Editing Second-order logic (section)

==Metalogical results==
It is a corollary of [[Gödel's incompleteness theorem]] that there is no deductive system (that is, no notion of ''provability'') for second-order formulas that simultaneously satisfies these three desired attributes:{{efn|The proof of this corollary is that a sound, complete, and effective deduction system for standard semantics could be used to produce a [[recursively enumerable]] completion of [[Peano arithmetic]], which Gödel's theorem shows cannot exist.}}

* ([[Soundness]]) Every provable second-order sentence is universally valid, i.e., true in all domains under standard semantics.
* ([[Completeness (logic)|Completeness]]) Every universally valid second-order formula, under standard semantics, is provable.
* ([[Decidability (logic)|Effectiveness]]) There is a [[automated proof checking|proof-checking]] algorithm that can correctly decide whether a given sequence of symbols is a proof or not.

This corollary is sometimes expressed by saying that second-order logic does not admit a complete [[proof theory]]. In this respect second-order logic with standard semantics differs from first-order logic; [[Willard Van Orman Quine|Quine]] pointed to the lack of a complete proof system as a reason for thinking of second-order logic as not ''logic'', properly speaking.{{sfn|Quine|1970|p=[https://books.google.com/books?id=S_NhnP0izA4C&pg=PA90 90–91]}}

As mentioned above, Henkin proved that the standard deductive system for first-order logic is sound, complete, and effective for second-order logic with [[Higher-order logic|Henkin semantics]], and the deductive system with comprehension and choice principles is sound, complete, and effective for Henkin semantics using only models that satisfy these principles.

The [[compactness theorem]] and the [[Löwenheim–Skolem theorem]] do not hold for full models of second-order logic. They do hold however for Henkin models.<ref>[[María Manzano|Manzano, M.]], ''Model Theory'', trans. Ruy J. G. B. de Queiroz ([[Oxford]]: [[Oxford University Press#Clarendon Press|Clarendon Press]], 1999), [https://books.google.com/books?id=Nc5uXx057JwC&pg=PR11 p. xi].</ref>