Axiomatic system
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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.[1]
Properties
[change | change source]- 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.[2]
Models
[change | change source]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
[change | change source]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.[3]
Related pages
[change | change source]References
[change | change source]- ↑ Weisstein, Eric W. "Complete Axiomatic Theory". mathworld.wolfram.com. Retrieved 2024-09-19.
- ↑ Weisstein, Eric W. "Zermelo-Fraenkel Axioms". mathworld.wolfram.com. Retrieved 2024-09-19.
- ↑ "Archived copy" (PDF). Archived from the original (PDF) on 2023-06-10. Retrieved 2024-09-19.
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