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All attempts at deriving the born rule in MWI have been shown to be circular in some way. So if it turns out that MWI cannot derive the born rule without some form of circularity, does that mean that the MWI is wrong?

Qmechanic
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A pretty standard presentation of the principles of quantum mechanics is as follows. MWI says

  1. We have a Hilbert space with
  2. unitary evolution of the wavefunction.

The Copenhagen interpretation (CI) adds additional postulates saying that

  1. measurements produce c numbers,
  2. probabilities are given by the Born rule, and
  3. measurement collapses the wavefunction.

All attempts at deriving the born rule in MWI have been shown to be circular in some way.

This just says that postulate 4 is independent of postulates 1-2. Why is that a problem? Actually, CI doesn't "derive" the Born rule, it just states it as a postulate.

So if it turns out that MWI cannot derive the born rule without some form of circularity, does that mean that the MWI is wrong?

Since the postulates of MWI are a subset of the postulates of CI, the only way that MWI can be wrong is if CI is wrong as well.

Note that if you want CI and you take some other rule than the Born rule, you will probably get nonconservation of probability. You can make this more formal using Gleason's theorem. Whether this counts as "deriving" the Born rule is a matter of opinion or preference.