Object Theory - A Formal System

A Formal System

In general, in mathematics a formal system or "formal theory" consists of "objects" in a structure:

• The symbols to be concatenated (adjoined),
• The formation-rules (completely specified, i.e. formal rules of syntax) that dictate how the symbols and the assemblies of symbols are to be formed into assemblies (e.g. sequences) of symbols (called terms, formulas, sentences, propositions, theorems, etc.) so that they are in "well-formed" patterns (e.g. can a symbol be concatenated at its left end only, at its right end only, or both ends simultaneously? Can a collection of symbols be substituted for (put in place of) one or more symbols that may appear anywhere in the target symbol-string?),
• Well-formed "propositions" (called "theorems" or assertions or sentences) assembled per the formation rules,
• A few axioms that are stated up front and may include "undefinable notions" (examples: "set", "element", "belonging" in set theory; "0" and " ' " (successor) in number theory),
• At least one rule of deductive inference (e.g. modus ponens) that allow one to pass from one or more of the axioms and/or propositions to another proposition.