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Could someone explain to me how Bell's theorem solves the EPR paradox and 'spooky action at a distance'?

From what I understand, when measuring a state, say spin up in the x direction, the wave function collapses and the other particle must be spin down in the x direction instantaneously no matter how far it is from its entangled partner.

I can't seem to put all the pieces together with local realism and local hidden variables.

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Bells inequality is derived by assuming 'local reality'. This inequality has been shown to be violated by experiment and by theory (quantum mechanics). So the conclusion is: nature doesnt always posess local reality. Assuming local reality made EPR a paradox, so without this it is resolved.

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