Decomposing ordinal numbers

Question

I believe every ordinal numbers $\alpha$ can be written as $\alpha=\lambda+n$ for a limit ordinal $\lambda$ and a natural number $n$. Is this true? Is it easy to see? Is there some terminology for the $\lambda$ and the $n$, something like limit part and natural part? (in analogy to integer and fractional part of reals)

Answer 1

Yes, yes, yes. The "$n$" is the part with zero exponent in the Cantor normal form of $\alpha$, while the "$\lambda$" is the part with positive exponents.

Answer 2

This is true, and can be shown by transfinite induction. This holds for $0=0+0$ provided that we consider $0$ a limit ordinal in this context. Further, if $\alpha=\lambda+n$ then $S(\alpha) = \lambda+(n+1)$. Finally, if $\alpha$ is a limit ordinal, then we may write $\alpha=\alpha+0$.

Another way to show this is to use the division algorithm for ordinals: given $\alpha$, we may write $\alpha = \omega\beta+n$, for some $\beta$, and where $n < \omega$. Note that $\omega \beta$ is a limit ordinal.