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1-1 partial recursive 1)-incomparable AE metatheorem AE sentences Algebra Applying the refinement Assume Boolean Algebra cardinal numbers classical theory contain a predecessor contradicting decision problem decomposition of linear definition Dekker-Myhill disjoint RE sets effective analogue elementary theory elements enumeration equation equivalence classes equivalence relation exist isols existence theorems existential quantifiers exists a set f is excluded f,ev finite complete H-chain finite set go to step H-linked H-versions highly decomposable isols Horn-reduct ic/o implies induction infinite RE subset infinitely many pairs isols satisfying lemma limit ordinal metatheoretic result multiple-free Nerode partial recursive function positive countable ordinal Proof of theorem refinement property results of Chapter satisfies the matrix satisfy the conclusion sequence of ideals set satisfying special AE formula step 8f successor ordinal Suppose theorem 3.8 theorem III.2.1 theory of addition Turing degree Xl,l