Mathematics contains a number of different systems, but each mathematical system, no matter how different it may be from another, has consistency as one of its goals. When a mathematical system is consistent, a statement and the opposite, or negation, of that same statement cannot both be proven true.

For example, in the familiar system of algebra, it is true that a+ 1 > a. Even if a is a negative number, or 0, the statement is true. For example, -3.5 + 1 is -2.5, and -2.5 is greater than -3.5 (because -2.5 is to the right of -3.5 on a number line). Because this system is consistent, it is not possible to prove that a+ 1 is less than a or equal to a.

Consistency is also important in the use of mathematical definitions and symbols. For example, (5)² equals 25 and (-5)² also equals 25. Is the square root of 25 equal to 5 or to 5? The...