 That such a reinterpretation strategy guarantees relative consistency can be seen via the following reasoning: If the set AX were inconsistent, then it would logically imply a contradiction. But as logical implication is independent of the specific meanings of such terms as “point” and “line”, AX would continue to imply a contradiction under its reinterpretation. But that is just to say that a set of theorems of B would imply a contradiction, hence that B itself would be inconsistent.