Philosophy of Mathematics seminar

Peter Fritz (University College London)

Higher-order metaphysical resolutions of the Continuum Hypothesis

I aim to draw a connection between higher-order metaphysics and the philosophy of mathematics, in particular set theory. Higher-order metaphysics means carrying out metaphysical debates in higher-order logic, using higher-order quantifiers to regiment talk of propositions, properties, and relations. A prominent topic in this area is grain science, the investigation of individuation conditions of propositions, properties, and relations. These topics seem purely metaphysical. But I will argue that they are intimately connected to questions in (the philosophy of) mathematics. In particular, I will argue that views about grain science can resolve the continuum hypothesis. To do so, I will present an example of such a view. I won’t argue for it, but I hope to motivate, first, that the view is attractive, or at least not implausible; second, that the view doesn’t obviously prejudge controversial questions in (the philosophy of) set theory; and third, that the view nevertheless settles the continuum hypothesis. The view assumes that sets obey the principles of ZFC set theory, and that propositions form a structure which corresponds to a particular complete Boolean algebra. Adapting standard forcing results using Boolean-valued models, we can show that this higher-order metaphysical view entails the failure of the continuum hypothesis.