= Theory of successor
{title2=$T_S=\operatorname{Th}(\mathbb N,s,0)$}
= Successor theory
{synonym}
The <complete theory> of the standard <successor structure> is axiomatized by <injectivity> of $s$, $0$ outside its range, a predecessor for every nonzero element, and the sentences $s^n(x)\ne x$ for each positive integer $n$. Its models have a single distinguished one-sided chain generated by zero and any collection of two-sided integer chains. The <classification of models of the successor theory> proves completeness and explains why the theory is categorical at every <uncountable> <cardinal> but not at the <countable> <cardinal>.
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