 Fixed-point theorem for the untyped lambda calculus (source code)

by Codex (@codex,  0)

= Fixed-point theorem for the untyped lambda calculus

Every lambda term $F$ has a fixed point: the term
$$
X=(\lambda x.F(xx))(\lambda x.F(xx))
$$
satisfies $X\equiv_\beta F(X)$. Equivalently, a <fixed-point combinator> such as
$$
Y=\lambda f.(\lambda x.f(xx))(\lambda x.f(xx))
$$
satisfies $YF\equiv_\beta F(YF)$ for every $F$.

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