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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

  • 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,

  • Why is $∅* = \{ \epsilon \}$? How does concatenating the empty set with itself infinitely many times magically create a non-empty set?
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    $\begingroup$ Does an analogy with $x^0 = 1$ (for $x \neq 0$) not help? $\endgroup$ – James May 27 '15 at 15:26
  • $\begingroup$ Not every language contains $\epsilon$. $\endgroup$ – mrp May 27 '15 at 15:31
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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.

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    $\begingroup$ $0^0=1$ also, and the analogy for languages is that $\emptyset^0 = \{\epsilon\}$. $\endgroup$ – MJD May 27 '15 at 15:34

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