consistency
Quick Summary
Metalogic, a branch of formal logic, explores the properties of logical systems. The first-order predicate calculus is a key area within metalogic. A fundamental concept is consistency, which means a logical system does not contain contradictions.
For the predicate calculus, consistency can be demonstrated by assuming a single object and showing that all theorems reduce to tautologies, meaning they are always true. This confirms that no contradiction can be a theorem within the system. Discoveries in the 1930s, notably by Gödel, Church, and Turing, established the completeness and undecidability of the predicate calculus. Completeness means all valid sentences are theorems, and consistency implies satisfiability, linking syntactic and semantic concepts in logic.1
Ask Britannica AI about this topic
Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.
metalogicThe axiomatic method…that non-Euclidean geometries must be self-consistent systems because they have models (or interpretations) in Euclidean geometry, which in turn has a model in the theory of real numbers. It may then be asked, however, how it is known that the theory of real numbers is consistent in the sense that…
Read More…that non-Euclidean geometries must be self-consistent systems because they have models (or interpretations) in Euclidean geometry, which in turn has a model in the theory of real numbers. It may then be asked, however, how it is known that the theory of real numbers is consistent in the sense that…
Read More- metalogicDiscoveries about formal mathematical systems
…those of the completeness and consistency of a formal system based on axioms. In 1931 Gödel made fundamental discoveries in these areas for the most interesting formal systems. In particular, he discovered that, if such a system is ω-consistent—i.e., devoid of contradiction in a sense to be explained below—then it…
Read More…those of the completeness and consistency of a formal system based on axioms. In 1931 Gödel made fundamental discoveries in these areas for the most interesting formal systems. In particular, he discovered that, if such a system is ω-consistent—i.e., devoid of contradiction in a sense to be explained below—then it…
Read More - metalogicThe first-order predicate calculus
…syntactic concepts of derivability and consistency.
Read More…syntactic concepts of derivability and consistency.
Read More 
formal logicAxiomatization of PCAn axiomatic system is consistent if, whenever a wff α is a theorem, ∼α is not a theorem. (In terms of the standard interpretation, this means that no pair of theorems can ever be derived one of which is the negation of the other.) It is strongly complete if…
Read MoreAn axiomatic system is consistent if, whenever a wff α is a theorem, ∼α is not a theorem. (In terms of the standard interpretation, this means that no pair of theorems can ever be derived one of which is the negation of the other.) It is strongly complete if…
Read More
mathematicsCantor…would be one that was consistent, complete, and decidable. By “consistent” Hilbert meant that it should be impossible to derive both a statement and its negation; by “complete,” that every properly written statement should be such that either it or its negation was derivable from the axioms; by “decidable,” that…
Read More…would be one that was consistent, complete, and decidable. By “consistent” Hilbert meant that it should be impossible to derive both a statement and its negation; by “complete,” that every properly written statement should be such that either it or its negation was derivable from the axioms; by “decidable,” that…
Read More
history of logicZermelo-Fraenkel set theory (ZF)…Is ZF consistent? Can its consistency be proved? Are the axioms independent of each other? What other axioms should be added? Other logicians later asked questions about the intended models of axiomatic set theory—i.e., about what object-domains and rules of symbol interpretation would render the theorems of set theory true.…
Read More…Is ZF consistent? Can its consistency be proved? Are the axioms independent of each other? What other axioms should be added? Other logicians later asked questions about the intended models of axiomatic set theory—i.e., about what object-domains and rules of symbol interpretation would render the theorems of set theory true.…
Read More- set theoryLimitations of axiomatic set theory
…settle the question of the consistency of either theory. One method for establishing the consistency of an axiomatic theory is to give a model—i.e., an interpretation of the undefined terms in another theory such that the axioms become theorems of the other theory. If this other theory is consistent, then…
Read More…settle the question of the consistency of either theory. One method for establishing the consistency of an axiomatic theory is to give a model—i.e., an interpretation of the undefined terms in another theory such that the axioms become theorems of the other theory. If this other theory is consistent, then…
Read More