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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
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Understanding Specker's disproof of the axiom of choice in New Foundations
I think the answer (to your first question) is "yes". I'm not very familiar with New Foundations, but I can describe how it works in ZF(C), and hopefully this will give some insight.
So, if you are …