Categoricity

Categoricity + Key Theorems

A theory $T$ is categorical is there exists a unique model of $\mathcal{M}\models T$ up to isomorphism, that is, if $\mathcal{M}\models T$ and $\mathcal{N}\models T$, then $\mathcal{M}\cong \mathcal{N}$.

This implies that all models of $T$ are of the same size and furthermore, are elementary equivalent (categoricity $⟹$ completeness and under most conditions $\kappa$-categoricity $⟹$ completeness, see Łoś–Vaught test in Classification Theory)

• There are no first-order categorical theories due to The Löwenheim-Skolem Theorem (there are no maximal infinite models)

Note: You can have first-order theories categorical in finite cardinalities (as the Löwenheim-Skolem Theorem does not apply), although these cases are often ignored, categoricity is only considered in infinite cardinals.

A theory $T$ is $\kappa -$categorical if there exists a unique model $\mathcal{M}\models T$ of cardinality $\kappa$ up to isomorphism $\to$ if $\mathcal{M}\models T,\mathcal{N}\models T$ and |$\mathcal{M}$| = |$\mathcal{N}$|=$\kappa$, then$\mathcal{M}\cong \mathcal{N}$ $\to$ This is equivalent to $I(T,\kappa )=1$, that is, there is exactly one isomorphism type of cardinality $\kappa$ of the theory T (see Classification Theory)

Morley’s Categoricity Theorem: If a first-order (${\mathcal{L}}_{\omega ,\omega }$)-theory $T$ in a countable vocabulary ($|\tau |\le {\aleph }_0$), then is $\kappa$-categorical in some uncountable cardinal $\kappa >{\aleph }_0$, then it $\lambda$-categorical for all $\lambda >{\aleph }_0$. That is, if $T$ is categorical in some uncountable cardinal, then it is categorical in all uncountable cardinals.

Shelah’s Generalization If a first-order (${\mathcal{L}}_{\omega ,\omega }$) theory $T$ in a vocabulary $\tau$ such that $|\tau |=\kappa \ge {\aleph }_0$, and if $T$ is $\lambda$-categorical for $\lambda \ge \kappa$, then it is $\mu$-categorical for all $\mu >\kappa$. That is, if $T$ is a first-order theory in an (uncountable) vocabulary of size $\kappa$, and is categorical in some cardinal $\lambda \ge \kappa$, then it is categorical in all cardinals greater than $\kappa$.

We say that a theory $T$ is $\omega$-categorical if it categorical in the cardinal ${\aleph }_0$. It can be shown, via Morley’s categoricity theorem, that if a first-order theory $T$ in a countable language is $\omega$-categorical and $\kappa$-categorical for some uncountable cardinal $\kappa$, then $T$ is categorical in all infinite cardinalities.

Other Notes

As mentioned previously, first-order theories cannot be categorical for infinite cardinalities. Although, for extensions of first-order logic ${\mathcal{L}}_{\omega ,\omega }$ such as higher-order logic can infinitary logic ${\mathcal{L}}_{\kappa ,\lambda },$ there can be as the Löwenheim-Skolem Theorem does not apply to these logics.

E.g., Consider second-order Peano Arithmetic, denoted $P{A}^2$. This theory is categorical, namely, all of it’s models are isomorphic to the standard structure $\mathcal{N}=⟨\mathbb{N},\mathcal{P}(\mathbb{N}),+,\cdot ,S(x),0⟩$. This means that all of it’s models have power $2^{{\aleph }_0}$ ($|\mathbb{N}|+|\mathcal{P}(\mathbb{N})|={\aleph }_0+2^{{\aleph }_0}=2^{{\aleph }_0}$).

Note: Obviously, categoricity in all (infinite) cardinals is not the same as (full) categoricity as you can be categorical in all infinite cardinals in first-order logic but a theory cannot be fully categorical in first-order logic.

In this sense, we can have maximal models (w/ respect to cardinality) as if all models of $T$ are isomorphic, then all models of $T$ have the same cardinality, and thus the cardinality of models of $T$ are bounded and not arbitrarily large.

References

Classification Theory The Löwenheim-Skolem Theorem Model Theory D.Marker.pdf