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I believe this has to do partially with the negative charge of electrons. They repel each other if in close proximity and if two electrons were to have the same four quantum numbers, they would theoretically be found in the same place with the same energy, same angular motion, and same spin direction. Each electron must have its own space and a difference in at least one of the quantum numbers dictates a slightly different space.
This refers to how there can be two electrons in an orbital but they are not moving in the same direction. There can be six electrons in the 2P energy level, but in every electron pair, they are drawn going in the opposite directions (one of the arrows points up and the other points down)
There are 4 different quantum numbers and each gives you an idea of where the electron is located in an atom. 2 electrons can't share more than 3 numbers because that would mean that they are in the same location which is not possible because they would repel each other due to like charges.
I think it makes sense to view it as the numbers helping us know where the electrons are located in space. Only two electrons of opposite spin can inhabit the same orbital. This would mean that all of their quantum numbers would be the same except for the spin magnetic quantum number. If all four quantum numbers were the same, this would mean that electrons of the same spin (two upward arrows or two downward arrows) were inhabiting the same space which cannot be allowed. Think of it like opposites attract, and same spin signs repel one another. I hope this helped!
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