Definable real number: Difference between revisions

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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.

All choices<ref>This is a consequence of a formal language being a set of finite sequences built from a finite (or countably infinite) alphabet.</ref> have in common that they lead to at most [[countably many]] definable real numbers, that is, that [[almost everywhere|almost every]] real number is undefinable.

== Constructible numbers ==

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==References==

{{Reflist}}

* {{Citation|last=Hamkins|first= Joel David|url=https://mathoverflow.net/q/44129 |title=Is the analysis as taught in universities in fact the analysis of definable numbers?|journal=MathOverflow|date=October 2010|accessdate=2016-03-05|authorlink=Joel David Hamkins}}.

* {{Citation|last=Hamkins|first=Joel David|last2=Linetsky|first2=David|last3=Reitz|first3=Jonas|title=Pointwise Definable Models of Set Theory|arxiv=1105.4597|journal=Journal of Symbolic Logic|volume=78|issue=1|pages=139–156|year=2013|doi=10.2178/jsl.7801090}}.

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* [[v:WikiJournal_Preprints/Can each number be specified by a finite text?|Can each number be specified by a finite text?]]

{{Number systems}}

[[Category:Set theory]]