There are several definitions of "finite", like Dedekind's and Tarski's (Thanks to A.K. for point out the latter - first time I've heard of it): From the Wikipedia entry on Finite Set:
(Richard Dedekind) Every one-to-one function from S into itself is onto.
(Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion.
And there are more.
Do these distinctions matter when considering definitions of fundamental categories like topological spaces or measure spaces? For example,
In measure theory: "A measure is continuous from above if [given measurable sets and closed under intersection] at least one set has finite measure," alternatively, in the definition of sigma-finite measures (same article).
Do the various definitions of finiteness lead to non-isomorphic or non-equivalent categories?
My question is not specifically about topology or measure theory, but these are basic definitions introduced at undergrad level, so I thought, better to understand the context via basic examples.