Definitions for "Formal system"
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
Keywords:  grammar
a formal grammar
Keywords:  prove, things, allows, system
a system that allows us to prove things