In computability theory, a set $A\subseteq\omega$ is called an index set if for all $x, y$

\begin{equation*} x\in A, \varphi_x=\varphi_y \Longrightarrow y\in A. \end{equation*} $\varphi_x$ stands for the partial function with Gödel number (or index) $x$

Thus, if $A$ is an index set and $\varphi_x=\varphi_y$ then either $x, y\in A$ or $x, y\not\in A$ Intuitively, if $A$ contains the Gödel index $x$ of a partial function $\varphi$ then $A$ contains all indices for the partial function. (Recall that there are $\aleph_0$ Gödel numbers for each partial function.)

It is instructive to compare the notion of an index set in computability theory with that of an indexing set.