Applications to mathematical logic: formalized languages of logic and arithmetic

Abstract

In [Church 56, sect.07] mathematical logic or symbolic logic, or logistics, is defined as “the subject of formal logic when treated by the method of setting up a formalized language”. Among formalized languages there are purely logical (propositional and predicate) languages, languages of arithmetic and languages of set theory. Predicate languages — elementary (=first-order) as well as non-elementary (of higher orders) — are used for formal descriptions of properties of mathematical structures (first of all algebraic structures). We can use them for an axiomatic description of different classes of structures. Languages of arithmetic describe the set of natural numbers (which hardly can be described axiomatically and must anyway be regarded as existing before any axiomatic considerations). Languages of set theory have no clear semantics and are intended to formulate various axiomatic theories.