Inertial System

An inertial system is a system with an inertial frame of reference in which Newton's laws are valid. For example if a body rests in an inertial system, it continues to rest for all times. In a non-inertial system this is not true. Take for example a system $S$ that accelerates with $a$ relative to an inertial system. From the perspective of system $S$ a particle that rests in the inertial system would accelerate with $-a$. An observer in the $S$ system would have to introduce a fictitious force $F_{fict.}$ that explains, why the particle accelerates. Without fictitious forces Newton's first law would not hold in $S$, since a body at rest does not continue to rest. Therefore $S$ is no inertial system.