I don't get the hang up people have with zero. It is not a counting number. It is a place-holder between numbers. When I write 1230 in decimal, the 0 being there only increments the power of 10's up by 1 for 1, 2 and 3 but that is the only purpose the zero has here.
When counting in unary/base(1) there is no need for 0, why? Because unless we're dealing with bizarro stuff, 1 to any power is 1. So, in unary, 111 = 10101 = 1000000010001000. Unless the 0's now represent time or are code from something else non-numerical, they've lost their meaning. 0's don't count anything in unary, thus in base 1 we only use 1's or tally marks. And, in base 0, all we can use is 0, since the only value expressed in base 0 is ironically the value I said wasn't a real number, 0. The problem when base 0 is that writing anything aside from just 0^1, or 0 is either redundant or undefined.
So, it isn't that base zero or non-whole number bases don't exist - what's important to note are questions such as: does such a base offer anything useful in counting, and if not, does it provide us with anything else useful?
I know I rambled on, but, does that make sense?