Real number - Construction and Completeness

Formal Constructions and Axiomatic Characterizations Introduction The real numbers $\mathbb{R}$ are not just a set of numbers we compute with—they have a precise mathematical definition based on axioms and formal constructions. This section covers two approaches: (1) what axioms $\mathbb{R}$ must satisfy, and (2) how to construct $\mathbb{R}$ from simpler objects like rational numbers. Understanding both approaches will give you a complete picture of what the real numbers are. Axiomatic Description of $\mathbb{R}$ The big idea: $\mathbb{R}$ is uniquely determined by being an ordered field with a special completeness property. An ordered field is a set with addition and multiplication operations, plus an ordering ($<$), satisfying the familiar arithmetic rules (associativity, commutativity, distributivity, etc.). Both $\mathbb{Q}$ (the rationals) and $\mathbb{R}$ are ordered fields. The key property that sets $\mathbb{R}$ apart is the least upper bound property (also called Dedekind completeness): > Every non-empty subset of $\mathbb{R}$ that has an upper bound must have a least upper bound (supremum) in $\mathbb{R}$. This single property, combined with the ordered field axioms, completely characterizes $\mathbb{R}$. In other words, if you start with axioms saying "$\mathbb{F}$ is an ordered field with the least upper bound property," then $\mathbb{F}$ is $\mathbb{R}$ (up to isomorphism—see below). Why this matters: The rationals $\mathbb{Q}$ are an ordered field too, but they violate this property. For example, the set $\{q \in \mathbb{Q} : q^2 < 2\}$ is bounded above (by 3, for instance), but it has no least upper bound in $\mathbb{Q}$ because $\sqrt{2}$ is irrational. Dedekind Completeness and Its Consequences Let's unpack the least upper bound property more carefully. Definition: A real number $U$ is an upper bound for a set $S \subseteq \mathbb{R}$ if $s \le U$ for all $s \in S$. The least upper bound (or supremum) of $S$ is an upper bound $M$ such that no smaller number is an upper bound. Dedekind Completeness states: If $S$ is a non-empty subset of $\mathbb{R}$ and $S$ has some upper bound, then $S$ has a least upper bound in $\mathbb{R}$. Key consequence—the Archimedean property: For any real number $x$, there exists an integer $n$ such that $n > x$. Why does this follow? Suppose, by contradiction, that some real number $x$ were an upper bound for all integers. Then the set of all integers would be a bounded, non-empty set. By Dedekind completeness, this set would have a supremum, say $M$. But then $M + 1$ is also an integer, and $M + 1 > M$, contradicting the fact that $M$ is an upper bound. Therefore, no such upper bound exists. This might seem obvious, but it's actually a deep fact: without completeness, you could have "exotic" ordered fields where integers don't eventually exceed every element. Uniqueness Up to Isomorphism Central theorem: Any two Dedekind-complete ordered fields are isomorphic. This means there is essentially one Dedekind-complete ordered field (up to relabeling). We call this field $\mathbb{R}$. What does "isomorphic" mean here? Two fields $F$ and $G$ are isomorphic if there is a bijection $\phi: F \to G$ that preserves both the field operations and the ordering: $\phi(a + b) = \phi(a) + \phi(b)$ $\phi(a \cdot b) = \phi(a) \cdot \phi(b)$ If $a < b$, then $\phi(a) < \phi(b)$ Moreover, this isomorphism is unique—there's only one way to match up the two fields. Why this matters: This explains why all the different ways to construct $\mathbb{R}$ (Dedekind cuts, Cauchy sequences, etc.) are valid: they all produce isomorphic copies of the same object. Any theorem proved in one construction transfers to all others. Construction via Dedekind Cuts One classical way to build $\mathbb{R}$ is to start with $\mathbb{Q}$ and "fill in the gaps." Definition: A Dedekind cut is a partition of the rational numbers $\mathbb{Q}$ into two non-empty sets $L$ (the "lower" set) and $U$ (the "upper" set) such that: Every rational in $L$ is less than every rational in $U$: if $\ell \in L$ and $u \in U$, then $\ell < u$ $L$ has no greatest element (no largest rational in $L$) $U$ has no least element (no smallest rational in $U$) Example: The cut corresponding to $\sqrt{2}$ has: $L = \{q \in \mathbb{Q} : q < 0 \text{ or } q^2 < 2\}$ $U = \{q \in \mathbb{Q} : q > 0 \text{ and } q^2 > 2\}$ Notice that $L$ captures all rationals "below" $\sqrt{2}$ (without including $\sqrt{2}$ itself, since $\sqrt{2} \notin \mathbb{Q}$). Construction: Each Dedekind cut defines a unique real number. We simply declare that the real number corresponding to a cut is the "boundary" between the lower and upper sets. The set of all Dedekind cuts becomes our model of $\mathbb{R}$. Addition and multiplication are defined on cuts in a natural way (you combine the elements), and the ordering is: cut $(L1, U1)$ is less than cut $(L2, U2)$ if $L1 \subsetneq L2$. Advantage of this construction: It makes completeness completely transparent—every cut by definition has a least upper bound (namely, the cut itself). Construction via Cauchy Sequences Another approach builds $\mathbb{R}$ from rational sequences with a special property. Definition: A sequence $(qn)$ of rational numbers is a Cauchy sequence if, no matter how small you make your tolerance $\varepsilon > 0$, eventually all terms get within $\varepsilon$ of each other. Formally: for every $\varepsilon > 0$, there exists an integer $N$ such that $|qn - qm| < \varepsilon$ whenever $n, m \ge N$. Intuition: A Cauchy sequence is a sequence that "wants to converge"—the terms cluster together. In $\mathbb{Q}$, some Cauchy sequences (like the sequence $1, 1.4, 1.41, 1.414, \ldots$ of approximations to $\sqrt{2}$) don't converge to any rational. Equivalence relation: Two Cauchy sequences $(qn)$ and $(rn)$ are equivalent if their termwise difference tends to 0: $$\lim{n \to \infty} |qn - rn| = 0$$ Construction: Each real number is defined as an equivalence class of Cauchy sequences. For instance, the real number $\sqrt{2}$ is represented by any Cauchy sequence that converges to $\sqrt{2}$ (e.g., $1, 1.4, 1.41, 1.414, \ldots$; or $2, 1.5, 1.42, 1.415, \ldots$; etc.). We define operations on equivalence classes by operating on representatives: $[(qn)] + [(rn)] = [(qn + rn)]$. Advantage of this construction: It emphasizes the connection between real numbers and convergent sequences, which is central to calculus. Equivalence of Constructions Both the Dedekind-cut and Cauchy-sequence constructions yield isomorphic copies of $\mathbb{R}$. This is not a coincidence—it reflects the fact that both constructions capture the same essential completeness property. There is a natural map between them: Given a Cauchy sequence $(qn)$, create the Dedekind cut where $L$ consists of all rationals smaller than infinitely many terms, and $U$ consists of all rationals larger than all but finitely many terms. Conversely, given a Dedekind cut $(L, U)$, you can construct a Cauchy sequence of elements of $L$ that approaches the cut. These maps are inverse isomorphisms, showing the two models are "the same" up to relabeling. Practical takeaway: If you prove a theorem about $\mathbb{R}$ using Dedekind cuts, the same theorem holds for the Cauchy-sequence model, and vice versa. The choice of construction is a matter of convenience, not substance. Metric (Topological) Completeness Now we shift perspective from algebra to analysis. This is a different (but related) notion of completeness. Definition: A sequence $(xn)$ of real numbers is a Cauchy sequence if for every $\varepsilon > 0$, there exists $N$ such that $|xn - xm| < \varepsilon$ whenever $n, m \ge N$. This is the same definition as before, but now the terms $xn$ are real numbers, not rationals. Metric Completeness: $\mathbb{R}$ is metrically complete, meaning every Cauchy sequence of real numbers converges to a real number. That is, if $(xn)$ is Cauchy, there exists a real number $L$ such that $\lim{n \to \infty} xn = L$. Contrast with Dedekind Completeness: These sound different because one is about suprema (upper bounds) and the other is about limits of sequences. However, they are intimately related: Dedekind completeness implies metric completeness. If $(xn)$ is a Cauchy sequence of reals, its set of terms is bounded, so it has a supremum $L$ by Dedekind completeness. With some care, you can show that $(xn)$ converges to $L$. Metric completeness (plus ordered field structure) implies Dedekind completeness. If $S$ is a bounded non-empty set, pick an increasing Cauchy sequence of rationals approximating the supremum. By metric completeness, this sequence converges to some real number, which is the supremum of $S$. In practice, many theorems in calculus and analysis rely on metric completeness (e.g., the fact that monotone bounded sequences converge). Understanding both notions helps you see the deep structure of $\mathbb{R}$. <extrainfo> Decimal and Positional Representation Structure of Decimal Expansions A non-negative real number $x$ can be represented in decimal notation as: $$x = dk d{k-1} \ldots d0 . d{-1} d{-2} d{-3} \ldots$$ where each digit $di$ is an integer from 0 to 9, $k$ is a non-negative integer (indicating the position of the leftmost digit), and the decimal point separates non-negative from negative powers of 10. Interpretation as an Infinite Series This decimal representation means: $$x = \sum{i=-\infty}^{k} di \cdot 10^i$$ For example, $35.7 = 3 \cdot 10^1 + 5 \cdot 10^0 + 7 \cdot 10^{-1}$. Construction via Truncations For each positive integer $n$, the truncation at place $n$ is the finite sum: $$\sum{i=-n}^{k} di \cdot 10^i$$ This is a rational number ("decimal fraction"). The real number $x$ is defined as the least upper bound of the set of all such truncations. For instance, if $x = 0.333\ldots$, the truncations are $0.3, 0.33, 0.333, 0.3333, \ldots$, and their supremum is $1/3$. Non-Uniqueness of Terminating Decimals A subtle fact: terminating decimals have two representations. For example: $$0.5 = 0.4999\ldots$$ More generally, any terminating decimal can be rewritten by replacing the last nonzero digit $d$ with $d-1$ and appending infinitely many 9's. The reason is that both representations describe the same supremum: Truncations of $0.5000\ldots$ are $0.5, 0.50, 0.500, \ldots$ with supremum $0.5$ Truncations of $0.4999\ldots$ are $0.4, 0.49, 0.499, 0.4999, \ldots$ also approaching $0.5$ For a non-terminating decimal (like $0.333\ldots$), the representation is unique. General Base Representation The same construction works for any base $b > 1$. Replace 10 with $b$, and replace the digit 9 with $b-1$. For instance, in base 2 (binary), a number like $0.1$ (in base 2) equals $0.0111\ldots$ (in base 2). </extrainfo> Summary The real numbers $\mathbb{R}$ can be characterized axiomatically as the unique Dedekind-complete ordered field. This unique characterization can be realized through multiple constructions (Dedekind cuts, Cauchy sequences), which yield isomorphic copies of $\mathbb{R}$. The key property is Dedekind completeness (every bounded set has a supremum), which ensures metric completeness (every Cauchy sequence converges). These foundational ideas underpin all of real analysis and calculus.