Suppose that I have a piece of paper with 0 on it (and nothing else). Suppose that, at each instant, I can either replace what is on the paper by writing either 0 or 1. I say that I change the value on the paper if I replace 0 by 1 or if I replace 1 by 0. Suppose I look at the paper after $\omega$-many instants. I will see either 0 or 1. Is there a way to understand if the values have changed unboundedly often?
Hamkins and Lewis in $Infinite$ $Time$ $Turing$ $Machines$ p. 572 seem to suggest to use an idea that may be adapted to this case. I am not sure I understand their idea, but I will try to explain what I understand: you put a lamp near the paper and every time you change the value on the paper you turn on and immediately off the lamp. After $\omega$-many steps the lamp will be on iff the values on the paper have changed unboundedly often.
Would this work? Why?
Other possible strategies?