Foundations of Mathematics
How are some parts of an axiomatic system further defined?
There are several terms that further define an axiomatic system. All of them are slightly intertwined, depending on the system.
The absence of contradiction—or the ability to prove a proposition (statement) and its negative are both true—is known as consistency. Independence is not necessary to an axiomatic system, but consistency is definitely necessary. The opposite of consistency in an axiomatic system is inconsistency.
An axiomatic system is called independent if no other axioms can be derived (or proved) from other axioms in the system; in other words, the entire axiomatic system will be termed independent if all of its underlying axioms are independent. The independence of a system is usually determined after the consistency. An axiomatic system that is dependent has some axioms that are redundant; this is also called redundancy.
An axiomatic system is complete if no additional axiom can be added to the system without making the new system dependent or inconsistent. In other words, the aim is to prove or disprove any statement about the objects in the system from the axioms alone. In complete systems, every true proposition about the defined and undefined terms can be proved from the axioms. Systems with the logic based on true or false propositions connected by “and,” “or,” and “not” are complete, as are those that include quantifiers. More complex systems, such as set theory, are not considered complete.