Too Restrictive Axiom- Example

An axiomatization is said to be categorical if all of its models are isomorphic; this is as restrictive as you can get. However, by the Löwenheim-Skolem theorem, no first-order theory with an infinite model is categorical (because any theory with at least one infinite model has models of every infinite cardinality, and models of different cardinalities can't be isomorphic).

Here are two ways of proceeding: $$ $$  (1)  Look at second-order logic. (In first-order logic, we can quantify only over members of the domain $M$ of the model in question. In second-order logic, we can quantify over relations on the model—in other words, we can quantify over subsets of $M^k$ for each natural number $k).$

There are important examples of categorical theories in second-order logic:

One is second-order Peano arithmetic; here the induction axiom applies to all subsets of the model (unlike first-order Peano arithmetic, where the induction axiom applies only to those subsets of the model that can be defined by a first-order formula). Second-order Peano arithmetic is categorical; its only model, up to isomorphism, is the usual model of natural numbers. So this second-order theory characterizes the structure of natural numbers.

Another example is the second-order theory of the real numbers, with a completeness axiom that says that every set of reals with an upper bound has a least upper bound (not just those subsets that are definable by a first-order formula). Again, this second-order theory is categorical; it characterizes the structure of the real numbers. $$ $$  (2)   Go back to first-order logic (which is much more tractable than second-order logic), and, instead of categoricity, look at categoricity in power. ("Power" here is used to mean "cardinality".)

If $\kappa$ is some cardinal number, a first-order theory $T$ with an infinite model is said to be $\kappa$-categorical if all models of $T$ of cardinality $\kappa$ are isomorphic. This is as restrictive as you can get in first-order logic.

A major example of this is the first-order theory of dense linear orderings without endpoints. Cantor proved that every countable model of this theory is isomorphic to the set of rational numbers with the usual ordering. So this theory is $\aleph_0$-categorical.

Morley proved a wonderful theorem on categoricity in power: If a countable first-order theory $T$ is $\kappa$-categorical for some uncountable cardinal $\kappa,$ then T is $\kappa$-categorical for every uncountable cardinal $\kappa.$

This leaves four possibilities for a countable first-order theory $T$ with infinite models:

(a) $T$ is not categorical in any infinite power;

(b) $T$ is $\aleph_0$-categorical but is not categorical in any uncountable power;

(c) $T$ is not $\aleph_0$-categorical but is categorical in every uncountable power;

(d) $T$ is $\kappa$-categorical in every infinite power.

There are examples of each of (a), (b), (c), (d) above.

Root systems might fit the bill. (See my blog for an alternative axiomatization.)

Finite crystallographic root systems (as described by the axioms in the article) are completely classified. There are four infinite families of root systems with these axioms (types $A_n$, $B_n$, $C_n$, and $D_n$) as well as five exceptional root systems ($E_6$, $E_7$, $E_8$, $F_4$, $G_2$). I don't know if it qualifies as "too restrictive" though. One could say it's just restrictive enough to classify everything while still remaining useful.