Two tiny additions: 1. In those proof assistants which don't have the axiom of choice built-in, you can still formalize proofs depending on this axiom by putti

Two tiny additions: 1. In those proof assistants which don't have the axiom of choice built-in, you can still formalize proofs depending on this axiom by putting it as an extra assumption. One repository using this style which I particularly like is the one by Martín Escardó: https://www.cs.bham.ac.uk//~mhe/TypeTopology/index.html https://www.cs.bham.ac.uk//~mhe/TypeTopology/index.html He even takes care to explicitly put axioms as visible assumptions which are almost always taken for granted, such as function extensionality. 2. You don't need the axiom of choice to construct the real numbers, in fact several constructions are available (Dedekind cuts, Cauchy sequences, Cauchy processes, ...) and all do their job. It's just that without the (countable) axiom of choice, you cannot prove that the Cauchy real numbers are complete. But the other constructions work fine even in the absence of choice.