# Formal System

A formal system is, broadly defined as any well-defined system of abstract thought based on the model of mathematics. Euclid's Elements is often held to be the first formal system and displays the characteristic of a formal system. The entailment of the system by its logical foundation is what distinguishes a formal system from others which may have some basis in an abstract model. Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory. A formal system need not be mathematical as such, Spinoza's Ethics for example imitates the form of Euclid's Elements.

Each formal system has a formal language, which is composed by primitive symbols. These symbols act on certain rules of formation and are developed by inference from a set of axioms. The system thus consists of any number of formulas built up through finite combinations of the primitive symbols—combinations that are formed from the axioms in accordance with the stated rules.

Formal systems in mathematics consist of the following elements:

1. A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite strings of symbols).
2. A grammar, which tells how well-formed formulas (abbreviated wff) are constructed out of the symbols in the alphabet. It is usually required that there be a decision procedure for deciding whether a formula is well formed or not.
3. A set of axioms or axiom schemata: each axiom must be a wff.
4. A set of inference rules.

A formal system is said to be recursive (i.e. effective) if the set of axioms and the set of inference rules are decidable sets or semidecidable sets, according to context.

Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation, for example, Paul Dirac's bra-ket notation.

### Other articles related to "formal system, formal, system, formal systems":

Formal System - Related Subjects - Formal Grammar
... In computer science and linguistics a formal grammar is a precise description of a formal language a set of strings ... The two main categories of formal grammar are that of generative grammars, which are sets of rules for how strings in a language can be generated, and that of analytic grammars, which are sets of rules for ...
Bew - Proof Sketch For The First Theorem - Proof of Independence
... Now assume that the formal system is ω-consistent ... Thus the system would be inconsistent, proving both a statement and its negation ... Thus on one hand the system supports construction of a number with a certain property (that it is the Gödel number of the proof of p), but on the other hand, for every specific number x ...
Algorithm Characterizations - 1936, 1963, 1964 Gödel's Characterization
... provable, definable, etc.) depend quite essentially on the system with respect to which they are defined." (p ... states (in a footnote) his belief that "formal systems" have "the characteristic property that reasoning in them, in principle, can be completely replaced by mechanical devices" (p ... a precise and unquestionaly adequate definition of the general notion of formal system can now be given a completely general version of Theorems VI and XI is now possible." (p ...
Metatheorem
... In logic, a metatheorem is a statement about a formal system proven in a metalanguage ... Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the ... A formal system is determined by a formal language and a deductive system (axioms and rules of inference) ...
Completed - Logical Completeness
... In logic, semantic completeness is the converse of soundness for formal systems ... A formal system is "semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas formulas that are ... See History of the Church–Turing thesis.) A formal system is consistent if for all formulas φ of the system, the formulas φ and ¬φ (the negation of φ) are not both ...

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