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Fermi's Theory of Beta Decay and Neutrinos
Pali originally proposed that this particle would be emitted. It wasn't really turned into a fully mathematical model until Fermi got his hands on it. He produced a theory of the weak interaction which would mediate this beta decay and produce these neutrinos. I think people did take it seriously because there was really no other way to understand what was happening in beta decay. But as you say, Pali apologized for having predicted the existence of a particle that could never be observed.