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; for example, Spinoza's Ethics 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 w.f.f.
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.
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