User blog comment:LittlePeng9/ITTM galore/@comment-10429372-20140317195302/@comment-1605058-20140317202710

1. In first ω steps machine writes nothing, so when entering limit stage for the first time, first cell is empty. Machine writes 1 there and right after that erases it (we are "flashing" the first cell). So later tape is again empty, so second time machine enters limit stage first cell is again empty, so we flash it again. Thus for every n we flash the flag at ω*n. So, before ω2 first cell flashes unboundedly many times, so there is 1 on that cell at stage ω2. Machine in limit stage notices it and halts.

2. If you mean machine halting in ω1CK, then it's impossible. This follows from theorem 8.8 from here (last theorem). Smallest nonrecursive ordinal we can clock is ω1CK+ω+1 (normal ITTM can cloch ω1CK+ω, but one tape machine can't, as it ends gap in clockable ordinals (theorem 3.2 here)).