I am hoping someone could explain, rather in-rigorously, the use of
the word inconsistent.
Essentially, in this context, inconsistent means the two theories give different, incompatible answers to the same question.
In this specific case, the question is:
- If light (an electromagnetic wave) is measured to propagate at speed $c$ in an inertial frame of reference (IFR), what speed is measured in a relatively moving IFR?
For simplicity, assume the relative moving IFR has speed $v$ in the direction of the light beam.
According to Newtonian relativity, and the Galilean coordinate transformation, the light will have a measured speed of $c - v$ in the relatively moving reference frame.
Maxwell's equations unambiguously predict the speed with which electromagnetic waves propagate in vacuum is $c$
To reconcile this inconsistency - these different, incompatible answers to the same question - requires accepting one of three possibilities:
- There is an ether - an absolute frame of reference in which the light propagates at speed $c$. This means that there is no relativity principle for electromagnetism. Experiments should be able to detect this absolute rest frame.
- Maxwell's equations are incorrect and must be modified to give an answer in accordance with Newtonian relativity.
- Newtonian mechanics is incorrect and must be modified to give an answer in accordance with Maxwell's equations.