User blog comment:Ikosarakt1/Hardy hierarchy up to psi(K)/@comment-25418284-20140617140300/@comment-5150073-20140617140842

So we don't let $$1+\omega = \omega$$. We can say that $$1+\omega \approx \omega$$ because $$H_{1+\omega}(n)$$ would be $$\approx H_\omega(n)$$. That's why I said that $$\omega$$ is just a symbol and useless without hierarchy.