Speaker
Hans Halvorson
Princeton University
Talks at this conference:
Friday, 15:40, J330 
Equivalence in foundations 
I survey results — recent as well as historical — about the equivalence or inequivalence of systems that could serve as the foundation of mathematics. Throughout I focus on clarifying which notion of equivalence is operative in the various results, in the hope of arriving at a more clear sense of which notions are relevant in mathematical practice. The known results that I discuss include:
I consider precise definitions of equivalence including:
While biinterpretability is, according to Hamkins, the “gold standard for equivalence”, there remains some unclarity in its definition — in particular, in the definition of “translation” on which biinterpretability is built. I show that two theories can be Morita equivalent without being biinterpretable, even under the most liberal interpretation of the latter notion. This mismatch between Morita equivalence and biinterpretability raises a puzzle about the classical result [2] that any theory formulated in manysorted logic can be replaced by an unsorted theory. While the result is true for Morita equivalence for theories with finitely many sorts, it fails for infinitely many sorts, and even more severely for biinterpretability. I attempt to bring order to this situation by considering various definitions of a translation between theories, and by comparing the resulting notion of equivalence with Morita equivalence. Bibliography
