## FANDOM

78 Pages

The Lemma on Constants is an elementary result, which says something like the following:

Lemma: Suppose $T \vdash \phi(c)$, where $T$ is a theory, $\phi(x)$ is a formula, x is a tuple of variables, and c is a tuple of constant symbols not appearing in T. Then $T \vdash \forall x : \phi(x)$.

Proof: If one interprets $T \vdash \psi$ to mean that $\psi$ holds in all models of T, then this follows from unwinding the definitions: $T \vdash \phi(c)$ means that whenever M is a model of T and c is a tuple from M, $M \models \phi(c)$. Then clearly $M \models \forall x: \phi(x)$!

On the other hand, if one interprets $T \vdash \psi$ to mean that $\psi$ can be proven from T, then this follows from the other version by Gödel's Completeness Theorem, which says that these two interpretations of $\vdash$ are the same. QED.

The Lemma on Constants is practically obvious. However, it is handy to have a name for this lemma when making technical arguments.