Henkin's completeness proof: forty years later

Hugues Leblanc, Peter Roeper, Michael Thau, George Weaver
1991 Notre Dame Journal of Formal Logic  
1 In his 1949 paper, "The completeness of the first-order calculus", Henkin developed what is now called the method of (individual) terms 1 to establish that every consistent set of statements of a first-order language L has a model of cardinality α, a the number of statements of L. The idea is to start with such a set S, construct a so-called term-extension L + of L by adding a new terms to the vocabulary of L, 2 extend 5 to a maximally consistent and term-complete set Soo of statements of L +
more » ... , 3 and construct a model of S& whose domain consists of the terms of L + . When restricted to L, the model in question automatically constitutes one of S. Henkin's result has come to be known as the Strong Completeness Theorem for First-Order L. Another, and more familiar, version of the theorem has it that if a statement A ofL is true in every model of a set S of statements ofL, then A is provable from S. Henkin himself did not bother to prove this. He merely proved the special case of it, known as the Weak Completeness Theorem for First-Order L 9 where S is 0. A model like the one Henkin constructed for his set S^, is commonly known as a Henkin model. It is the kind of model in which each member of the domain "has a name". Henkin accomplished this by making each member of his domain a name of itself, a radical move at the time. In consequence, though, the restriction of his model to L does not constitute a Henkin model of S, a pity in the event that S does have such a model. To commemorate the publication of Henkin's paper, we offer here two new completeness proofs for first-order Z,. 4 The language considered by Henkin had an unspecified number of terms to start with, but those played no special role in his proof. The one we construct in Section 2 has denumerably many, and these will play a crucial role in our proofs. In the first of them, begun in Section 3 and concluded in Section 5, no new terms will be added; in the second, presented in Section 5 and relating to truth-value semantics, 5 denumerably many will be. The proofs are sharpenings of proofs of Leblanc's in [10]. They have two cases each, Case One minding the consistent sets of statements of L that extend without the
doi:10.1305/ndjfl/1093635746 fatcat:6mpiydkek5dtjcov22ushkf4ni