Theorem - Theorems in Logic

Theorems in Logic

The neutrality of this section is disputed. Relevant discussion may be found on the talk page. Please do not remove this message until the dispute is resolved.

Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language. The statements of the language are strings of symbols and may be broadly divided into nonsense and well-formed formulas. A set of deduction rules, also called transformation rules or rules of inference, must be provided. These deduction rules tell exactly when a formula can be derived from a set of premises. The set of well-formed formulas may be broadly divided into theorems and non-theorems. However, according to Hofstadter, a formal system will often simply define all of its well-formed formula as theorems.

Different sets of derivation rules give rise to different interpretations of what it means for an expression to be a theorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the most common examples use first-order logic. Other deductive systems describe term rewriting, such as the reduction rules for λ calculus.

The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas. The most famous result is Gödel's incompleteness theorem; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory.

Read more about this topic:  Theorem

Famous quotes containing the word logic:

    Histories make men wise; poets witty; the mathematics subtle; natural philosophy deep; moral grave; logic and rhetoric able to contend.
    Francis Bacon (1561–1626)