> You can define counting as defining an injective function from your set to the natural numbers, and then you need to have some element going to 0 - as per the definition of a countable set[0].

You can define counting as an injection into other (equivalent) sets just as rigorously.

And it's not even universally agreed on that the natural numbers should include 0. Wikipedia mentions the different conventions: https://en.wikipedia.org/wiki/Natural_number

(I like my natural numbers to start with 0. But that's just because 0 is my second most favourite number. Starting with 1 is legitimate.)