Formalized theory

In classical mathematics, the formalized theory describes a procedure whereby the propositions of a theory are derived from the axioms by logical conclusions .

Colloquial language is used to formalize the axioms and the proofs. But you can also do without colloquial language and express the axioms (using predicate logic and well-defined terms) in a formula language. Only certain formal inference rules are then allowed for proof.

This creates a formalized theory that can be the subject of metamathematic investigations. The formalized theory is then a theory whose means of expression, especially meaningful statements and sentences, are exactly delimited by a formalized language or formula language constituted for this purpose ( calculus ). In most cases, the means of expression in a formalized theory are characterized as a series of symbols made up of certain basic symbols, based on special structural rules.

One speaks of a formalized theory with a semantically defined set of sentences if its sentences ( theorems ) are understood as the statements that are true for a certain interpretation. One speaks of a formalized theory with a syntactically defined set of sentences if its sentences (theorems) are understood as statements that can be proven from a certain axiom system according to precisely defined inference rules.

A theory that is formalized within the framework of the first-level predicate calculus is called a formalized first-level theory or elementary theory. A theory that is formalized in a higher-level predicate calculus is called a formalized theory of the corresponding level.

The formalization of a theory is an important tool in basic mathematical research. It is only through them that general epistemological questions such as

• Consistency ,
• Completeness ,
• Axiomatizability ,
• Independence,
• Decidability u. a.

accessible to an exact mathematical treatment. The treatment of such problems for a particular formalized theory forms the subject of the metatheory of that theory.

literature

• Hans Hermes : An axiomatization of general mechanics . Research on logic and the foundations of the exact sciences, Volume 3, Leipzig 1938.