Axiomatic System

An Axiomatic System is a formal system composed of an axiom set that can be used in conjunction to logically derive theorems.

AKA: Axiom System, Axiomatic Mathematical Framework.
Context:
- It can range from being a Consistent Axiomatic System to being an Inconsistent Axiomatic System.
- …
Example(s):
- Peano's Axioms.
See: Axiom, Formal Language, Mathematical System, Likelihood Principle, Deductive System.

References

(Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/axiomatic_system Retrieved:2017-1-19.
- In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical proof within a formal system.

(Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/axiomatic_system#Properties Retrieved:2017-1-19.
- An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its denial from the system's axioms.
  In an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other axioms in the system. A system will be called independent if each of its underlying axioms is independent. Although independence is not a necessary requirement for a system, consistency is.
  An axiomatic system will be called complete if for every statement, either itself or its negation is derivable.

(Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/axiomatic_system#Axiomatization Retrieved:2017-1-19.
- In mathematics, axiomatization is the formulation of a system of statements (i.e. axioms) that relate a number of primitive terms in order that a consistent body of propositions may be derived deductively from these statements. Thereafter, the proof of any proposition should be, in principle, traceable back to these axioms.

(Weber, 2002) ⇒ Keith Weber. (2002). “Beyond proving and explaining: Proofs that justify the use of definitions and axiomatic structures and proofs that illustrate technique." For the learning of mathematics.
- QUOTE: … The proof that 1 + 1 = 2 is not a proof that provides knowledge about this mathematical truth; this proof provides information about the axiomatic system in which one is working and about how one can generate proofs within that system. …