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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?

MatteoBianchetti
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  • Related: http://en.wikipedia.org/wiki/Thomson's_lamp – Quinn Culver Oct 22 '14 at 16:20
  • Thanks Quinn. It would be great if our lamp can work in this way: every time that the situation S occurs turn the lamp on and off (immediately); after $\omega$-many stages, check the lamp: it is on iff S occurred unboundly often. I do not think that Thomson's lamp does the job, but Hamkins and Lewis seem to suggest that something similar will work. But I still do not see exactly what will work. Many thanks – MatteoBianchetti Oct 23 '14 at 18:28

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