I've been browsing through Jech's and Levy's texts on set theory, and the ideas of ordinals come up fairly quickly. The idea of a limit ordinal is introduced, which is an ordinal with no maximum element. My question is, can any infinite ordinal be written as the sum of a limit ordinal and a finite ordinal, possibly unique?
My thinking was, if $\alpha$ is an infinite ordinal with no maximum, it is a limit ordinal, so $\alpha=\alpha+0$. Otherwise, suppose $\alpha$ has some order type $\{a_0,a_1,\ldots, b\}$, so $\alpha=\omega+1$. Similarly, if $\alpha$ has order type $\{a_0,a_1,\ldots,b,c\}$, we could write it as $\omega+2$.
(Thanks to Arturo Magidin, for pointing out that the following example I gave is not an ordinal.) But what about an order type like $\{a_0,a_1,a_2,\ldots, b_2,b_1,b_0\}$, this has order type $\omega+\omega^*$, would it still be possible to write is a sum of a limiting ordinal and a finite ordinal? Thanks.