Finished Set Lemma/Corollary

Corollary to Finished Set Lemma

Let $\Delta$ be a finished set of WFFs of propositional logic.

Then $\Delta$ has a model.

Proof

Note that the set of basic WFFs in $\Delta$ has at least one model.

Let $v$ be the boolean interpretation defined as follows:

$\quad \map v p = \begin{cases} \T & : p \in \Delta \\ \F & : p \notin \Delta \end{cases}$

Because $\Delta$ is finished, it is not contradictory, and hence $\map v p = \F$ if $\neg p \in \Delta$.

Thus $v$ is a model of the basic WFFs of $\Delta$.

The result follows by the Finished Set Lemma.

$\blacksquare$

Sources

• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.9$: Finished Sets: Lemma $1.9.1$