Wow you just disproved all of academic mathematical foundations and philosophy! Congrats!

We make mathematical definitions to do math. We can define 0! any way we want but we defined it to be equal to 1 because it fits in nicely with the way the factorial function works on other numbers.

Literally the only reason why mathematicians define stuff is because it’s easier to work with definitions than to do everything from elementary tools. What the elementary tools are is also subjective. Mathematics isn’t some objective truth, it’s just human made structures that we can expand and better understand through applying logic in the form of proofs. Sometimes we can even apply them to real world situations!