Categoricity in cardinality

$\kappa$

The property of a class of algebraic systems (models) requiring that all systems of cardinality $\kappa$ of the class be isomorphic. A first-order theory $T$ is said to be categorical in cardinality $\kappa$ if all models of cardinality $\kappa$ of $T$ are isomorphic. For a countable complete theory $T$, categoricity in countable cardinality (in $\aleph_0$) holds if and only if there exists, for any natural number $n$, a finite set $F_n$ of formulas of the language of $T$ with free variables $x_1,\dotsc,x_n$ such that any formula of the language with free variables $x_1,\dotsc,x_n$ is equivalent in the theory $T$ to one of the formulas of $F_n$. The collection of axioms:

1) $x<y\to\neg(y<x)$,

2) $(x<y\mathbin{\&}y<z)\to x<z$,

3) $x<y\lor x=y\lor y<x$,

4) $\exists u\exists v\exists t(x<y\to u<x<v<y<t)$, defines a theory $T_0$ of dense linear orderings which is categorical in $\aleph_0$, but is non-categorical in all uncountable cardinalities. The theory $T_1$ of algebraically closed fields of characteristic zero is categorical in all uncountable cardinalities, but is not categorical in $\aleph_0$. The following general theorem holds: If a first-order countable theory $T$ is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. This result generalizes to uncountable theories $T$ on replacing the condition of uncountable cardinality by cardinality greater than that of $T$. By a quasi-identity one means the universal closure of a formula

$$(Q_0\mathbin{\&}\dotsb\mathbin{\&}Q_n)\to P,$$

where $Q_i$ and $P$ are atomic formulas. For countable theories $T'$ which are axiomatized by means of quasi-identities, there is an even smaller distribution of possible categoricities: If such a theory $T'$ is categorical in countable cardinality, then it is categorical in all infinite cardinalities. If one appends to the axiom of the theory $T_0$ axioms for constants $c_i$:

$5_i$) $c_i<c_{i+1}$, where $i$ runs through the natural numbers, then the theory $T_3$ so obtained has exactly three countable models (up to isomorphism), since only three cases are possible: the set $\{c_0,c_1,\dotsc\}$ has no upper bound, has an upper bound but no least upper bound, or has a least upper bound. If for two countable models $M_1$ and $M_2$ of the theory $T_3$ the same one of the above three cases applies, then $M_1$ is isomorphic to $M_2$. Among theories which are categorical in uncountable cardinalities it is impossible to obtain an analogue of the above example. Thus, if a first-order theory $T$ is categorical in uncountable cardinality, then the number of countable models of $T$ (up to isomorphism) is either 1 or infinite.

References

[1]	G.E. Sacks, "Saturated model theory" , Benjamin (1972)
[2]	E.A. Palyutin, "Description of categorical quasivarieties" Algebra and Logic , 14 (1976) pp. 86–111 Algebra i Logika , 14 (1975) pp. 145–185
[3]	S. Shelah, "Categoricity of uncountable theories" , Proc. Tarski Symp. , Proc. Symp. Pure Math. , 25 : 2 (1974) pp. 187–203

Comments

The definition of a quasi-identity can also be found in Algebraic systems, quasi-variety of.

The "general theorem" mentioned in the text was conjectured by J. Łoś [a1], to whom the term "categoricity" is due, and proved by M.D. Morley [a2].

References

[a1]	J. Łoś, "On the categoricity in power of elementary deductive systems and some related problems" Colloq. Math. , 3 (1954) pp. 58–62
[a2]	M. Morely, "Categoricity in power" Trans. Amer. Math. Soc. , 114 (1965) pp. 514–538
[a3]	C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973)
[a4]	S. Shelah, "Classification theory and the number of non-isomorphic models" , North-Holland (1978)