Main Question or Discussion Point
Imagine a system consisting of two hydrogen atoms with the electrons sitting in the ground state with the same spin. Surely there is nothing stopping us from constructing such a system. The system is completely symmetric - each electron experiences the same potential. The wavefunction of the system is thus symmetric under interchange of electron labels, but this violates the exclusion principle, which suggests that we could not contruct such a system. But this doesnt make sense, surely there can be more than one hydrogen atom in the universe with an electron in the ground state.