is used to establish the semantics of recursive definitions. While the fixed-point theorem is applied to the "same" function (from a logical point of view), the development of the theory is quite different. The same definition of recursive function can be given, in computability theory, by applying Kleene's recursion theorem. These results are not equivalent theorems; the Knaster–Tarski theorem is a much stronger result than what is used in denotational semantics. However, in light of the Church–Turing thesis their intuitive meaning is the same: a recursive function can be described as the least fixed point of a certain functional, mapping functions to functions. The above technique of iterating a function to find a fixed point can also be used in set theory; the fixed-point lemma for normal functions states that any continuous strictly increasing function from ordinals to ordinals has one (and indeed many) fixed points. Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place. Every involution on a finite set with an odd number of elements has a fixed point; more generally, for every involution on a finite set of elements, the number of elements and the number of fixed points have the same parity. Don Zagier used these observations to give a one-sentence proof of Fermat's theorem on sums of two squares, by describing two involutions on the same set of triples of integers, one of which can easily be shown to have only one fixed point and the other of which has a fixed point for each representation of a given prime (congruent to 1 mod 4) as a sum of two squares. Since the first involution has an odd number of fixed points, so does the second, and therefore there always exists a representation of the desired form. List of fixed-point theorems Atiyah–Bott fixed-point theorem Banach fixed-point theorem Bekić's theorem Borel fixed-point theorem Bourbaki–Witt theorem Browder fixed-point theorem Brouwer fixed-point theorem Caristi fixed-point theorem Diagonal lemma, also known as the fixed-point lemma, for producing self-referential sentences of first-order logic Discrete fixed-point theorems Earle-Hamilton fixed-point theorem Fixed-point combinator, which shows that every term in untyped lambda calculus has a fixed point Fixed-point lemma for normal functions Fixed-point property Fixed-point theorems in infinite-dimensional spaces Injective metric space Kakutani fixed-point theorem Kleene