Definable real number: Difference between revisions

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[[File:Square root of 2 triangle.svg|thumb|200px|The [[square root of 2]] is equal to the length of the [[hypotenuse]] of a [[right triangle]] with legs of length 1 and is therefore a '''constructible number''']]

Informally, a '''definable real number''' is a [[real number]] that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a [[formal language]]. For example, the positive square root of 2, <math>\sqrt{2}</math>, can be defined as the unique positive solution to the equation <math>x^2 = 2</math>, and it can be constructed with a compass and straightedge.

Different choices of a formal language or its interpretation can give rise to different notions of definability. Specific varieties of definable numbers include the [[constructible number]]s of geometry, the [[algebraic numbers]], and the [[computable number]]s. Because formal languages can have only [[countably many]] formulas, every notion of definable numbers has at most countably many definable real numbers. However, by [[Cantor's diagonal argument]], there are uncountably many real numbers, so [[almost everywhere|almost every]] real number is undefinable.