Can Russell's paradox, Halting problem and Godel's Incompleteness theorem be generalized?

Yes: see Lawvere's fixed-point theorem.

There is an expository paper by Yanofsky covering exactly the results you mentioned (and more): A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points.

See also this MathOverflow question.

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Set Theory

Model Theory

Computational Mathematics

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