Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2010 / B / 5 / a

A term-generated model is a structure in which every element is the value of a closed term of the original language. Equivalently its substructure generated by the empty set, including interpretations of named constants, is the entire structure. This is the relevant meaning of “term model” for a model of an arbitrary complete theory, rather than a claim about one particular syntactic construction.

Each closed term $t$ gives the parameter-free defining formula $x=t$. ThusThe next parts show that pointwise definability implies atomicity, and countable atomicity is equivalent to primeness. Therefore property (i) implies all three other properties.

An element $a$ is parameter-free definable if some formula $\theta (x)$ of the original language has $a$ as its unique solution in the structure. Saying this for every element makes the model pointwise definable.

If a finite tuple has individually defining formulas, their conjunction uniquely defines that tuple. Its complete type is isolated by this conjunction: for each formula $\phi$, the sentence saying the unique tuple does or does not satisfy $\phi$ is decided by the complete theory. ThusIt follows that the countable model is also prime. The converse implication to property (i) fails. In $(\mathbb{N},<,0)$ every element is defined by having exactly $n$ predecessors, but the only closed term in this language is the constant $0$. This infinite model is pointwise definable and is not term-generated.

An atomic model is a model in which every finite tuple realizes an isolated complete type over the empty set. For a complete theory in a countable language, every countable atomic model is prime. The underlying argument is a back-and-forth method in the one-sided form needed for an embedding: enumerate the atomic model, and extend a finite partial elementary map one element at a time using an isolating formula for the enlarged tuple. The target model has the required witness because the already matched tuple has the same complete type.

Thus property (iii) implies property (iv). It need not imply pointwise definability or term-generation. The countable dense linear order without endpoints with a single named point, for example $(\mathbb{Q},<,0)$, is atomic: quantifier elimination for dense linear orders without endpoints isolates a tuple by its finite order-and-equality pattern relative to zero. But order structure automorphisms fixing zero move every nonzero rational. Such elements are not parameter-free definable, and no closed term names them. This supplies the failed implications (iii)$\nRightarrow$(ii) and (iii)$\nRightarrow$(i).

A prime model admits an elementary embedding into every model of its complete theory. In the countable setting a prime model is atomic. If a tuple in the prime model had a nonisolated complete type, the omitting types theorem would give a model omitting that type; an elementary embedding of the prime model into it would realize the omitted type, a contradiction.

Combining the four parts, all positive implications are exactly those generated byThe structure $(\mathbb{N},<,0)$ refutes (ii)$\Rightarrow$(i). The structure $(\mathbb{Q},<,0)$ is a countable atomic and hence prime model of its complete theory, while failing both (i) and (ii); it refutes every remaining reverse arrow, including (iv)$\Rightarrow$(i) and (iv)$\Rightarrow$(ii). The counterexamples therefore cover all ordered pairs of distinct properties not forced by the boxed chain.