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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.
2
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Finding the largest integer describable with a string of symbols of predefined length
Scott Aaronson has written a lovely essay on just this question, although focusing on computation instead of logic. http://www.scottaaronson.com/writings/bignumbers.html