Inductive definitions via a monotone operator

Question

Aczel* defines the term the set inductively defined by $\varphi$ as follows (p. 744).

Let $\varphi:\mathrm{Pow}(A)\rightarrow\mathrm{Pow}(A)$ where $\mathrm{Pow}(A)$ denotes the set of all subsets of $A$. The operator $\varphi$ is monotone if $X\subseteq Y\subseteq A$ implies $\varphi(X)\subseteq\varphi(Y)$. [...] we write $I(\varphi)$ for $\bigcap\{X\subseteq A\mid \varphi(X)\subseteq X\}$ and call it the set inductively defined by $\varphi$.

Won't $I(\varphi)$ always be the empty set?

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* Peter Aczel, An Introduction to Inductive Definitions, Editor(s): Jon Barwise, In Studies in Logic and the Foundations of Mathematics, Elsevier, Volume 90, 1977, Pages 739-782. (link)

Answer 1

How about $A=\Bbb Z$ and $\phi(X)=X\cup\{0\}$? I reckon $I(\phi)=\{0\}$.