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real_number [2014/03/21 11:11] 127.0.0.1 external edit |
real_number [2015/09/30 22:43] nikolaj |
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| ==== Set ==== | ==== Set ==== | ||
| | @#FFBB00: definiendum | @#FFBB00: $ r \in \mathbb R $ | | | @#FFBB00: definiendum | @#FFBB00: $ r \in \mathbb R $ | | ||
| - | + | | @#FFFDDD: for all | @#FFFDDD: $x,y\in \mathbb Q$ | | |
| - | | $x,y\in \mathbb Q$ | | + | | @#FFFDDD: for all | @#FFFDDD: $ r\subset \mathbb Q, r\neq \emptyset $ | |
| - | + | ||
| - | | @#55EE55: postulate | @#55EE55: $ r\subset \mathbb Q $ | | + | |
| - | | @#55EE55: postulate | @#55EE55: $ r\neq \emptyset $ | | + | |
| | @#55EE55: postulate | @#55EE55: $ y\in r\implies x\in r $ | | | @#55EE55: postulate | @#55EE55: $ y\in r\implies x\in r $ | | ||
| | @#55EE55: postulate | @#55EE55: $ \neg\ \exists (b\in r).\ \forall (a\in r).\ a<_{\mathbb Q}b $ | | | @#55EE55: postulate | @#55EE55: $ \neg\ \exists (b\in r).\ \forall (a\in r).\ a<_{\mathbb Q}b $ | | ||
| + | ----- | ||
| + | === Discussion === | ||
| + | {{ are you for reals.jpg?X200}} | ||
| Remark: We distinguish between "$\subset$" and "$\subseteq$", i.e. the above definition implies $ r\neq \mathbb Q $. | Remark: We distinguish between "$\subset$" and "$\subseteq$", i.e. the above definition implies $ r\neq \mathbb Q $. | ||
| - | ==== Discussion ==== | ||
| Each real number is modeled as a Dedekind cut of $ \mathbb Q $ into two pieces. The axioms above say that $r$ is a $<_{\mathbb Q}$-complete set with no upper bound. We define the total order of the reals via $s<r\equiv s\subset r$, see [[Order structure of real numbers]]. | Each real number is modeled as a Dedekind cut of $ \mathbb Q $ into two pieces. The axioms above say that $r$ is a $<_{\mathbb Q}$-complete set with no upper bound. We define the total order of the reals via $s<r\equiv s\subset r$, see [[Order structure of real numbers]]. | ||
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| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Real_number|Real number]], [[http://en.wikipedia.org/wiki/Construction_of_the_real_numbers|Construction of the real numbers]], [[http://en.wikipedia.org/wiki/Dedekind_cut|Dedekind cut]] | Wikipedia: [[http://en.wikipedia.org/wiki/Real_number|Real number]], [[http://en.wikipedia.org/wiki/Construction_of_the_real_numbers|Construction of the real numbers]], [[http://en.wikipedia.org/wiki/Dedekind_cut|Dedekind cut]] | ||
| - | ==== Parents ==== | + | |
| + | ----- | ||
| === Subset of === | === Subset of === | ||
| [[Complex number]], [[Extended real number line]], [[Real coordinate space]] | [[Complex number]], [[Extended real number line]], [[Real coordinate space]] | ||
| === Context === | === Context === | ||
| [[Rational number]] | [[Rational number]] | ||
