Axiomatic system

The English used in this article or section may not be easy for everybody to understand. You can help Wikipedia by reading Wikipedia:How to write Simple English pages, then simplifying the article. (September 2024)

An axiomatic system in mathematics and logic is a set of axioms or primitive notions from which theorems are logically derived. A system is usually part of a formal theory, which is a collection of sentences closed under logical implication.

Properties

• Consistency: An axiomatic system is consistent if it does not contain any contradictions. Inconsistent systems allow for any statement to be proven (principle of explosion).
• Independence: An axiom is independent if it cannot be proven using the other axioms of the system. A system is independent if all its axioms are independent.
• Completeness: A system is complete if every statement can either be proven true or false using the axioms.

Models

A model provides interpretations of the undefined terms in an axiomatic system and proves the system's consistency. Models can be concrete (with real-world objects) or abstract (based on other axiomatic systems).

Relative consistency

Relative consistency refers to the ability to define the undefined terms of one system within another, such that the axioms of the first system become theorems of the second.

Related pages

• Mathematical logic

References

1. ↑ Weisstein, Eric W. "Complete Axiomatic Theory". mathworld.wolfram.com. Retrieved 2024-09-19.
2. ↑ Weisstein, Eric W. "Zermelo-Fraenkel Axioms". mathworld.wolfram.com. Retrieved 2024-09-19.
3. ↑ "Archived copy" (PDF). Archived from the original (PDF) on 2023-06-10. Retrieved 2024-09-19.{{cite web}}: CS1 maint: archived copy as title (link)