Seeking explanation for this apparent paradox shaking the grand pillars of mathematics.
If I stand on the unit circle (of circumference $2\pi$) and I take steps of arclength $s$, then what is required of $s$ such that I eventually come within length $\delta$ of any point?
And I have a device that makes a gong at time t_0=0 seconds, then at t_1=1/2 sec, next at t_2=3/4 sec, next at t_3=7/8 sec, the n'th gong happening at time $t_n=1-1/2^n$ sec. And at the $n$'th gong i take f(n) steps of length $s$. After 1 second has passed, with f(n)=1, is there an $s$ such that I visited every point on the circle? If not, which points did I not visit?
Same questions at t=1, except now I take f(n)=3^n steps at the n'th gong.
Same questions at t=1, except now at the n'th gong, I take f(n)=n^n*(number of subsets of the natural numbers (1 to n)) steps.
Apparently, already after 1 second, I have made a step for every (finite and infinite) subset of the natural numbers, which has cardinality continuum!?