# $\mathbb{R}$ as set of Dedekind cuts on $\mathbb{Q}$

-- "let $A,B \in \mathbb{Q}$, $A < B$ if $A\leq B$ and $A \neq B$

-- "let $\preceq$ be a ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall b \in B(a \preceq b \to a \in B )$"

the following definitions are correct:

-- "let $\leq$ be a ordering of $\mathbb{Q}$, and $B \subsetneqq \mathbb{Q}$, $B$ is Dedekind cut on $\mathbb{Q}$ if $B$ initial segment of $A$ under $\leq$ and $\forall C \in B, \exists D \in B (C < D)$"

-- "$\mathbb{R}:=\{B \subsetneqq \mathbb{Q}|B\text{ is Dedekind cuts on } \mathbb{Q}\}$"

??

I want to know if the definitions of Dedekind cut and $\mathbb{R}$ are written correctly.. – mle Aug 14 '13 at 10:29
Yep, these are correct. Think of Dedekind cuts on $\Bbb Q$ as sets of the form $S_{\alpha } = \{x \in \Bbb Q: x \lt \alpha \}$, where $\alpha \in \Bbb R$. The above definition is the identification $S_{\alpha } = \alpha$. – walcher Aug 14 '13 at 10:34