Set isomorphisms

There doesn’t seem to be any standard adjective to describe the property of a pair of sets having the same cardinality. Some authors use ‘equivalent’, but this adjective could be used for any equivalence relation; it might be useful to reserve it for the equivalence relation which is currently most prominent in the context. Other use ‘equinumerous’, and I guess that’s fine. There’s a natural counterpart of this adjective for the property of one set having cardinality less than or equal to the other, too, viz. ‘subnumerous’, although I don’t know if anyone actually uses this term. Anyway, if I was going to write a textbook on set theory, I think I would go for the adjective ‘isomorphic’. Consider the definition of a structure in abstract algebra, which can be stated as follows.

Definition 1. Structures are ordered tuples of the form $(A, R_0, R_1, \dotsc, R_{n - 1})$, where $A$ is a set called the underlying set of the structure, and for every natural number $i$ such that $i < n$, there is a natural number $k$, called the arity of $R_i$, such that $R_i$ is a subset of the Cartesian product $A^k$. $R_i$ is called the $i$th relation of the structure.

A set can be seen as a degenerate structure with no relations. Now, consider the definition of a homomorphism:

Definition 2. For every pair of structures $(A, R_0, R_1, \dotsc, R_{n - 1})$ and $(B, S_0, S_1, \dotsc, S_{n - 1})$ with the same number of relations and with the property that for every natural number $i$ such that $i < n$, $R_i$ and $S_i$ have the same arity, homomorphisms are mappings $f$ from $A$ to $B$ such that for every natural number $i$ such that $i < n$ and every $k$-tuple of members $a_0$, $a_1$, … and $a_{k - 1}$ of $A$ (where $k$ is the common arity of $R_i$ and $S_i$), $(a_1, a_2, \dotsc, a_{k - 1}) \in R_i$ if and only if $(f(a_1), f(a_2), \dotsc, f_(a_{k - 1})) \in S_i$.

In the trivial case where the two structures have no relations, a homomorphism between the structures is simply a function between their underlying sets. Accordingly, an isomorphism between the structures is simply a bijection between their underlying sets. Since we say that two structures are isomorphic if and only if there is an isomorphism between them, two sets are isomorphic if and only if there is a bijection between them, i.e. they have the same cardinality. That’s the reasoning behind using the adjective ‘isomorphic’. Since injective homomorphisms are sometimes called monomorphisms, and surjective homomorphisms are sometimes called epimorphisms, for every pair of sets $A$ and $B$, we can also say that $A$ is monomorphic to $B$ if and only if $|A| \le |B|$ and $A$ is epimorphic to $B$ if and only if $|B| \le |A|$. The only problem I can see with this terminology is that since structures are often referred to by their underlying set when it is clear from context which relations should be taken as part of the structure; for example, people might speak of the ring $\mathbb Z$ of integers and the ring $\mathbb Q$ of rational numbers. So the statement that $\mathbb Z$ and $\mathbb Q$ are isomorphic might be taken to mean the rings $\mathbb Z$ and $\mathbb Q$ are isomorphic, which is false. Of course, this problem can be resolved just by saying ‘the sets $\mathbb Z$ and $\mathbb Q$ are isomorphic’ or ‘$\mathbb Z$ and $\mathbb Q$ are isomorphic as sets’.