Wouldn’t this also mean that 1!=0!, why is this true?
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In mathematics we can define things however we want to. That’s one of the beauties of it. We define things and explore the consequences, looking for patterns.
When we define things we are looking for things that are interesting, useful and consistent with all our other rules (ideally all three of them, but hopefully at least two.
We could define 0! to be something other than 1. That is a perfectly valid thing to do in maths. But it turns out not to be very useful, interesting or consistent.
Whereas defining 0! to be 1 lines up with some other neat patterns and rules, and works out quite well.
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