A set of rules for manipulating strings of symbols. Almost all notation in maths forms a formal system, they are the basic tools for doing maths.

a hierarchical grouping of sets of symbols or complete formulae, out of which other formulae can be generated, which can be accepted as valid

a rigidly defined kind of game that specifies rules for manipulating tokens

a system in which one manipulates generic symbols through set rules to generate new combinations of symbols

A mathematical formalism in which statements can be constructed and manipulated with logical rules.

In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. A formal system may be formulated purely abstractly, for its own sake, or it may be intended to serve as a description of some domain of real phenomena or some aspect of objective reality.