= Projective determinacy collapses the inner-model omega-one
If $M$ is projectively well-ordered and <projective determinacy> holds, then $\omega_1^M$ is countable in the ambient universe. Otherwise, selecting with the projective well-order the least $M$-code for each countable ordinal produces an uncountable projective set of unique well-order codes. It has no perfect subset by the <boundedness theorem for well-order codes>, contradicting the <perfect set property> implied by projective determinacy.
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