Errata in Henkin’s 1950 type-theory completeness paper

Errata in Henkin’s 1950 type-theory completeness paper. The first paragraph of Leon Henkin’s influential and widely-read article [1] reads as follows. The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system. As it is written the second sentence says that each formula is a provable logical truth: a patent absurdity. Of course, what was intended was more like following, which moves the clause beginning ‘which becomes’ so it modifies ‘formula of the calculus’. Roughly speaking, this proof demonstrates that each formula of the calculus which becomes a true sentence under every one of a certain intended class of interpretations of the formal system is a formal theorem. Taking more advantage of the disclaimer ‘roughly speaking’, he could say: Roughly speaking, this proof demonstrates that each formula which is true under every interpretation is a theorem. Almost 20 years later, Jaakko Hintikka [2] reprinted [1] along with other important articles. Hintikka noted errata in another Henkin article but not in [1]. There are other mistakes in [1]. Moreover, it contains several passages that, if not actual mistakes, are regrettable. We list and analyse glitches in [1] in order to facilitate study of this landmark article. We also discuss the old question of whether, as logicians including Burton Dreben have charged [personal communication], [1]’s title is misleading in that it seems to contradict an easy consequence of Gödel’s 1931 incompleteness paper. The consequence in question is explicitly mentioned in the second paragraph of [1]. [1] LEON HENKIN, Completeness in the theory of types. Journal of Symbolic Logic, vol. 15 (1950) pp. 81–94. [2] JAAKKO HINTIKKA, Philosophy of Mathematics, Oxford UP, 1969. Acknowledgements: Mark Brown, William Demopoulos, Thomas Drucker, Rodolfo Ertola, William Frank, Richard Grandy, Idris Samawi Hamid, Allen Hazen, Leonard Jacuzzo, Miguel León, Kristo Miettinen, Joaquin Miller, Frango Nabrasa, David Plache, Sriram Nambiar, José Miguel Sagüillo, Michael Scanlan, Stewart Shapiro, James Smith, Roberto Torretti, Alasdair Urquhart, Albert Visser, George Weaver, George Williams, Stanley Ziewacz, and others—some of which disagree with some of my points or with my way of making them. In particular, distinguished and accomplished logicians including Allen Hazen, Christopher Menzel, Alasdair Urquhart, and Albert Visser think that Henkin’s expression of completeness is correct as it is—awkwardness notwithstanding. To be clear, they believe and argue that I am wrong to imply that this is a mistake on Henkin’s part. WHAT IS YOUR VIEW? ...Read more