Goodstein's theorem can be proved (using techniques outside Peano arithmetic, see below) as follows: Given a Goodstein sequence "G"("m"), we construct a parallel sequence "P"("m") of ordinal numbers which is strictly decreasing and terminates. Then "G"("m") must terminate too, and it can terminate only when it goes to 0. A common misunderstanding of this proof is to believe that "G"("m") goes to 0 "because" it is dominated by "P"("m"). Actually, the fact that "P"("m") dominates "G"("m") plays no role at all. The important point is: "G"("m")("k") exists if and only if "P"("m")("k") exists (parallelism). Then if "P"("m") terminates, so does "G"("m"). And "G"("m") can terminate only when it comes to 0.