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wuncidunci comments on Proofs, Implications, and Models - Less Wrong
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My data point: as an undergraduate mathematician at Oxford, the mathematical logic course was one of the most popular, and certainly covered most of this material. I guess I haven't talked a huge number of mathematicians about logic, but I'd be pretty shocked if they didn't know the difference between syntax and semantics. YMMV.
Another data point: in Cambridge the first course in logic done by mathematics undergraduates is in third year. It covers completeness and soundness of propositional and predicate logic and is quite popular. But in third year people are already so specialised that probably way less than half of us take it.