The sequence Fk used in this proof corresponds to the Kleene chain in the proof of the Kleene fixed-point theorem.
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Fixed-point combinators, which are used in lambda calculus for the same purpose as the first recursion theorem.
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This restriction is similar to the restriction to continuous operators in the Kleene fixed-point theorem of order theory.
Kleene's recursion theorem, also called the fixed point theorem, in computability theory
Liouville's theorem | Chinese remainder theorem | Shannon–Hartley theorem | Recursion | Quillen–Suslin theorem | Nyquist–Shannon sampling theorem | Hahn–Banach theorem | Fermat's Last Theorem | Buckingham π theorem | Thue–Siegel–Roth theorem | Szemerédi's theorem | Stephen Cole Kleene | Schottky's theorem | Riemann-Roch theorem | Pythagorean theorem | Nash embedding theorem | Müntz–Szász theorem | Malgrange–Ehrenpreis theorem | Kleene fixed-point theorem | Kakutani fixed-point theorem | Gauss–Bonnet theorem | Doob's martingale convergence theorem | Dirichlet's theorem on arithmetic progressions | Denjoy theorem | Birch's theorem | Wilkie's theorem | Wick's theorem | Whitney extension theorem | Weierstrass theorem | Wedderburn's little theorem |
Those most popular in the literature are three-valued (e.g., Łukasiewicz's and Kleene's, which accept the values "true", "false", and "unknown"), the finite-valued with more than three values, and the infinite-valued, such as fuzzy logic and probability logic.