I understand you meant that in jest but base 1 is possible, albeit very cumbersome:

1 = 1

2 = 11

3 = 111

4 = 1111 ...etc.

Sorry for the nitpick, but that's not really "base 1".

It is "unary", which is completely different encoding than the "positional notations" such as decimal, octal, binary, etc. The positional notation doesn't work with base 1, as in that notation there would be only 0, and 0, 00, 000, etc., which all mean the same number: zero.

That I agree on, it is conceptually different (and more primitive); the whole conversation just reminded me of GEB.

Really? How do I write zero in your system? (as in the number of apples I have if I have none at all)

If octal uses the digits 0-7, and binary uses the digits 0-1, then shouldn't base one use only the digit 0?

0 = 1

1 = 11

2 = 111

3 = 1111, etc. {1,2,3...} and {0,1,2...} are isomorphic.

But surely that's not the way it works in any other base. In any other base there is a digit '0' such that:

0 = 00 = 000 = 0000 = the number of apples I have if I don't have any.

To dig in a little further, when we say base N, we mean that we are expressing an integer as a sum of the values N^i (for i being every nonnegative integer), each multiplied by a weight that is an integer between 0 and N-1 inclusive.

Why do we do this? Because every integer has exactly one unique representation in this system. We could have picked any values we wanted for the places (rather than N^i) but unless we pick carefully, either some numbers aren't representable, or some numbers have multiple representations.

Base one has a problem right from the start, which is that 1^i = 1 for any value of i - when all places have the same place value, numbers aren't going to have a unique representation. 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.

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.

"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