Bulletin of Symbolic Logic. 12 (2006) 143-67. John Corcoran, Complete enumerative inductions. Philosophy, University at Buffalo, Buffalo, NY 14260-4150 E-mail: corcoran@buffalo.edu This largely expository paper explores analogues of ω-completeness and ω-inconsistency, starting with "finite cases". For each positive integer n, let Ln be a first-order language having exactly n [individual] constants, say the digits '0', '1', etc. of a base-n arithmetic notation. An instance of a universal sentence is the result of deleting the quantifier and replacing every occurrence of the freed-up variable by one and the same constant. The complete induction sentence [for Ln] is the sentence expressing "for every object x, x is 0 or x is 1 or etc.", for L3, in symbols, Ax(x = 0 V x = 1 V x = 2). A set of sentences is ncomplete iff it [deductively] yields every universal sentence each of whose instances it yields. Theorem COM: In order for a set to be n-complete it is necessary and sufficient for it to yield complete induction. A set of sentences is n-inconsistent iff it [deductively] yields the negation of some universal sentence each of whose instances it yields. Theorem INC: in order for a set to be n-inconsistent it is necessary and sufficient for it to yield the negation of complete induction. In that both of these theorems reduce a condition concerning infinitely many deductions to the deducibility of a single sentence, they are syntactic, or proof-theoretic, results analogous to semantic, or model-theoretic, results about ω-completeness and ω-inconsistency in second-order languages announced in my abstract "Semantic omega properties and mathematical induction", Bulletin of Symbolic Logic 3 (1997) 280. When we leave either the "finite" case or the second-order case to consider intermediate cases such as sublanguages (involving constants for zero and successor) of the usual first-order languages of number theory, we find that they do not admit of ω-completeness or ω-inconsistency being reducible to the deducibility of a single sentence.