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human
Clarification about the Pauli exclusion principle

This is something that has been bugging me for a while. The Pauli exclusion principle claims "no two fermions can be in exactly the same quantum state." Most qm books I have encountered describe this equivalently as "no two fermions can have exactly the same set of quantum numbers" and then proceed to list a finite set of them that describe the complete state: J, L, S, ml, ms, N, etc (I might be forgetting some). But an electron in the ground state of hydrogen will always have L = 0, S = 1/2, J = 1/2, ml = 0, ms = 1/2, N = 1 by definition. So if a state can be completely characterized by this set of quantum numbers, and an electron in the ground state of hydrogen will only take these values, this seems to imply that there can only be ONE electron in the ground state of hydrogen in the entire universe. Clearly, this is not true, but I'm having a hard time seeing how it is not implied by the Pauli exclusion principle. Can someone clarify the Pauli exclusion principle for me to help me resolve my misunderstanding? Thanks!
gpt
To answer your immediate question, an electron in the ground state of hydrogen can have ms = +1/2 or -1/2.  

Slightly more interesting, perhaps, is that the PEP you know is a bit simplified.  The actual statement of the PEP says that when you put two identical fermions together, their combined wavefunction must be antisymmetric with respect to exchanging the two particles.  You can get the simplified version from the complete statement by seeing that if both fermions have the same quantum numbers, exchanging them won't change the sign of the wavefunction, so this configuration is symmetric and not allowed.