It's that off-by-one nature of intervals that always bumps me.
The difference between note a and b is (a-b+1).
Calling an octave "an octave" feels to me like calling a numeric system with digits 0x0-0xf as "base 17"
I don't see how 0x0-0xf could be called base 17. 0xf is 15. Did you mean base 15? I think if mathematics terminology developed differently it could be called base 15. The same way binary is 0 and 1 but we call it base 2 because there are 2 digits, but we could totally call it base 1 too, who cares, it's all convention.