User blog comment:Vel!/1+w/@comment-5982810-20141011203204/@comment-2033667-20141011211228

In fact it can even be proven without assuming the Wainer hierarchy. \(\omega[n]\) is always finite, so \(1 + \omega[n] = \omega[n]\) implies that \(a = a + 1\) for some \(a\). Consequently \(0 = 1\).