Definable real number: Difference between revisions

Revision as of 07:34, 5 July 2021

Revision as of 21:17, 5 July 2021

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Georg Cantor in his 1874 paper "[[Georg Cantor's first set theory article|On a Property of the Collection of All Real Algebraic Numbers]]".

Non-algebraic numbers are called [[transcendental numbers]]. Specific examples of transcendental numbers include [[Pi|<math>\pi</math>]] and {{nowrap|[[Euler's number]] <math>e</math>.}}

== Computable real numbers ==

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A real number <math>a</math> is '''first-order definable in the language of set theory, without parameters''', if there is a formula <math>\varphi</math> in the language of [[set theory]], with one [[free variable]], such that <math>a</math> is the unique real number such that <math>\varphi(a)</math> holds.{{r|kunen}} This notion cannot be expressed as a formula in the language of set theory.

All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as [[Zero|0]], [[One|1]], [[Pi|&<math>\pi;]]</math>, [[E (mathematical constant)|''<math>e'']]</math>, et cetera, along with all algebraic numbers. Assuming that they form a set in the model, the real numbers definable in the language of set theory over a particular model of [[Zermelo–Fraenkel set theory|ZFC]] form a field.

Each set [[Model theory|model]] <math>M</math> of ZFC set theory that contains uncountably many real numbers must contain real numbers that are not definable within <math>M</math> (without parameters). This follows from the fact that there are only countably many formulas, and so only countably many elements of <math>M</math> can be definable over <math>M</math>. Thus, if <math>M</math> has uncountably many real numbers, one can prove from "outside" <math>M</math> that not every real number of <math>M</math> is definable over <math>M</math>.