User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-5529393-20130619110333/@comment-5529393-20130620101749

I hate to keep saying it, but it still doesn't work. One counterexample is \(\alpha_1 = \omega + 1, \alpha_2 = \omega\). This disproves the addition, multiplication, and exponentiation rules. If you want limit ordinals, you can take \(\alpha_1 = \omega^2 + \omega, \alpha_2 = \omega^2\).