[Math Lair] Mathematical Systems

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A mathematical system (the terms formal system or axiom system are also used) is a structure that consists of:

  1. A list of undefined terms or symbols
  2. A set of rules or concepts for determining whether something is a statement of the system
  3. A list of postulates.

Any statement formed using the rules or concepts (#2) is called a statement of the system, whether it can be proven, disproven, or neither. Any statement that can be proven is called a theorem.

The axioms or postulates composing the system have to be consistent; a "system" with contradictory axioms would be of no value. It is possible that, even if the postulates do not appear to be contradictory, they can be used to deduce two contradictory statements. Such a system, which would be of no value either, is called inconsistent. On the other hand, a consistent system is one that is free of contradictions.

An abstract system can be subject to many interpretations. Such an interpretation is called a model. A model is formed by assigning meanings to the undefined terms and verifying the truth of the postulates with respect to those meanings. A system that admits of a model is called satisfiable, and must also be consistent.

Sources used (see bibliography page for titles corresponding to numbers): 51.