A consistent first-order theory is complete when it decides every sentence in its language.
Compactness gives a countable model elementarily equivalent to that contains a nonzero element divisible by every positive standard integer. It is not isomorphic to , so is not aleph-zero-categorical.
The complete theory of infinite-dimensional vector spaces over a fixed finite field is aleph-zero-categorical: every countably infinite model has countably infinite dimension and is therefore isomorphic to every other such model.
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Complete theory is a concept from model theory, a branch of mathematical logic. In this context, a theory \( T \) in a given language \( L \) is said to be complete if every statement (or sentence) in the language \( L \) is either provably true or provably false from the axioms of the theory \( T \).