User blog comment:Chronolegends/Tetration on ordinals/@comment-1605058-20160524190634/@comment-1605058-20160524194656

Binomial formula for $$(a+b)^3$$ isn't $$(a+b)(a^2-ab+b^2)$$, you are confusing this with $$a^3+b^3$$. We have $$(a+b)^3=a^3+3a^2b+3ab^2+b^3$$, no subtraction. Of course this formula isn't guaranteed to work for ordinals (indeed, $$(\omega+1)^2=\omega^2+\omega+1$$, not $$\omega^2+\omega+1$$ as the binomial formula suggests), I just point out you've got the formula wrong.

Also, I believe this definition of tetration is fairly well-behaved - as long as $$\alpha\geq\omega$$ I believe we have (using your definition $$^{\beta+1}\alpha=(^\beta\alpha)^{(^\beta\alpha)}$$, as expected, $$^3\alpha=\alpha^{\alpha^\alpha},^4\alpha=\alpha^{\alpha^{\alpha^\alpha}},\dots$$. I didn't bother checking details, but I am fairly certain this is right.