1 December 2017, Cool Logic, Yvette Oortwijn
Michael Dummett has a variety of arguments for why we should favour intuitionistic over classical logic. Most of his arguments attack the complete realism one needs to believe in bivalence, but there is one argument concerning mathematics specifically, based on a phenomenon he calls indefinite extensibility. We see what this phenomenon is and why it matters for the foundation of mathematics.
Now, most of us think that getting rid of naive comprehension got us out of the biggest problems of naive set theory. With this we abandon the possibility of forming a set of all sets and dodge all kinds of paradoxes. According to Dummett, though, this is not enough. We got rid of a symptom, but there still exists an underlying problem. He argues that the only sensible thing to do is to adopt intuitionistic logic. But is this actually the case? And is it really sensible to claim that unrestricted quantification should be possible?
We will look into a different solution: the potential hierarchy of sets, as formulated by Linnebo. This view of sets gives an explanation of why unrestricted quantification is not possible, instead of merely restricting it. This account of the hierarchy of sets also sheds new light on the abandonment of naive set theory.