Epistemic Arithmetic

This course examines epistemic extensions of formal arithmetic. We will begin by reviewing formal theories of arithmetic, including Peano Arithmetic, and Gödel's incompleteness theorems. With this foundation in place, we will study extensions of Peano Arithmetic that introduce an operator meant to represent the knowledge of an ideal mathematician. One of the key topics covered in the course is the Knower Paradox, which arises when the knowledge operator is treated as a predicate. We will analyze different versions of the paradox and evaluate proposed solutions. The course also explores attempts to formalize Gödel's disjunction, which asserts that either no algorithm can fully capture human mathematical reasoning or there are absolutely undecidable problems. While it will not be possible to cover all formal details of Gödel's theorems, our goal is to provide students with a clear understanding of the central ideas, the significance of the Knower Paradox, and the philosophical implications of the incompleteness theorems.