The first part of the video takes a look at how mathematicians assert different size infinities via “set theory”.
The second part (starts at 5:47) considers how infinity isn’t part of science, and includes an example which considers infinite speed in reality via a laser's visible dot moving across a screen (or wall), as the angle of the laser beam approaches parallel to the screen’s face.
If the screen is facing towards us and is assumed to increase in size as it travels further away (so that the dot from the laser remains on the screen when moving left to right), then as the laser rotates back and forth, the speed at which the dot moves across the screen increases, the further away the screen is, because the radius of the arc increases.
Cheating is cool: Think of the dot's motion as kind of cheating, because it's effectively an animation on a computer screen, i.e., if computer-controlled signals light up two pixels, 1 at each end of a screen, at almost exactly the same time, the observed (in principle) speed will approach (hypothetically) infinity.
As stated in Sabine’s video – at some distance away the screen would be far enough such that the laser’s dot could be observed (in principle) to be moving faster than light, even though of course the photons themselves are fixed at the speed of causality.
However, the screen would need to be infinitely far away for the dot to be observed travelling infinitely fast, but because the photons could never reach the screen, such an observation would not be possible (not even in principle) – in reality it is impossible for something to exist an infinite distance away, because space is finite, and infinity is a misconception.
Consider the sensitivity: increasing the “laser rotation speed” or the “screen distance away” – causes an increase in speed at which the laser’s dot travels across the screen’s face. Additionally, if the screen is face-on (not tilted), then although the sensitivity of speed increase of the laser’s dot remains at its lowest, the maximum speed possible is at its maximum, i.e., infinite... Note: for a virtually infinite current laser rotation speed, the sensitivity cannot be increased, and therefore tilting the screen will simply cause the dot to slowdown as explained below.
Tilting the screen: although indeed the “sensitivity of speed increase” approaches its maximum (for a “current laser rotation speed” of virtually 0) as the screen approaches side-on (90 degrees), the maximum possible speed of the laser’s dot actually decreases to the speed of light – a direct result of subsequent photons having to travel further – simply imagine that as each photon is emitted from the laser, it is assigned its own ID so that you can identify each photon individually as it travels across the screen.
Instantaneous rotation: imagine two photons being fired at the exact same time, equivalent to instantaneous (hypothetically) rotation of the laser, slightly angled apart from each other. If the screen is not tilted, the observed speed of the laser’s dot will be infinite, because the photons would hit the screen at the same time... but instead if the screen is tilted, the second photon must travel further and therefore it arrives later.
Perplexing threshold angle: imagine the laser is rotating very slowly at just 1 degree per second, and that the screen, which is central to the arc of rotation, is itself tilting/rotating between being face-on and side-on (90 degrees) to the laser – the speed of the laser’s dot will be at its maximum (virtually the speed of light) when the screen is at 90 degrees as the dot passes over it, but if the laser was to rotate ultra-fast, the maximum dot speed would occur when the screen is face-on.
Now consider once again, all the above – it follows that for the laser’s dot:
Example: as the screen becomes almost exactly parallel (side-on) to the laser, 1 photon will hit the face at the near edge, but the next photon travels at the speed of light for ages (almost parallel to the screen’s face) until it eventually hits the screen much further along – i.e., in this scenario the photons are effectively skimming the surface of the screen at the speed of light, which would indeed be the observed (in principle) speed of the laser’s dot.
Alternatively: use two lasers (for example) to each fire 1 photon at the same time – if both photons hit the screen at the same time (regardless of how far away the screen is), the laser’s dot will have travelled (theoretically) infinitely fast – but in reality, the speed of the laser’s dot is limited by the definition (as applicable) of movement, i.e., any photon trailing another needs to hit the screen at a different position and at a later time.
Velocity Example: consider the very 1st photon that is emitted at the start of the laser’s angular swing, and also the very last photon that is emitted at the end of its swing. Now imagine the swing taking 1 whole second, and that the two photons are now travelling towards the large screen which is very far away – when the two photons hit the screen, the observed speed of the laser’s dot will be whatever the distance is between the 2 photons / 1 second (because they arrive 1 second apart).
Visual Resolution: of course, by having only 2 photons there wouldn’t be much of an existence of motion to observe. If the screen consisted of photo sensitive pixels, then the smoothness of motion of the laser’s dot would depend on: