Predicate calculus in the context of "Finitary relation"

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory ${\displaystyle T}$ is consistent if there is no formula ${\displaystyle \varphi }$ such that both ${\displaystyle \varphi }$ and its negation ${\displaystyle \lnot \varphi }$ are elements of the set of consequences of ${\displaystyle T}$. Let ${\displaystyle A}$ be a set of closed sentences (informally "axioms") and ${\displaystyle \langle A\rangle }$ the set of closed sentences provable from ${\displaystyle A}$ under some (specified, possibly implicitly) formal deductive system. The set of axioms ${\displaystyle A}$ is consistent when there is no formula ${\displaystyle \varphi }$ such that ${\displaystyle \varphi \in \langle A\rangle }$ and ${\displaystyle \lnot \varphi \in \langle A\rangle }$. A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent meant in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

In a sound formal system, every satisfiable theory is consistent, but the converse does not hold. If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete. The completeness of the propositional calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of (first order) predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.

In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science. It consists of a set of formal statements known as axioms that are used for the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical theory is an expression used to refer to an axiomatic system and all its derived theorems.

A proof within an axiomatic system is a sequence of deductive steps that establishes a new statement as a consequence of the axioms. By itself, the system of axioms is, intentionally, a syntactic construct: when axioms are expressed in natural language, which is normal in books and technical papers, the nouns are intended as placeholder words. The use of an axiomatic approach is a move away from informal reasoning, in which nouns may carry real-world semantic values, and towards formal proof. In a fully formal setting, a logical system such as predicate calculus must be used in the proofs. The contemporary application of formal axiomatic reasoning differs from traditional methods both in the exclusion of semantic considerations, and in the specification of the system of logic in use.