[stuffihavemade]

"When you write 1111 = 4 you are sneakily using the length of the number on the page to encode the value. The length of the number on the page isn't supposed to matter. In base 10, 17 = 017 = 0017 = 00017."

I couldn't find a rigorous definition of a base system offhand. If you require that number N prepended with additive identity I, so IN = N, I agree with you. But every number in the system I described certainly has a unique representation. Also, this is incorrect: "The other problem is that the only number you can write in base one is zero, because the only possible weight (the only integer between 0 and N-1) is 0." Try taking a look at: http://en.wikipedia.org/wiki/Peano_arithmetic