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forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.

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Essential reads in the philosophy of mathematics and set theory

I think since the fundamentals of the philosophy of mathematics have been well-established above, I'd like to submit Albert Lautmann's Mathematics, Ideas and the Physical Real and the more recent (and …