The fallacy of attaining infinity

I made a lot of trips to The Hindu office on Mount Road this past week, and I made all of them using the suburban railways. When standing still inside a train that's moving at around 80 km/hr, I'm also moving at the same speed in the same direction. When walking ahead inside the train, I'm moving faster than the train inside the train. When walking toward the back of the train, I'm moving slower than the train is. All this is boring relative-motion stuff. How about when I'm moving sideways inside the train?

When I'm moving sideways at a speed of, say, 2 m/min, the train will have moved forward a distance of 22.22 m in that same minute. If there were an imaginary path that I was inscribing on the ground, then my sideways-one won't be perpendicular to the train's path: it will be adjacent and separated by an angle (like in the diagram shown below).

In the diagram, b is 22.22 m long, a is 2 m long, C is 90°, and A is tan(a/b) = 0.0015°. Now, the time taken by the train to traverse 22.22 m is 1 s. Let's keep that fixed; instead, in that same second, let's move faster and faster from point A to point B (i.e., my sideways motion). If I move 3 m instead of 2, the angle A becomes 0.0022°. If I move 5 m, the angle's value climbs to 0.0039°. At some point, where the train's speed is too high, the value of A has to move toward around 0°, and c, toward b. In other words, if I move really fast along the breadth of the train and the train has also sped up to a great velocity, I can get from one side of the train to the other as if I simply vanished at this point and materialized at that.

For that to happen, let's make some hypothetical modifications to the train: let the breadth be 2 km instead of a few metres, and let it be accelerating toward around 2,000 km/hr. Assuming that at some point of time the train has stopped accelerating and attained a constant velocity of 2,000 km/hr, if I move 2 m sideways in 1 s, the value of A stands at 0.0000628° and c at 555.5536 m. To make c smaller, let's say the train has sped up to 2,100 km/hr and I move at 1 m/s. This makes A 0.0000298° and c, 583.3309 m. If I move so much as 0.1 m, A becomes 0.00000298° and c, 583.3300002 m. At this stage, A is as good as 0° and c almost equal to b.

For someone watching me move from inside the train, I will have moved sideways at 0.1 m/s. However, for someone looking at the imaginary path (i.e., from a relativistic reference frame), it will be non-existent! This is because I will have moved from A to B in a train so fast that my path will become a, as if there were two parallel lines (one beginning at A and the other at B) and I moved from one to the other along a path that is parallel to both lines. This situation is a mathematical improbability, and thus must correspond to an improbable assumption in the real-world. What is it?

The simplest wrong assumptions are always associated with almosts and nearlies. Saying 583.33 m is almost equal to be 583.3300002 m is different from saying 583.33 m is equal to 583.3300002 m. In the real world, for as long as we don't hit relativistic velocities (i.e., those close to that of light), there will always be these extremely small but furiously persistent inconsistencies - they might seem valid for mathematical and practical approximations but they will always translate into very real differences.