Educated thinking people can be divided into a myriad of groupings. One of the more obvious is the "Liberal Arts" crowd and the "Science and Engineering" crowd. Interestingly, members of these groups tend to walk to a different drum from way back, so that by the time they reach adulthood, they have developed their own way of looking at the world, even - to some extent - their own language.
A quick example of this difference in language: The Liberal Arts crowd uses "theoretical" as a synonym for "hypothetical." The Science and Engineering crowd uses "theoretical" as an antonym to "empirical." Liberal Arts theory differentiates assumption from reality. Scientific theory differentiates model-derived from empirical solutions.
One of the consequences of the differentiation between Liberal Arts types and Science types is their intuitive understanding of how matter behaves. This, in turn, results in ideas that appear logical and, on the face of it, to present a conundrum.
For example, an acquaintance, Ed, asks you to imagine a rotating light source shining a beam onto a circular wall surrounding the light source. If you move the wall sufficiently far away, the spot of light will move faster than light speed along the wall - right?
Let's review the source of the statement that nothing can exceed the speed of light.
Einstein actually developed two theories of relativity: the Special Theory of Relativity in 1905, and ten years later the General Theory of Relativity in 1915. (Now, remember: as used here, "theory" does not mean hypothetical. It means a model derived solution.) According to the Special theory, matter, the stuff that makes up everything around us - air, furniture, ground, water, cars, etc. - behaves quite differently when it moves at high speed than when it is at rest.
When you fire a bullet from a gun, although it is not at all obvious, the speeding bullet gets heavier. The amount is so small that it cannot be measured by any laboratory device we have, but this is only because a speeding bullet really is moving quite slowly, when compared to the speed of light. If you were to accelerate the bullet so that it was moving at some significant fraction of light speed, its increase in mass would be very apparent. Furthermore, if you were to accelerate it to the speed of light, its mass would become infinite, an obvious impossibility, since it would take an infinite amount of energy to get this infinite mass to light speed.
This characteristic, strange as it seems, is one of the fundamental facts of the universe - things that move fast increase their mass; they get heavier. This becomes a practical matter in a cyclotron where sub-atomic particles are accelerated to very high speeds. As their speed becomes a significant fraction of light speed, they become very much heavier so that the accelerating magnets have to be given a great deal more power just to keep things going.
Another strange effect at high speed is that time slows down for a rapidly moving object. Several years ago, the amount of this slowing was physically measured when a satellite was orbited containing a highly accurate atomic clock, while the twin of the clock remained on the earth's surface. Even though the satellite's speed still was slow when compared to light speed, it was sufficiently fast for the slowing of the satellite's time to be measured by the two identical clocks as their synchronized times began to move apart.
If an object were to move at the speed of light, time would entirely cease to exist for it, another obvious impossibility.
We see this effect clearly in sub-atomic particle research, where once again these effects have practical significance. If you fire a radioactive particle with a half-life of a microsecond at a target, you can move the target far enough away from the source that the particle will decay before it hits the target. But if you fire the particle with a speed that is a significant fraction of light speed, because time slows down for the fast moving particle, it arrives at the target before it decays, whereas calculations that ignore this effect indicate that it should have decayed before arriving at the target. In other words, the "time-dilation" effect must be considered when running experiments using radioactive high-speed particles.
A third interesting effect for objects moving at very high speed is that a speeding object gets thinner in the direction it is moving. As before, this effect can only be observed when the object is moving at a significant fraction of light speed. If you were to move an object at the speed of light, it would become infinitely thin, a third impossibility.
This effect also has practical implications in high-speed particle research. A very fast moving particle of known size actually appears thinner than when it is at rest. Sensors must be calibrated to take this into account, or they can't even see the particles.
Each of these effects becomes more pronounced as the object's speed approaches light speed. In fact, at light speed, an object's mass becomes infinite, time stops, and it becomes infinitely thinthings which obviously cannot happen in any real universe.
Hence, in the world in which we really live, nothing can exceed the speed of light.
So why does the rotating light thought experiment seem to contradict this?
Simply stated, here is Ed's argument again: A rotating beam of light produces a moving light spot that will exceed light speed if the surface on which the spot moves is sufficiently far from the light source.
Ed's error is in his definition of what is moving. The light "beam" is not really a thing. Neither is the "spot." Photons depart the source at light speed, traverse the distance to the wall, and reflect off the wall to the observing or measuring instrument. Under ideal circumstances the beam is invisible, since no photons are scattered during transit to the wall - they all reach the wall. Your mental image, therefore, should be a light source and a spot on the wall.
As the source rotates, the spot appears to move, but in reality, your eyes (or whatever sensing mechanism you are using) simply receive the reflected photons, all traveling at exactly light speed.
In Ed's thought experiment, there are always photons that have left the source but not reached the wall. At the same time, while these photons are moving, the source is rotating, so that successive photons stream out ever further around the circle. Once the photons hit and reflect from the wall, they still have to travel to your eyes, still at light speed.
Since the source is constantly moving, the space between the source and the wall eventually will be completely filled with photons moving toward the wall, and what you will see is a band or ring of light stretching all the way around the wall.
If you could see the beam, since even the scattered photons you use to see it are subject to light speed, the band would appear to be bending backward twisting upon itself or winding up as it were, so that the "end of the beam," which you see as a moving spot of light on the wall, would never exceed the speed of light. Furthermore, as the rotation speed increased, and thus the apparent speed of the spot, the spot would appear to stretch backward until it stretched completely around the circular wall, forming the band or ring mentioned earlier.
The real world in which we live differs dramatically from how we sometimes perceive it. Since our everyday perceptions are limited to things that never move sufficiently fast compared to light speed, we do not intuitively take the bizarre relativity effects into account in our normal living.
This is why Ed's could make his cogent but incorrect argument, because he didn't account for how things behave when they move very fast.