Past exam of the logic and set theory course of the University of Oxford / 2010 / A / 3 / c

In the Henkin construction, first extend the consistent sentence set to a complete consistent theory with witnesses in an expanded language containing fresh constants. Its term model has closed terms modulo provable equality as elements:Functions are interpreted by applying their symbols to representative terms, and an atomic predicate holds of term classes exactly when the corresponding atomic sentence belongs to $T$. Equality axioms make these definitions independent of representatives. The Henkin witness property handles the existential step in the truth lemma, so the resulting structure satisfies $T$ and hence the original $\Sigma$.

An arbitrary sentence set need not already have enough closed terms or witnesses. Calling this a term model for $\Sigma$ means a term-model construction of a model satisfying $\Sigma$, usually through that witnessed completion; it does not assert that every arbitrary $\Sigma$ itself already supplies the construction.

The downward Löwenheim-Skolem theorem says that a structure in a countable language has a countable elementary substructure containing any specified finite or countable set of its elements. More generally the size bound is $\max (|\mathcal{L}|,|C|,{\aleph }_0)$ for a required parameter set $C$. In particular any satisfiable theory in a countable language has a model of size at most countable.

To see the last assertion directly from the completeness construction, add only countably many witness constants in the Henkin construction. There are then countably many finite closed terms, and taking their quotient cannot increase cardinality. The resulting term model is therefore countable. For the elementary-substructure version, perform the witness construction inside the given structure, choosing actual witnesses and closing under the function symbols. The induced substructure is elementary by the Tarski-Vaught test. This is the same witness mechanism as the completeness construction, with the ambient structure supplying the chosen witnesses.

The upward companion states that a theory with an infinite model has models in every sufficiently large infinite cardinal, at least $|\mathcal{L}|+{\aleph }_0$. Add that many pairwise unequal constants, apply compactness, and then the size-bounded construction. The countable consequence is the part needed below.