Therefore there are an infinite number of possible values of 1+1. Therefore the probability that 1+1=2 equals zero. We express this identity as $P^1_1(2)=0$. Indeed $P^1_1(x)=0$ for all x. However in the "real" numbers we must have $\int_{-\infty}^\infty P^1_1(x)\,dx=1$, a contradiction by the Dominated Convergence Theorem.

The obvious resolution is to introduce enough infinite and infinitesimal numbers to the so-called "real" numbers that the line ceases to be second countable. However the simplest and most elegant solution is simply to notice that infinity does not exist: that there is at most a potential (rather than completed) infinitude of real numbers, and the value of 1+1 is an unknown, possibly changing, value among them.