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How does the Pauli exclusion principle limit the number of electrons in an orbital?

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The Pauli exclusion principle limits the number of electrons in an orbital to a maximum of two.12

This limitation arises because an orbital comprises only two distinct quantum states, and the principle asserts that no two electrons in an atom can occupy the same state simultaneously. Specifically, electrons possess an intrinsic property called spin, which is restricted to two possible orientations: $+ 1 / 2$ and $- 1 / 2$ .12

If an orbital is occupied by two electrons, the Pauli exclusion principle requires that their spins must be paired—meaning one electron must have a spin of $+ 1 / 2$ and the other a spin of $- 1 / 2$ . Once these two states are filled, the orbital is considered full, and no additional electrons can enter it until one of the pair vacates the space. Consequently, no two electrons in an atom can share the same set of all four quantum numbers.123