Why is $T^*$ a companion of $T$?

Question

If $T$ and $T^*$ are two first order theories over the same language $\mathcal{L}$ such that $T \subseteq T^*$, $T^*$ is complete and $T$ and $T^*$ prove the same quantifier free $\mathcal{L}$ formulas then why can every model of $T$ be extended to a model of $T^*$? I'm at a loss. Any hints would be appreciated.

Answer 1

I notice that you used the phrase "$T$ and $T^{\star}$ prove the same quantifier free $\mathcal{L}$ formulas". That phrase doesn't quite make sense on its own, since theories only prove sentences, not formulas that might have free variables. The distinction is important in this case: whether or not your claim is true depends on how precisely you change that phrase into one that makes sense.

Option 1

We assume that $T$ and $T^\star$ prove the same quantifier free sentences. The claim is then false.

Let $L$ be the language with a single unary predicate, $P$, $T$ be the empty theory, and $T^\star$ be the complete theory asserting that the universe is infinite and everything is in $P$. This language has no quantifier free sentences (except the "always true sentence" and the "always false sentence", if your formalism allows these (which it should, in my opinion)), so we've satisfied our assumption. However, there are models of $T$ that have elements not in $P$; these don't extend to any model of $T^\star$.

Option 2

The interpretation that yields a true theorem is that we should assume $T$ and $T^\star$ prove the same implications between quantifier free formulas. That is, if $\varphi(\overline{x})$ and $\psi(\overline{x})$ are quantifier free formulas, $$ T \models (\forall \overline{x}) \varphi(\overline{x}) \rightarrow \psi(\overline{x}) \iff T^\star \models (\forall \overline{x}) \varphi(\overline{x}) \rightarrow \psi(\overline{x}) $$

The example above does not satisfy this requirement, since $x = x \rightarrow P(x)$ in $T^\star$ but not in $T$.

Here's a sketch of how to prove the theorem under this assumption:

• Given a model $M$ of $T$, consider the atomic diagram of $M$. That is, consider the quantifier free theory of $M$ in a language expanded with new constants for the elements of $M$.
• Use the compactness theorem and the assumption to show that the atomic diagram is consistent with $T^\star$.
• Take a model $M^\star$ of $T^\star$ together with the atomic diagram.
• Observe that the constant symbols from the language expansion allow you to find a copy of $M$ inside $M^\star$.

Note also that we don't need to assume $T^\star$ is complete or even that it extends $T$.