In the paper "Infinite Time Turing Machines", the following information is given:
To set up such a limit ordinal configuration, the head is plucked from wherever it might have been racing towards, and placed on top of the first cell. And it is placed in a special distinguished limit state. Now we need to take a limit of the cell values on the tape. And we will do this cell by cell according to the following rule: if the values appearing in a cell have converged, that is, if they are either eventually 0 or eventually 1 before the limit stage, then the cell retains the limiting value at the limit stage. Otherwise, in the case that the cell values have alternated from 0 to 1 and back again unboundedly often, we make the limit cell value 1. This is equivalent to making the limit cell value the lim sup of the cell values before the limit (pp. 569).
Moreover, the authors make use of a technique in which we can detect whether the machine has entered the limit state infinitely many times, i.e., whether the machine has run for $ω^2$ steps. The basic idea is to use a cell as a "flag" such that, once every limit stage, the machine writes a 1 to the cell and then immediately writes a 0 to the same cell (the cell's initial value is 0). Using this technique, once we hit $ω^2$ steps, the value of the flag cell will be a 1 in the limit state of the machine for the first time.
My question is this: Can this flag cell ever again be a 0 at any future limit state? It seems like it cannot be. If we write a 0 to that cell, then the next time the machine enters the limit state (at $ω^2 + ω$ steps), the cell value will be updated to 1. And this will be true at every future limit state. Is this reasoning correct? If it isn't correct, then I'm struggling to see how the flag cell technique could work. On the other hand, if it is correct, then I'm struggling to see how the following from the above mentioned paper works (this is part of the proof of Theorem 2.2):
There is a slight complication at the compound limits (limits of limits) since at such stages we will have a lot of garbage on the scratch tape, but because we were gradually erasing elements from the field of the relation coded by the real on the input tape, the input tape has stabilized to the intersection of those relations, which is exactly what we want there. By flashing a flag on and then off again every time we reach a limit stage, we can recognize a compound limit as a limit stage in which this flag is on, and then in $ω$ many steps wipe the scratch tape clean before continuing with the algorithm (ibid., pp. 572, emphasis added).
I don't understand how we can "wipe the scratch tape clean" given that some of the cells on the scratch tape could have flipped from 0 to 1 and back again unboundedly often. For if there are cells like this, then as soon as we finish wiping the scratch tape clean in $ω$ steps, the machine will enter the limit state and all of those cells will be flipped back to 1, resulting in a scratch tape that is no longer "clean." Is this correct? I'm very confused about this.