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We work with the usual infinite-model convention for an omega-categorical theory: it has a unique countably infinite model up to isomorphism. The three conditions are the type-space and automorphism formulations of the Ryll-Nardzewski theorem; the following arguments prove them rather than invoke that theorem. The standard hypothesis that $T$ has infinite models is understood. If finite models are allowed instead, uniqueness among models of size at most countable is the matching convention for condition (i).

We first prove (i) implies (ii). Suppose some type space $S_n(T)$ were infinite. An infinite compact space cannot have every point isolated: its cover by singleton open sets would have no finite subcover. Thus it has a nonisolated complete type $p$. The omitting types theorem gives a countable model omitting $p$. On the other hand, compactness theorem gives a model realizing $p$, and the Downward Lowenheim-Skolem theorem gives a countable elementary submodel containing its realizing tuple. These countable models cannot be isomorphic, since an isomorphism preserves realization of $p$. This contradicts omega-categoricity. Hence every $S_n(T)$ is finite.

Yes. By the Ryll-Nardzewski theorem, aleph-zero-categoricity of $T$ says that, for every $n$, only finitely many $L$-formulas in $n$ variables exist modulo equivalence over $T$. The $L_0$-formulas form a subcollection. Moreover, two $L_0$-formulas are equivalent modulo $T_0$ exactly when their universal equivalence sentence belongs to $T_0$, equivalently when it follows from $T$. Thus there are only finitely many $L_0$-formulas modulo $T_0$ in each arity. Applying Ryll-Nardzewski again proves that $T_0$ is aleph-zero-categorical. This is the reduct of an aleph-zero-categorical theory property.

Suppose first that $T$ is aleph-zero-categorical. If a type in some $S_n(T)$ were nonisolated, the omitting types theorem would produce a countable model omitting it, while a countable elementary submodel of a model realizing it would be another countable model. This contradicts categoricity. Thus every type is isolated. The compact Stone space $S_n(T)$ is then discrete and therefore finite.

Conversely, if every $S_n(T)$ is finite, every type is isolated. Every countable model is consequently atomic, and part i says that any two countable models are isomorphic. This proves the Ryll-Nardzewski theorem.

The Ryll-Nardzewski theorem says that for a complete theory $T$ in a countable first-order language with infinite first-order models, the following are equivalent: $T$ is ${\aleph }_0$-categorical; for every positive integer $n$, the set $S_n(T)$ of complete parameter-free $n$-types is finite; every such finite-arity complete type is isolated by a logical formula. The countable-language hypothesis and the requirement for every finite arity are important.

The omitting types theorem says that, for a first-order consistent complete theory in a countable first-order language, a countable family of nonprincipal partial types in finite tuples of logical variables can all be omitted in a countable first-order model. Here nonprincipal means that no logical formula first-order consistent with the first-order theory implies every logical formula of the partial type. Omission means that no tuple in the constructed first-order model satisfies all logical formulas of that partial type. The complete one-type version is the special case of one partial type in one logical variable.

Let $F_n$ be the formulas whose free variables are among $x_1,\dots ,x_n$. Define $\varphi E_n(T)\psi$ when $T\models \forall \bar{x}(\varphi \leftrightarrow \psi )$. The n-variable Lindenbaum algebra quotient is a Boolean algebra, with meet, join and complement induced by conjunction, disjunction and negation; its zero and one are contradiction and truth classes. Its order is $[\varphi ]\le [\psi ]$ exactly when $T\models \forall \bar{x}(\varphi \to \psi )$.

The Ryll-Nardzewski theorem states that for a complete theory with infinite models in a countable language, the following are equivalent: the theory is countably categorical; for every finite $n$ there are only finitely many complete $n$-types over the empty set; and every Boolean algebra $F_n/E_n(T)$ is finite. Equivalently each such type is isolated. The countable-language and completeness hypotheses are essential parts of this formulation.

Suppose $T$ is an omega-categorical theory. By the Ryll-Nardzewski theorem, every finite-arity complete type over the empty set is isolated. Let $p(x,a)$ be a complete type over a finite tuple in the unique countable model $M$, and take an elementary extension realizing it at $b$. The joint type of $(b,a)$ is isolated by a formula $\theta (x,y)$.

The complete type of $a$ contains $\exists x\theta (x,y)$, since this formula is true of $a$ in the extension. Hence it is true of $a$ in $M$, and some $b^{\prime }\in M$ satisfies $\theta (b^{\prime },a)$. Isolation of the joint type implies that $b^{\prime }$ realizes $p$. Thus the countable model is an omega-saturated model.

The converse fails. The complete successor theory from question 4 has a countable modelwith countably infinitely many two-sided successor components. It is omega-saturated. Over finitely many parameters, a complete one-variable type either fixes a finite successor/predecessor translate of a parameter or zero, or asks for a point in no component represented by those parameters and zero. The first kind is realized in the named component, and the second in an unused integer component. The quantifier-elimination description of classification of models of the successor theory shows there are no other cases.

Nevertheless this theory is not omega-categorical: $M$ and its standard model $\mathcal{N}$ are countable and not isomorphic. Thus existence of a countable saturated model is strictly weaker than countable categoricity.

For the additive first-order structure, letIt defines $n\mathbb{Z}$. These sets are pairwise distinct: if $m>n>0$, the integer $n$ belongs to $n\mathbb{Z}$ but not to $m\mathbb{Z}$. Thus there are infinitely many inequivalent one-variable first-order formulas modulo the complete theory of the additive integers. The Ryll-Nardzewski theorem implies that first-order theory is not aleph-zero-categorical.

For the multiplicative first-order structure define divisibility by $d(y,x):=\exists z(yz=x)$. For $k\ge 1$, the first-order formulaasserts the existence of at least $2k$ distinct integer divisors. The integer $2^{k-1}$ has exactly $2k$ such divisors, namely $\pm 2^j$ for $0\le j\le k-1$, so it satisfies ${\psi }_k$ but not any ${\psi }_{\ell }$ with $\ell >k$. Again there are infinitely many inequivalent one-variable first-order formulas. The Ryll-Nardzewski theorem therefore implies the first-order theory of the multiplicative integers is not aleph-zero-categorical.

An aleph-zero-categorical theory has, up to isomorphism, exactly one first-order model of cardinality ${\aleph }_0$.

The Ryll-Nardzewski theorem says that, for a complete theory $T$ in a countable first-order language, with infinite first-order models, the following conditions are equivalent: $T$ is aleph-zero-categorical; for every positive integer $n$, its type space $S_n(T)$ of complete types over the empty set is finite; and, for every $n$, there are only finitely many $L$-formulas in $n$ free variables modulo equivalence over $T$. Here $\phi$ and $\psi$ are equivalent over $T$ when $T\models \forall \mathbf{x}(\phi \leftrightarrow \psi )$.

Every reduct of an aleph-zero-categorical theory to a sublanguage is aleph-zero-categorical. There can only be fewer formulas in each arity, so the finite-formula characterization in the Ryll-Nardzewski theorem is preserved.