User blog comment:Nayuta Ito/A simple way to compare ordinals/@comment-1605058-20150331145509

The problem here is ambiguity. We can say that $$f_1(n)=2n$$ and then $$f_1(\omega)=2\omega=\omega$$, or we can say that $$f_1(n)=n\cdot 2$$ and then $$f_1(\omega)=\omega 2$$.

Another problem is that most of the functions defined for natural numbers aren't defined for transfinite ordinals, or can be defined in more than one way.