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{{Short description|Real number uniquely specified by description}}
[[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''']]
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[[File:Algebraicszoom.png|thumb|Algebraic numbers on the [[complex plane]] colored by degree (red=1, green=2, blue=3, yellow=4)]]
A real number <math>r</math> is called a real [[algebraic number]] if there is a [[polynomial]] <math>p(x)</math>, with only integer coefficients, so that <math>r</math> is a root of <math>p</math>, that is, <math>p(r)=0</math>.
Each real algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial <math>q(x)</math> has 5 real roots, the third one can be defined as the unique <math>r</math> such that <math>q(r)=0</math> and such that there are two distinct numbers less than <math>r</math> at which <math>q</math> is zero.
All rational numbers are
The real algebraic numbers form a [[field extension|subfield]] of the real numbers. This means that 0 and 1 are algebraic numbers and, moreover, if <math>a</math> and <math>b</math> are algebraic numbers, then so are <math>a+b</math>, <math>a-b</math>, <math>ab</math> and, if <math>b</math> is nonzero, <math>a/b</math>.
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Every computable real number is arithmetical, and the arithmetical numbers form a subfield of the reals, as do the analytical numbers. Every arithmetical number is analytical, but not every analytical number is arithmetical. Because there are only countably many analytical numbers, most real numbers are not analytical, and thus also not arithmetical.
Every computable number is arithmetical, but not every arithmetical number is computable. For example, the limit of a [[Specker sequence]] is an arithmetical number that is not computable.
The definitions of arithmetical and analytical reals can be stratified into the [[arithmetical hierarchy]] and [[analytical hierarchy]]. In general, a real is computable if and only if its Dedekind cut is at level <math>\Delta^0_1</math> of the arithmetical hierarchy, one of the lowest levels. Similarly, the reals with arithmetical Dedekind cuts form the lowest level of the analytical hierarchy.
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* [[Berry's paradox]]
* [[Constructible universe]]
* ''[[Entscheidungsproblem]]''
* [[Ordinal definable set]]
* [[Richard's paradox]]
* [[Tarski's undefinability theorem]]
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<ref name=kunen>{{Citation | last1=Kunen | first1=Kenneth | author1-link=Kenneth Kunen | year=1980 | title=[[Set Theory: An Introduction to Independence Proofs]] | publisher=North-Holland | ___location=Amsterdam | isbn=978-0-444-85401-8 | page=153}}</ref>
<ref name=tsirelson>{{Citation | last1=Tsirelson | first1=Boris | author1-link=Boris Tsirelson | year=2020 | title=Can each number be specified by a finite text? | periodical=WikiJournal of Science | volume=3 | issue=1 | page=8 | doi=10.15347/WJS/2020.008 | doi-access=free | arxiv=1909.11149 | s2cid=202749952 }}</ref>
<ref name=turing>{{Citation | last1=Turing | first1=A. M. | author1-link=Alan Turing | year=1937 | title=On Computable Numbers, with an Application to the Entscheidungsproblem | journal=[[Proceedings of the London Mathematical Society]] | series=2 | volume=42 | issue=1 | pages=230–65 | doi=10.1112/plms/s2-42.1.230 | s2cid=73712 | url=http://www.abelard.org/turpap2/tp2-ie.asp }}</ref>
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