Model Theory - Categoricity

Categoricity

As observed in the section on first-order logic, first-order theories cannot be categorical, i.e. they cannot describe a unique model up to isomorphism, unless that model is finite. But two famous model-theoretic theorems deal with the weaker notion of κ-categoricity for a cardinal κ. A theory T is called κ-categorical if any two models of T that are of cardinality κ are isomorphic. It turns out that the question of κ-categoricity depends critically on whether κ is bigger than the cardinality of the language (i.e. + |σ|, where |σ| is the cardinality of the signature). For finite or countable signatures this means that there is a fundamental difference between -cardinality and κ-cardinality for uncountable κ.

A few characterizations of -categoricity include:

For a complete first-order theory T in a finite or countable signature the following conditions are equivalent:

1. T is -categorical.
2. For every natural number n, the Stone space S_n(T) is finite.
3. For every natural number n, the number of formulas φ(x₁, ..., x_n) in n free variables, up to equivalence modulo T, is finite.

This result, due independently to Engeler, Ryll-Nardzewski and Svenonius, is sometimes referred to as the Ryll-Nardzewski theorem.

Further, -categorical theories and their countable models have strong ties with oligomorphic groups. They are often constructed as Fraïssé limits.

Michael Morley's highly non-trivial result that (for countable languages) there is only one notion of uncountable categoricity was the starting point for modern model theory, and in particular classification theory and stability theory:

Morley's categoricity theorem

If a first-order theory T in a finite or countable signature is κ-categorical for some uncountable cardinal κ, then T is κ-categorical for all uncountable cardinals κ.

Uncountably categorical (i.e. κ-categorical for all uncountable cardinals κ) theories are from many points of view the most well-behaved theories. A theory that is both -categorical and uncountably categorical is called totally categorical.