The complete theory of the standard successor structure is axiomatized by injectivity of , outside its range, a predecessor for every nonzero element, and the sentences for each positive integer . 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.
New to topics? Read the docs here!