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A famous paper by Leon Henkin ("Completeness in the theory of types") begins 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."

I do not understand the latter sentence. Indeed a non-provable formula has no reason to be a formal theorem and no reason to become true under every "sensible" interpretation. For me, it should be something like that: "this proof demonstrates that each formal theorem of the calculus becomes a true sentence under every one of a certain intended class of interpretations of the formal system." Since I trust Leon Henkin as a logician, I guess that I am missing something; maybe is my English the problem and do Henkin's sentence and my sentence have the same meaning?

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    $\begingroup$ In ordinary English, it would be clearer to say that the completeness theorem shows that "each formula which is true in every intended interpretation, is, in fact, a formal theorem." The converse, which seems to be what you are suggesting, also holds but is known as "soundness" rather than "completeness" and is much simpler to prove, as it only requires that the inference rules and logical axioms preserve truth. The completeness proof requires that there are enough axioms and rules to formally prove every logical consequence of the axioms. $\endgroup$ – Ned Feb 3 '16 at 1:41
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    $\begingroup$ @MauroALLEGRANZA and Rob Arthan: And, as said by Ned, it would be a statement of the trivial direction (the "soundness"), which does not reflect the deep content of Gödel's theorem (which is rather the converse). $\endgroup$ – user251130 Feb 3 '16 at 20:30
  • $\begingroup$ @user251130: you are right. Mauro's suggestion does not fix the sentence. $\endgroup$ – Rob Arthan Feb 3 '16 at 23:06
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As a native English speaker with some knowledge of Henkin's subject matter, I find Henkin's phrasing very odd. Given that he is talking about completeness his wording makes sense if you read "which" as "if it". (Your reading would be appropriate to soundness rather than completeness.)

However, even though "which" is a word that has been used differently in the past from the way it's used now, I'd be surprised if reading "which" as "if it" here was standard 1950s academic US English usage.

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    $\begingroup$ I'm glad I'm not the only person who read this and went " . . . Bwahuh?" $\endgroup$ – Noah Schweber Feb 3 '16 at 2:39
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    $\begingroup$ @NoahSchweber: I've been racking my brain trying to think of other examples of "which" used as Henkin seems to use it and came up with one: maybe Henkin had been reading Milton: "they also serve who only stand and wait" uses "who" in something close to the way Henkin apparently uses "which". $\endgroup$ – Rob Arthan Feb 3 '16 at 2:58
  • $\begingroup$ I'm not a native English speaker and I was utterly confused by his way of writing as well, I was afraid it was solely because of my terrible English. $\endgroup$ – YoTengoUnLCD Feb 3 '16 at 4:43

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