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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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"Axiom of global choice"
I think that you should place yourself in ZFC+ existence of strongly innaccessible cardinals.
Then the existence of a strongly inaccessible cardinal provides you a universe as in Borceux's Handbook of …