Does Godel's completeness theorem stands for formulas which are not sentences?

Question

Godel completeness theorem (as I am familiar with): $\Sigma\models \alpha \Rightarrow \Sigma\vdash \alpha$. In class (A year ago) we proved the next claim: $\Sigma$ is Theory - a set of sentences - is consistent $\Rightarrow$ $\Sigma$ has a model, And proved: the last claim $\Rightarrow$ Godel completeness theorem.

My question is , does the completeness theorem stands for formulas which are not sentences?

I was also searching in Mendelson's and Enderton's books, but they demonstrate the prove in slightly different way than the lecturer did ,and I am not sure whether I fully understood it.

Answer 1

Yes, see Enderton, page 135 :

COMPLETENESS THEOREM (Gödel, 1930)

(b) Any consistent set of formulas is satisfiable

provided that the semantical relation $\vDash$ of satisfaction in an interpretation is defined for formulas: see page 83.

But there is a different approach: see Dirk van Dalen, Logic and Structure, Springer (5th ed. 2013), page 67) where the definition of meaning and truth value is limited to sentences, i.e. "closed" formulas.

In this case, the following convention is adopted:

$\mathfrak A \vDash \varphi$ iff $\mathfrak A \vDash \text{Cl}(\varphi)$,

where $\text{Cl}(\varphi)$ is the universal closure of $\varphi$.

Having said that, we have that $\Gamma \vDash \psi \text { iff (if } \mathfrak A \vDash \Gamma \text {, then } \mathfrak A \vDash \psi)$, where $\Gamma$ and $\psi$ are sentences.

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As pointed by Carl Mummert, the two conventions disagree about satisfiability, also if they agree about validity : under both approaches the formula $x ≠ y$ is not valid.

In a nutshell, Enderton's approach adopts as basic the satisfaction relation (following Tarski) while van Dalen's one adopts as basic the true (in a model) relation.