I’ve talked too much with mathematicians

You are talking to one right now :) (not sure if a bachelors degree is enough to call yourself one)

you can’t have 0 apples

You can actually. In fact, right at this moment I have 0 apples. If 0 is not natural, then you have no way of describing the number of apples I have.

There are a lot of concepts (degree of a polynomial, dimension of a space, cardinality of a set, in a graph) where 0 is a natural possibility.

So I think 1 indexing is fine, I use it all the time, but to me 0 belongs with the natutals. I will say tho, that 0 does not make sense to me as an ordinal. “He finished the race in 0-th place”???

I guess it depends on the place. But the arguments for not including seem futile, when

• we use 0 to even write the other natural numbers
• we define almost all of our algebraic objects (groups, rings/fields, modules/vector spaces, algebras) to include 0
• we don’t do modular arithmetic with {1,…,n} that would be crazy

Of course 0 vs no 0 only matters if you actually do arithmetic with it. If you only index you could just as well start with 5.

(The only reasons I can think of to start at 1 is that 1 is the 1-st element then and the sequence (1/n) is defined for all natural n)

Idk if you are trolling, but in most cases 0 is considered a part of the natural numbers. And there is a huge difference between the naturals and the integers: the naturals are for induction, the integers are for algebra.