I would be pleased if someone could point out to me where I go wrong in the following sequence of statements:

1. One model of quantum mechanics identifies states of a particle with normalized vectors (up to a "phase factor") in a **separable** complex Hilbert space.
2. In a separable space any any orthonormal basis is countable
3. $\psi(x)$, the wave function for the position of a particle on a line seems to treat $x \in R$ as a basis for the space.
4. If a particle is at $x\_1$ it is there with probability 1, and it is not at $x\_2$, so $\langle x\_1|x\_2\rangle = 1$ if $x\_1 = x\_2$ and zero otherwise.
5. So, $\{x: x \in R\}$ is orthonormal and a basis, but it isn't countable which contradicts (2)..