Yes, you can definitely create a system with {0, 1, 2} and this is in base 3. Binary is in base 2 that uses only {0, 1} and this is machine-readable with an off (0) and an on (1) switch.

We humans use the decimal system which uses {0, 1, …, 9}. And in fact there are even higher bases such as the hexadecimal which uses the same as decimal but with additional A, B, …, F to represent 10, 11, …, 15 respectively. The decimal and hexadecimal are bases 10 and 16 respectively.

The base 3 thing is just not as used unlike the base 2, 10, and 16. But you can for sure define them, as well as for any other higher base