Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2009 / B / 4 / c

The complete theory $\mathrm{Th}(A)$ is the set of all $L$-sentences true in $A$. The existential theory of a structure ${\mathrm{Th}}_{\exists }(A)$ consists of those true sentences of the form $\exists \bar{x}\theta (\bar{x})$ with quantifier-free $\theta$. The universal theory ${\mathrm{Th}}_{\forall }(A)$ uses $\forall \bar{x}\theta (\bar{x})$ with quantifier-free matrix. Allow equivalent formulas and empty quantifier blocks, or insert dummy variables in nonempty domains. These are sentences in the original language, distinct from diagrams, which add constants naming Elements of the structure.

If a model $B$ of $T$ contains $A$ as a substructure, every existential sentence true in $A$ remains true in $B$: use the same witness tuple and preservation of quantifier-free truth. Hence $B$ satisfies the required joint theory.

For the existential-theory embedding criterion, conversely suppose that joint theory has a model. Expand the language by a constant $c_a$ for each $a\in A$, and add the atomic and negated atomic diagram $D(A)$ to $T$. Any finite collection of its literals is a quantifier-free conjunction $\theta (\bar{c})$ true of the corresponding tuple in $A$. Replacing those named constants by variables and existentially closing gives a sentence in ${\mathrm{Th}}_{\exists }(A)$. A model of the assumed joint theory can supply its witnesses, and hence an expansion satisfying those finitely many diagram requirements and any finite subset of $T$.

Compactness therefore gives a model of $T+D(A)$. Its named Elements define an embedding of $A$ into its $L$-reduct: inequalities ensure injectivity, and all function and relation literals ensure preservation and reflection. Replace the ambient model by an isomorphic copy identifying this image with $A$ itself. Thus a model of $T$ containing $A$ exists. The diagram step explains how parameter-free existential information nevertheless yields an embedding of the whole structure.

If $B\models T$ and $B\subseteq A^{\ast }\equiv A$, every universal sentence of $A$ holds in $A^{\ast }$ and is preserved by passage to the substructure $B$. Thus $B$ witnesses satisfiability of the joint theory.

For universal theory and substructures of elementarily equivalent models, conversely let $B$ satisfy $T$ together with the universal theory of $A$. Every existential sentence true in $B$ must be true in $A$: if an existential $\sigma$ were false in $A$, its universal negation would belong to ${\mathrm{Th}}_{\forall }(A)$ and hold in $B$, a contradiction. Hence ${\mathrm{Th}}_{\exists }(B)\subseteq \mathrm{Th}(A)$. In particular $\mathrm{Th}(A)\cup {\mathrm{Th}}_{\exists }(B)$ is satisfiable, with model $A$. Apply the preceding embedding result to the theory $\mathrm{Th}(A)$ and structure $B$. It supplies a structure $A^{\ast }\models \mathrm{Th}(A)$ containing $B$. Therefore $A^{\ast }\equiv A$ and $B\subseteq A^{\ast }$, as required. Elementary equivalence here does not assert that $A$ itself contains a copy of $B$.

The natural-number successor chain embeds into the integer successor chain by ordinary inclusion. Thus $\mathbb{Z}$ witnesses the existential joint satisfiability in the first assertion. The two chains are not elementarily equivalent: every integer has a predecessor, whereas zero has none in the natural-number chain.

For the universal assertion, construct an elementary extension of the natural-number chain containing a two-sided successor chain. To its elementary diagram add distinct constants $d_j$ for $j\in \mathbb{Z}$ with $s(d_j)=d_{j+1}$ and require them to differ from all named natural numbers. Every finite subset is realized in the natural-number chain: place the finitely many required consecutive blocks sufficiently far to the right, away from the finitely many forbidden named Elements. Compactness yields an elementary extension $A^{\ast }$ with the required two-sided chain. That chain is a substructure isomorphic to the integer successor structure and hence is a model of its complete theory.

Consequently both joint theories in the two assertions are satisfiable, though the integer chain cannot embed into the natural-number chain itself: repeated predecessors would give an infinite descending sequence of natural numbers. The second assertion correctly uses an elementarily equivalent ambient structure rather than the original one.