Definable real number: Difference between revisions

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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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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 algebraicconstructible, and all constructible numbers are algebraic. There are numbers such as the cube root of 2 which are algebraic but not constructible.

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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* ''[[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>