Imagine a particle, w, located in a continuous three-dimensional space. The space is infinite in all directions, and does not loop back in on itself at any point or distance. W starts at P, a constant, fixed point in the space. W is always moving at a constant speed relative to P, but its velocity can change arbitrarily fast and infinitely often; it can turn on a point, reverse direction, and generally trace out any path in the space, as long as its speed is constant.

Describe the cardinality of the set of all possible paths P can take in the space as a function of t, the time elapsed.

Prove that the length of P's worldline is always proportional to t.