What are the most important questions or areas of study in the philosophy of mathematics? This question is intended to complement What mathematical questions or areas have philosophical implications outside of mathematics?

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*Formal Undecidability and related topics (Tarski truth definability, Godel incompleteness, The hydra problem, Matiyasevich's negative solution of the 10th problem)

*Proof Theory (Gentzen's consistency proof, Girard's linear logic, Barendregt's Lambda Cube)

*The Continuum (Cantor's diagonalization, Cohen and Godel's independence proof, Robinson's analysis)

