In ZF without the axiom of infinity are there transitive sets possible?

Question

Somebody told me that ZF without the axiom of infinity has no transitive sets. Probably it depends on the definition/axiom of the `emptyset' $\emptyset$? The sets $\emptyset$ and $\{\emptyset\}$ are transitive, is it?

Answer 1

Maybe what they meant is that we cannot prove that there are inductive sets, or infinite transitive sets. The finite ordinals can all be constructed in $T=\mathsf{ZF}-\textrm{Infinity}$, and they are all transitive.

That said, whether we can prove in $T$ that every set (or even any hereditarily finite set) admits a transitive closure depends on how precisely we axiomatize the theory $T$. See here, and the references I give there, showing that a naive axiomatization of $T$ does not prove this.