The notion of ordinal computability is defined by generalising standard Turing computability on tapes of length $\omega$ to computations on tapes of arbitrary ordinal length. The fundamental theorem on ordinal computability states that a set $x$ of ordinals is ordinal computable from ordinal parameters if and only if $x$ is an element of the constructible universe $\mathbf{L}$. In this paper we present a new proof of this theorem that makes use of a theory SO axiomatising the class of sets of ordinals in a model of set theory. The theory SO and the standard Zermelo–Fraenkel axiom system ZFC can be canonically interpreted in each other. The proof of the fundamental theorem is based on showing that the class of sets that are ordinal computable from ordinal parameters forms a model of SO.