathematical statements A. Also, already from early on set theorists were aware of the influence this new technique could have on problems related to the philosophy of set theory. Most prominently among these are the search for new axioms, the related question about the existence of a unique set-theoretic universe and the truth value of independent statements.

In my research project I aim to provide a comprehensive discussion of the role of forcing in set theory including mathematical, philosophical and socio-historical aspects. Although some of these questions, like criteria for accepting axioms, have been widely discussed, others, like the universe/multiverse debate, only recently gained influence. This is due to larger research programs in the philosophy of set theory led by settheorists like Hugh Woodin, Joel Hamkins and Sy-David Friedman and work by philosophers like Peter Koellner. My contribution to the debate starts from the novel observation that the different directions in contemporary philosophy of set theory are not just triggered by the results forcing provides us with, but also by the way in which these results are obtained. Building on this observation, I claim that the method of forcing itself and the way(s) in which it is used by set theorists is one of the differentiating factors responsible for the philosophical conclusions that they draw in these programs.

Institutionen
  • Zukunftskolleg
  • FB Philosophie
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Laufzeit: 01.07.2016 – 31.03.2019