What is an isomorphism of sets?
I know in general an isomorphism is a structure-preserving bijective map between two algebraic structures. But what algebraic structure does a set have? Does a function between sets needs to be anything more than bijective to be an isomorphism of sets?
The reason I ask is that the term is used on PlanetMat's Finite Field page in explaining why a field $F$ of characteristic $p$ has cardinality $p^r$, where $r$ is the degree of the extension $F/ \mathbb{F}_p$. It says "Since $F$ is an $r$-dimensional vector space over $\mathbb{F}_p$ for finite $r$, it is set isomorphic to $\mathbb{F}_p^r$ and so has cardinality $p^r$."