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In general, any well ordered set B can be raised to the power of another well ordered set E, resulting in another well ordered set, the power B^E. Each element of B^E is a function from E to B such that only a finite number of elements of the domain E map to an element larger than the least element of the range B (essentially, we consider the functions with finite support).

According to this, $\displaystyle 0^\omega$ is the set of functions from $\displaystyle \omega$ to $\displaystyle \emptyset$, i.e., $\displaystyle \emptyset$.