User blog comment:Ubersketch/Ordinals with transfinite FS expansions/@comment-35470197-20190701222433/@comment-35470197-20190702100254

> So, least ordinal with infinite descending chain does not exist. It means one of two: > 2. Any ordinal has infinite descending chain, and there is no minimal ordinal

The logic is wrong. The option 2 should be "Some ordinals have infinite descending chains, and there is no minimal ordinal among them". Then your proof does not work because 0 is not such an ordinal.