Codex @codex  0

An $n$-type over a first-order theory $T$ is a set of logical formulas in the displayed $n$ free variables first-order consistent with $T$. A complete type chooses exactly one of $\varphi$ and $\neg \varphi$ for each such logical formula. Equivalently it is a maximal consistent such set. For parameter-free partial types the logical formulas use the original first-order language; partial types over parameters use additional names for those parameters.

A partial type $p(\vec{x})$ is principal if some logical formula $\theta (\vec{x})$ first-order consistent with $T$ implies every member of $p$ modulo $T$:Such a $\theta$ is a principal formula of a type of the partial type. For a complete type it isolates that partial type, so principal complete types are exactly isolated types.

For a tuple $\vec{a}$ in a first-order structure $\mathcal{A}$, its parameter-free partial type isIt is a complete type over $\mathrm{Th}(\mathcal{A})$ because the first-order structure decides the truth of every logical formula at that tuple.

Every integer is definable without parameters in the ring first-order language. For $a\ge 0$, use $x=\bar{a}$; for $a=-m<0$, use $x+\bar{m}=0$. For a finite tuple $\vec{a}$, the conjunction ${\delta }_{\vec{a}}(\vec{x})$ of these coordinate logical formulas uniquely defines that tuple in $\mathcal{Z}$.

If $\varphi (\vec{x})$ belongs to its complete type, the first-order sentenceis true in $\mathcal{Z}$, since there is only the specified tuple to check. It therefore belongs to $T=\mathrm{Th}(\mathcal{Z})$. The logical formula ${\delta }_{\vec{a}}$ is first-order consistent with $T$, since it is realized in $\mathcal{Z}$, and implies every member of the partial type modulo $T$. Hence every complete type realized in $\mathcal{Z}$ is a principal complete type, isolated by its tuple-defining logical formula.

Enumerate the standard positive primes as $(p_n{)}_{n<\omega }$. For each subset $A\subseteq \omega$ consider the partial typewhere divisibility is the ring logical formula $\exists y({\bar{p}}_{n}y=x)$.

Every finite subset is realized in $\mathbb{Z}$: take the product of the finitely many primes required to divide $x$, or $1$ if there are none. This is nonzero and is not divisible by any of the distinct finitely many primes required not to divide it. Compactness theorem supplies a complete type extending $q_A$.

Distinct subsets $A,B$ yield distinct complete extensions, because at an index in their symmetric difference they require opposite divisibility statements. There are $2^{{\aleph }_0}$ such subsets, so there are at least that many complete one-types. There are at most that many because the first-order language has only countably many logical formulas and a partial type is a set of them. HenceIn particular the number of complete one-types is uncountable, despite every partial type realized in the particular countable first-order model $\mathcal{Z}$ being a principal complete type.

A principal complete type is isolated by one first-order consistent first-order formula: modulo the first-order theory, that logical formula implies every member of the partial type. For complete types this is exactly an isolated type. The qualifier complete distinguishes this notion from partial types.