The Real Numbers

In the setting of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals, let $C$, $\le_{C}$, $u^{*}$ for $u\in\mathbb{Q}$, $\oplus$, $\ominus$ and $\odot$ be as in Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts.

¶ The set of real numbers is $\mathbb{R}=C$, the set of Dedekind cuts (Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §set), and its order $\le$ is $\le_{C}$, a total order on $\mathbb{R}$ by Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §order.

¶ For $x,y\in\mathbb{R}$, the sum is $x+y=x\oplus y$. Let $|x|_{C}=x$ if $0^{*}\subseteq x$, and $|x|_{C}=\ominus x$ otherwise. The product is $x\cdot y=|x|_{C}\odot|y|_{C}$ if both or neither of $0^{*}\subseteq x$ and $0^{*}\subseteq y$ hold, and $x\cdot y=\ominus(|x|_{C}\odot|y|_{C})$ otherwise. All of these lie in $\mathbb{R}$, and $0^{*}\subseteq|x|_{C}$, by Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §sum, Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §negative, Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §absolute and Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §product; so $+$ and $\cdot$ are binary operations on $\mathbb{R}$ by Maps and Relations Given by Formulas §binary.

¶ $0_{\mathbb{R}}=0^{*}$ and $1_{\mathbb{R}}=1^{*}$, which lie in $\mathbb{R}$ by Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §rational.