I do not know if this has to do with extending rules. For instance, you cannot extend properties of prime numbers to all integers, even though we hope elementary number theory is consistent.
Consistency in mathematics has to do with how a given statement $a$ relates to a set $X$ of other statements. $a$ is consistent with $X$ provided there is no proof of $\neg a$ using the statements in $X$.
$X$ is self-consistent if you cannot prove the opposite of one of its statements using the others. This is equivalent to saying you will never be able to prove both a statement and its negation by assuming only the statements in $X$.