Editing Metalogic (section)

== Results ==
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Results in metalogic consist of such things as [[formal proof]]s demonstrating the [[consistency]], [[completeness (logic)|completeness]], and [[Decidability (logic)|decidability]] of particular [[formal system]]s.

Major results in metalogic include:

* Proof of the [[uncountability]] of the [[power set]] of the [[natural number]]s ([[Cantor's theorem]] 1891)
* [[Löwenheim–Skolem theorem]] ([[Leopold Löwenheim]] 1915 and [[Thoralf Skolem]] 1919)
* Proof of the consistency of truth-functional [[Propositional calculus|propositional logic]] ([[Emil Leon Post|Emil Post]] 1920)
* Proof of the semantic completeness of truth-functional propositional logic ([[Paul Bernays]] 1918),<ref name="reflections">Hao Wang, [https://books.google.com/books?id=wLLePwhDOMYC Reflections on Kurt Gödel]</ref> (Emil Post 1920)<ref name="metalogic"/>
* Proof of the syntactic completeness of truth-functional propositional logic (Emil Post 1920)<ref name="metalogic"/>
* Proof of the decidability of truth-functional propositional logic (Emil Post 1920)<ref name="metalogic"/>
* Proof of the consistency of first-order [[Monadic predicate calculus|monadic predicate logic]] ([[Leopold Löwenheim]] 1915)
* Proof of the semantic completeness of first-order monadic predicate logic (Leopold Löwenheim 1915)
* Proof of the decidability of first-order monadic predicate logic (Leopold Löwenheim 1915)
* Proof of the consistency of first-order predicate logic ([[David Hilbert]] and [[Wilhelm Ackermann]] 1928)
* Proof of the semantic completeness of first-order [[predicate logic]] ([[Gödel's completeness theorem]] 1930)
* Proof of the [[cut-elimination theorem]] for the [[sequent calculus]] ([[Gerhard Gentzen|Gentzen]]'s ''Hauptsatz'' 1934)
* Proof of the undecidability of first-order predicate logic ([[Entscheidungsproblem|Church's theorem]] 1936)
* [[Gödel's incompleteness theorems#First incompleteness theorem|Gödel's first incompleteness theorem]] 1931
* [[Gödel's incompleteness theorems#Second incompleteness theorem|Gödel's second incompleteness theorem]] 1931
* [[Tarski's undefinability theorem]] (Gödel and Tarski in the 1930s)