I have read that fermions cannot exist in the same state simultaneously. I understand why indistinguishable particles with an antisymmetric superposition of states can't exist in the same state simultaneously, but why must fermions have an antisymmetric superposition of states?
The only characterising property I know of fermions having besides antisymmetry is spin, for which they have half-integer. I understand that this is the case simply because particles with half-integer spin and particles with zero or integer spin were defined as fermions and bosons respectively.
My perusal of the Wikipedia page on the spin statistics theorem leaves me under the impression that spin has nothing to do with the wavefunction's symmetry properties:
Naively, neither has anything to do with the spin, which determines the rotation properties of the particles, not the exchange properties.
Are antisymmetric wavefunctions simply classified as fermions, in the way half-integer spin particles were? I don't see how this could be the case, as, if spin and symmetry were independent, half-integer spin particles with symmetric wavefunctions (and antisymmetric integer spin particles) would be possible.