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I was skimming Odifreddi's: Classical Recursion Theory some days ago. Here:

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He writes this about these structures. Given a set of axioms just as the ones given by Dedekind in this case, can we know all possible structures attending these conditions in a systematic way? I'm just curious about what is done to accomplish this sort of task.

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    As long as there is exactly _one_ chain with a starting point in one end and infinite elements in the other direction, there can be as many other chains (with infinite elements in both directions) and cycles as you want, and the given axioms are still satisfied. I think those are all the possibilities, but I don't know. – 2017-01-23

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Associate each structure in the language to a graph where every element is a vertex and there is an edge $a \to b$ iff $S(a) = b$. Now just chase the 'edges' using the axioms, to see that any model must be a disjoint union of exactly one semi-infinite chain and any number of doubly-infinite chains and/or cycles. Note that every vertex has a unique out-edge, and following the out-edges will yield either an infinite path or a repeated vertex. In the latter case, the repeated vertex cannot be $0$, so starting from $0$ yields the semi-infinite chain. The repeated vertex also cannot have different in-edges, so it must be in a cycle. Similarly, any vertex not in the chain starting from $0$ has a unique in-edge, so following the in-edges backward will yield the other half of the doubly-infinite paths.

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    Note that such simplistic analysis will completely fail for more complicated axioms, such as those for Peano Arithmetic. – 2017-01-24