The asymmetry of mathematical determinacy: understanding why arithmetic is determinate and set theory indeterminate.

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Most mathematicians and philosophers implicitly accept that arithmetic, the mathematical investigation of natural numbers and their basic operations, is determinate, i.e. all of its statements are either true or false. Nonetheless, many of them also suspect that some statements of set theory, the part of mathematics that deals with collections of objects as well as infinity, could be indeterminate, i.e. lack a truth value. In my project, I try to ground these intuitions by providing philosophical and technical arguments whose ultimate goal is to support a form of set-theoretic pluralism, that is, the idea that, unlike arithmetic, set theory presents a multiplicity of universes. And, on the way there, I hope to grasp what it means for something to be determinate and, more importantly, to reveal some of the hidden features of our everyday use of mathematics and some of the wonders of the infinite.