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computable sets and functions, Turing degrees, c.e. degrees, models of computability, primitive recursion, oracle computation, models of computability, decision problems, undecidability, Turing jump, halting problem, notions of computable randomness, computable model theory, computable equivalence relation theory, arithmetic and hyperarithmetic hierarchy, infinitary computability, $\alpha$-recursion, complexity theory.

3 votes

What are the most attractive Turing undecidable problems in mathematics?

Type inference for sufficiently powerful type systems, e.g. System F
0 votes

A subset of all languages which is uncountable?

The Turing degrees are an uncountable collection of sets of languages (equivalence classes under Turing reduction). So if you choose one representative (canonical in some way) language from each degr …
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