In formal languages, why is $L^0 = \{ \epsilon \}$? Why isn't it the empty set ∅? It just doesn't make logical sense to me that a language to the power of $1$, is itself, but to the power of $0$ is only a tiny part of itself 


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*Wouldn't it would make much more sense if $L^0 = ∅$. 


In English that would say, if you have this language in zero magnitude, you've got absolutely nothing.
Also, a related question I have is, 


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*Why is $∅* = \{ \epsilon \}$? How does concatenating the empty set with itself infinitely many times magically create a non-empty set?


You want $L^nL = L^{n+1}$ (law of indices). You also clearly want $L^1 = L$. Combining them gives $L = L L^0$, so $L^0 = \{\epsilon\}$.
By the way, the same sort of reasoning explains why defining $n^0$ to equal $1$ when $n$ is an integer is a good idea.
