Topic summary
Axiom system
Extracted from the Wikipedia article Axiomatic system.
Properties
Four important properties of an axiom system are consistency, relative consistency, completeness and independence. An axiomatic system is said to be consistent if it lacks contradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion). Relative consistency comes into play when we can not prove the consistency of an axiom system. However, in some cases we can show that an axiom system A is consistent if another axiom set B is consistent.