In logic and mathematics, together with the allied branches of computer science, information theory, and statistics, a formal system is an idealized and abstract language or formal grammar used for modeling purposes. Formalization is the act of creating a formal system, in an attempt to capture the essential features of a real-world or conceptual system in formal language. A model is a structure that can be used to provide an interpretation of the abstract or symbolic language that is embodied in the formal system.
Formal systems have the virtue that absolutely certain proofs—or absolutely certain knowledge—can be found within them, and nowhere else. If the formal system is itself consistent, then it is possible to prove, within the given system, that a given conclusion follows from given premises. This means that, within that system, if the given premises are true, then the conclusion cannot be false. Thus we have absolutely certain knowledge that this conclusion is true if the premises are true. This is, of course, a highly restricted notion of certain (absolute) knowledge, but it is the only place or domain in which genuine or unassailable or unquestionable absolute knowledge exists, despite all claims to the contrary.
In mathematics, formal proofs are the product of formal systems, consisting of axioms and rules of deduction. Theorems are then recognized as the possible 'last lines' of formal proofs. The point of view that this picture encompasses in mathematics has been called formalism. That term has been used both approvingly and pejoratively. On the other hand, David Hilbert founded metamathematics as a discipline designed for discussing formal systems; it is not assumed that the metalanguage in which proofs are studied is itself less informal than the usual habits of mathematicians suggest. To contrast with the metalanguage, or language in which the formal system is itself stated, the language described by a formal grammar is often called an object language (i.e., the object of discussion—this distinction may have been introduced by Carnap).
It has become common to speak of a formalism, more-or-less synonymously with a formal system within standard mathematics invented for a particular purpose. This may not be much more than a notation, such as Dirac's bra-ket notation.
Mathematical formal systems consist of the following:
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