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real_number [2014/03/21 11:11] 127.0.0.1 external edit |
real_number [2014/04/04 15:58] 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 ==== | ||
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]]. | ||