Formal proof

Overview

**formal proof**or

**derivation**is a finite sequence of sentences (called well-formed formula

Well-formed formula

In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...

s in the case of a formal language

Formal language

A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...

) each of which is an axiom

Axiom

In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

or follows from the preceding sentences in the sequence by a rule of inference

Rule of inference

In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion...

. The last sentence in the sequence is a theorem

Theorem

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

of a formal system

Formal system

In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...

. The notion of theorem is not in general effective

Effective method

In computability theory, an effective method is a procedure that reduces the solution of some class of problems to a series of rote steps which, if followed to the letter, and as far as may be necessary, is bound to:...

, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists.