He is best known for the development of the lambda calculus, his 1936 paper that showed the existence of an "undecidable problem" in it. This result preempted Alan Turing's famous work on the halting problem which also demonstrated the existence of a problem unsolvable by mechanical means. Supervising Turing's doctoral thesis, they then showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities, and subsequently demonstrated a variety of alternative "mechanical processes for computation" had equivalent computational abilities. This resulted in the Church-Turing thesis, which is also known as Church's Thesis and Turing's Thesis as there is dispute about who proposed it first.

Church's other doctoral students included Stephen Kleene.

Church remained a professor of mathematics at Princeton until 1967, when he moved to California.

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