Is $\sqrt[\omega]{\omega}$ an omnific integer?

Question

I heard that $\omega^n$ (for any positive real, n) is an omnific integer, but dose this property also extend to numbers such as $\sqrt[\omega]{\omega}$ (aka $\omega^{1/\omega}$)?

Answer 1

Assuming you mean to define $\omega^{1/\omega}$ by the definition of powers of $\omega$ in Conway's writings that is featured on Wikipedia, then yes, $\omega^{1/\omega}$ is an omnific integer. By definition, $\omega^{1/\omega}=\{\mathbb R^+\mid r\omega^{1/n}\}$ where $r$ ranges over the positive reals and $n$ ranges over the positive naturals (or the powers of $2$, say). Subtracting $1$ or adding $1$ won't go beyond those bounds (the same is true of any positive power of $\omega$), so $\omega^{1/\omega}$ is indeed an omnific integer by the simplicity theorem.