Maybe learn some algebra before you start being pedantic. You can't divide by zero and any notation that does is shorthand for something else. There is no way to define 6/0 in the same way as 6/3 and the fact that you need a whole-ass metric space so you can have convergent sequences shows that.
And n0 is not just arbitrary but the definition of an empty sequence of an operation is that it yields the neutral element. The empty sum yields 0, the empty product yields 1 and n0, in the most naive and direct interpretation, Is n multiplied by itself no times, hence an empty product and therefore yields 1 (there is a caveat for 00 but that's like a whole lecture). Sure that is arbitrary in the sense that any axiomatic system is arbitrary but not in the /0 sense where it keeps changing based on context and no self-consistent axiomatic system is possible where /0 is assigned a value.
Maybe learn some algebra before you start being pedantic. You can't divide by zero and any notation that does is shorthand for something else. There is no way to define 6/0 in the same way as 6/3 and the fact that you need a whole-ass metric space so you can have convergent sequences shows that.
And n0 is not just arbitrary but the definition of an empty sequence of an operation is that it yields the neutral element. The empty sum yields 0, the empty product yields 1 and n0, in the most naive and direct interpretation, Is n multiplied by itself no times, hence an empty product and therefore yields 1 (there is a caveat for 00 but that's like a whole lecture). Sure that is arbitrary in the sense that any axiomatic system is arbitrary but not in the /0 sense where it keeps changing based on context and no self-consistent axiomatic system is possible where /0 is assigned a value.