linguistic:paraphrasing, structural:scholar_edition
Example

As an example, observe the following axiomatic system, based on first-order logic with additional semantics of the following countably infinitely many axioms added (these can be easily formalized as an axiom schema):

Informally, this infinite set of axioms states that there are infinitely many different items. However, the concept of an infinite set cannot be defined within the system — let alone the cardinality of such as set.

The system has at least two different models – one is the natural numbers (isomorphic to any other countably infinite set), and another is the real numbers (isomorphic to any other set with the cardinality of the continuum). In fact, it has an infinite number of models, one for each cardinality of an infinite set. However, the property distinguishing these models is their cardinality — a property which cannot be defined within the system. Thus the system is not categorial. However it can be shown to be complete.