The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness

¶ This setting extends The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals: its conventions and notation are in force, as extended by the clause identification below.

¶ $\mathbb{R}$, its order $\le$, its operations $+$ and $\cdot$, and its elements $0$ and $1$ are as in The Real Numbers §reals, The Real Numbers §operations and The Real Numbers §constants. $\mathbb{R}$ is an ordered field by The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum §ordered-field, so the notation of Commutative Rings, Fields and Ordered Fields: Standard Notation applies to it.

¶ As for $\mathbb{N}_{0}$ in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion §numbers-only, a real number is used as a number only, never as a set.

¶ Let $\kappa:\mathbb{Q}\to\mathbb{R}$ be the canonical embedding. It is injective and preserves $0$, $1$, sums, products, the order in both directions, negatives, reciprocals and absolute values, by The Rational Numbers Embed in Exactly One Way into Every Ordered Field §injective, The Rational Numbers Embed in Exactly One Way into Every Ordered Field §zero, The Rational Numbers Embed in Exactly One Way into Every Ordered Field §homomorphism, The Rational Numbers Embed in Exactly One Way into Every Ordered Field §order, The Rational Numbers Embed in Exactly One Way into Every Ordered Field §negative, The Rational Numbers Embed in Exactly One Way into Every Ordered Field §reciprocal and The Rational Numbers Embed in Exactly One Way into Every Ordered Field §absolute; hence it preserves finite sums and products and powers, by Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §homomorphism and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals §reindexing. The element of $\mathbb{R}$ that an $n\in\mathbb{N}_{0}$ denotes by Commutative Rings, Fields and Ordered Fields: Standard Notation §numerals is $\kappa(n)$, by Natural Numbers Read in an Ordered Field Agree with the Canonical Embedding of the Rationals §numerals.

¶ A rational number, and a set introduced as a subset of $\mathbb{Q}$, is identified with its image under $\kappa$, and so, through The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals §identification, is an element of $\mathbb{N}_{0}$ or of $\mathbb{Z}$ and a set introduced as a subset of either, with its image under $\kappa\circ j\circ\iota_{0}$ or $\kappa\circ j$. This is done wherever a real number or a subset of $\mathbb{R}$ is required, in the senses listed in The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals §identification with $\mathbb{R}$ as the larger set; conversely, where an element or a subset of one of the smaller sets is required, a real number or a subset of $\mathbb{R}$ of this form stands for the element or subset of which it is the image. In this sense $\mathbb{N}\subseteq\mathbb{N}_{0}\subseteq\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$.

¶ By the clause embedding, The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals §agreement, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §union, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §composition, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §membership, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §intersection, Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §difference and Images of Unions, Intersections, Differences and Subclasses under a Function, and under an Injective Function §inclusion, this is well defined, and equality, $0$, $1$, sums, products, finite sums and products, powers, negatives, differences, reciprocals, quotients, absolute values, the order, membership in subsets, unions, intersections, set differences and inclusions agree whether they are formed in the smaller set or in $\mathbb{R}$. Bounds, suprema and infima of a subset read in $\mathbb{R}$ are formed in $\mathbb{R}$, as in Sets and Maps: Ordinary Notation §orders.

¶ Every nonempty subset of $\mathbb{R}$ that is bounded above has a supremum, by The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum §supremum. Hence Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema applies to $\mathbb{R}$, whose canonical embedding is the identification above, and gives the clauses infimum, archimedean, density and rational-supremum below.

¶ Every nonempty subset of $\mathbb{R}$ that is bounded below has an infimum, by Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §infimum.

¶ For every $x\in\mathbb{R}$ there is $n\in\mathbb{N}$ with $x<n$, the two readings of $n$ in $\mathbb{R}$ agreeing by the clause embedding, by Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §archimedean.

¶ For $x,y\in\mathbb{R}$ with $x<y$ there is $u\in\mathbb{Q}$ with $x<u<y$, by Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §density.

¶ $x=\sup\{v\in\mathbb{Q}:v<x\}$ for every $x\in\mathbb{R}$, by Infima, the Archimedean Property, Density of the Rationals and Rational Approximation from Below in an Ordered Field Whose Nonempty Sets Bounded Above Have Suprema §rational-supremum.

¶ Every ordered field in which every nonempty subset that is bounded above has a supremum is isomorphic to $\mathbb{R}$ by exactly one isomorphism of ordered fields, by Ordered Fields in Which Every Nonempty Set Bounded Above Has a Supremum Are Unique up to Unique Isomorphism §unique and Ordered Fields in Which Every Nonempty Set Bounded Above Has a Supremum Are Unique up to Unique Isomorphism §isomorphism.