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Craig's theorem says that a theory with a computably enumerable axiomatization has an equivalent decidable axiomatization. The Craig trick gives a constructive proof by encoding an enumeration index in repeated conjunctions of the enumerated axiom.

The Craig trick converts an enumeration $A_1,A_2,\dots$ of axioms into a decidable equivalent set by replacing $A_i$ with a syntactically self-indexing repetition containing $i$ copies of $A_i$. A candidate of length $m$ can only encode one of the first $m$ axioms, so membership is decidable after computing that finite prefix.

The construction proves Craig's theorem: every computably enumerable axiomatizable theory has an equivalent decidable axiom set.