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:

- A finite set of symbols (i.e. the alphabet), that can be used for constructing formulas (i.e. finite strings of symbols).
- 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. - A set of axioms or axiom schemata: each axiom must be a wff.
- 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.

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### Famous quotes containing the words system and/or formal:

“We find ourselves under the government of a *system* of political institutions, conducing more essentially to the ends of civil and religious liberty, than any of which the history of former times tells us.”

—Abraham Lincoln (1809–1865)

“That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through *formal* instruction.”

—Fred Rogers (20th century)