TITLE: Is $\omega^\frac{1}{\omega} > n \forall n \in \mathbb{N}$?
QUESTION [8 upvotes]: I was thinking about $log(\omega)$ which appears to be $\{\mathbb{N}|\omega^{\frac{1}{n}}\}_{n\in\mathbb{N}}\stackrel{?}{=}\omega^\frac{1}{\omega}$. Intuitively, there's the idea that, if the highest power of $\omega$ is greater for one surreal number, then it is greater, but this seems incorrect for two reasons:
$$n<\omega^\frac{1}{\omega}\rightarrow n^\omega <\omega$$
The statement on the right is not true, hence the question. Additionally, from a real analysis perspective,
$$\lim_{n\to\infty}{n^{\frac{1}{n}}}=1$$

REPLY [8 votes]: Every surreal number has a unique representation as a Hahn series of $\omega$ with coefficients being real numbers and exponents being themselves surreal numbers, and they are compared lexicographically as such Hahn series.  (This form is known as a "Conway normal form".  See, e.g., Norman Alling, Foundations of Analysis over Surreal Number Fields (North-Holland Math Studies 141 (1987)), esp., around §6.50, although pretty much the whole book really.)  So $\omega^x$ is greater than all real numbers for any $x>0$, including $1/\omega$.