If the proof's core strategy is salvageable under modest technical revision, the fault is in the execution rather than the foundational claim that reducibility is dispensable for induction.
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induction(Offered as the mechanism behind empirical universality.)
The empirical method by which observations are generalized into rules; yields only comparative or assumed universality, not strict universality.
proof(Frege's formal system; the definition still used by logicians today)
Any finite sequence of statements such that each statement is either an axiom of the formal system or follows from previous members of the sequence by a valid rule of inference.
reducibility(Computability theory; used to compare the relative difficulty of decision problems.)
Problem P1 is reducible to problem P2 if a solution of P2 immediately yields a solution of P1.