metamathematics the study and establishment, by restricted (and, in particular, finitary) means, of the consistency or reliability of the various systems of classical mathematics. The term was apparently introduced, with pejorative overtones relating it to ‘metaphysics’, in the 1870s in connection with the discussion of non-Euclidean geometries. It was introduced in the sense given here, shorn of negative connotations, by Hilbert (see his ‘Neubegründung der Mathematik. Erste Mitteilung,’ 1922), who also referred to it as Beweistheorie or proof theory. A few years later (specifically, in the 1930 papers ‘Über einige fundamentale Begriffe der Metamathematik’ and ‘Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I’) Tarski fitted it with a somewhat broader, less restricted sense: broader in that the scope of its concerns was increased to include not only questions of consistency, but also a host of other questions (e.g. questions of independence, completeness and axiomatizability) pertaining to what Tarski referred to as the ‘methodology of the deductive sciences’ (which was his synonym for ‘metamathematics’); less restricted in that the standards of proof were relaxed so as to permit other than finitary – indeed, other than constructive – means.
On this broader conception of Tarski’s, formalized deductive disciplines form the field of research of metamathematics roughly in the same sense in which spatial entities form the field of research in geometry or animals that of zoology. Disciplines, he said, are to be regarded as sets of sentences to be investigated from the point of view of their consistency, axiomatizability (of various types), completeness, and categoricity or degree of categoricity, etc. Eventually (see the 1935 and 1936 papers ‘Grundzüge des Systemenkalkül, Erster Teil’ and ‘Grundzüge der Systemenkalkül, Zweiter Teil’) Tarski went on to include all manner of semantical questions among the concerns of metamathematics, thus diverging rather sharply from Hilbert’s original syntactical focus. Today, the terms ‘metatheory’ and ‘metalogic’ are used to signify that broad set of interests, embracing both syntactical and semantical studies of formal languages and systems, which Tarski came to include under the general heading of metamathematics. Those having to do specifically with semantics belong to that more specialized branch of modern logic known as model theory, while those dealing with purely syntactical questions belong to what has come to be known as proof theory (where this latter is now, however, permitted to employ other than finitary methods in the proofs of its theorems). See also CATEGORICITY , COMPLETENESS, CONSISTENCY , MODEL THEORY , PROOF THE – OR. M.D.