Example of a categorical axiom system (i.e. one that fixes its models "up to an isomorphism")

Question

From page 252 of Logic: The Laws of Truth:

In general, the more axioms we add, the greater the constraints will be on the values of the nonlogical symbols featured in the axioms. Note, however, that we can never determine a unique model by adding more axioms. This follows from the isomorphism lemma (Chapter 9, n. 3): if we have a model that makes all our axioms true, we can always define a different model that still makes them all true by switching all objects in the domain for new objects (without making any other changes; i., if $x$ is in the extension of a certain predicate or is the referent of a certain name before the switch, then its replacement is in the extension of that predicate or is the referent of that name after the switch). Therefore, the most that a set of axioms can do is fix a model “up to isomorphism.” [From note 15: An axiom system that fixes its models up to an isomorphism is said to be categorical] Note that not all axiom systems fix their models up to isomorphism. For example, we listed four models that make true all the axioms in our example system, but none of these models can be derived from any of the others just by switching objects in the domain (while holding referents and extensions fixed — relatively speaking — in the way discussed above). In other words, no two of these four models are isomorphic.

The example system referenced in the excerpt is the following set of axioms:

1. $∀x(Ax ∨ Bx)$
2. $¬∃x(Ax ∧ Bx)$
3. $∀x(Cx → Ax)$

,,,and the following four models are presented as examples of models that satisfy the above system:

1. Domain: { 1 } Extensions: A: {1} B: ∅ C: {1}
2. Domain: {1, 2} Extensions: A: {1, 2} B: ∅ C: {2}
3. Domain: {1, 2} Extensions: A: { 1 } B: { 2 } C: { 1 }
4. Domain: {1, 2, 3} Extensions: A: {1, 2} B: { 3 } C: { 2 }

Questions:

1. So, a categorical axiom system is one in which all possible models that satisfy the axioms are ismorphic to one another. The example system above is not categorical because there exist pairs of models that satisfy the system that are not isomorphic to each other (e.g. 3 and 4). Is my understanding correct?
2. If my understanding is correct, then how is it possible for an axiom system to fix its models up to a single isomorphism. Wouldn't we have to fix the size of the domain in order to do this? For if the size of the domain was allowed to vary, then we'd have an infinite set of isomorphic models, one for each domain size.
3. Is it possible to make the example system above categorical by adding more axioms? Otherwise, I'd be grateful if someone could provide an example of a categorical system.

Answer 1

Different logics (basically, different formal languages with appropriate notions of semantics) can achieve different levels of categoricity.

The most common logic - first-order logic - can pin down any (finite-language) finite structure up to isomorphism with a single sentence. In particular, the cardinality issue you mention can be handled using equality: consider sentences like $$\exists x,y\forall z[\neg(x=y)\wedge (z=x\vee z=y)].$$ Historically, though, first-order logic didn't always "come with equality built in" - and equality-free first-order logic can't pin down any structure up to isomorphism besides the one-element structure. On the other hand, pinning down any infinite structure up to isomorphism is impossible for first-order logic even with equality; this is a consequence of the compactness theorem, arguably the first "substantial" theorem in logic.

There are logics much stronger than first-order logic, though; second-order logic, for example, has no trouble pinning down structures like $(\mathbb{N};+,\times)$ or $(\mathbb{R};+,\times)$. Indeed, it's an interesting fact that every "naturally-occurring" structure seems to be capturable up to isomorphism by a single second-order sentence.

However, these stronger logics are hard to use: there's an inherent trade-off between expressive power and tameness. The general study of logics beyond first-order logic, and of their various properties and trade-offs, is called abstract model theory; the first couple chapters of the Barwise-Feferman volume provide a good introduction to the subject, but it's not likely to be comprehensible (let alone well-motivated!) until you've gotten some experience with first-order logic itself.