  It has been shown that \(\mathbf{GL}\) is adequate for provability in the following sense. Let a sentence of \(\mathbf{GL}\) be always provable exactly when the sentence of arithmetic it denotes is provable no matter how its variables are assigned values to sentences of \(\mathbf{PA}\). Then the provable sentences of \(\mathbf{GL}\) are exactly the sentences that are always provable. This adequacy result has been extremely useful, since general questions concerning provability in \(\mathbf{PA}\) can be transformed into easier questions about what can be demonstrated in \(\mathbf{GL}\).