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From a philosophical standpoint, $0!=1$ as it is possible to arrange nothing in one way--there isn't.

Are there any rigorous proofs showing this concept?

Masacroso
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uncertainty
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    There are lots of ways to see that $0!=1$ makes sense, but ultimately, the factorial is a tool that humans made up because it is useful, so we are free to define $0!$ in whatever way is most useful. The fact that $0!=1$ is so useful in so many ways and in so many contexts is the kind of thing that makes math so cool to me. – Plutoro Apr 20 '17 at 01:59

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The factorial function is defined to count the number of bijections between finite element sets, namely $n!=$ # $ f: \{1,..,n\} \to \{1,..,n\}$ that are bijective. There is precisely one bijective map $f : \emptyset \to \emptyset$, namely f is the empty map.

I recognize this is a very odd concept. Hope this helps.

ADA
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  • The factorial can be defined in such a way; there are similar, but non equivalent definitions of $n\mapsto n!$ which are valid. – YoTengoUnLCD Apr 20 '17 at 03:11