 \(\mathbf{HA}\) and \(\mathbf{PA}\) are proof-theoretically equivalent, as will be shown in Section 4. Each is capable of (numeralwise) expressing its own proof predicate. By Gödel’s famous Incompleteness Theorem, if \(\mathbf{HA}\) is consistent then neither \(\mathbf{HA}\) nor \(\mathbf{PA}\) can prove its own consistency. 