An Introduction to Real Analysis
Section 1.1 notes
2025-10-06
This section begins “The natural numbers are the positive integers: \(1, 2, 3, \dots\) .” However, the symbols \(1, 2, 3, \dots\) are the natural numbers in their radix-\(10\), or base-\(10\), representation. Most of the properties of natural numbers, and indeed real numbers, as proven by Katznelson and Katznelson (2024, chaps. 1–2) and Schröder (2008, chap. 1) are independent of the choice of radix; the numbers in any given proposition or theorem are typically represented by letters and without reference to any radix-\(b\) representation.
A natural number \(a \in \symbb{N}\) and its radix-\(b\) representation are related yet distinct concepts, and this distinction ought to be clarified. However, doing so requires the integers \(\symbb{Z}\), which are not introduced by Katznelson and Katznelson (2024) until Section 1.2. This is in part because the definition of the radix-\(b\) representation of \(a\) involves \(0\), which is not part of \(\symbb{N}\). Because of this, the exposition of radix-\(b\) representations of natural numbers is located in the extra Section 1.2.4 notes for Katznelson and Katznelson (2024).
Katznelson and Katznelson (2024, sec 1.1.1) take the natural numbers \(\symbb{N}\) and their arithmetic properties as axiomatic. These can in fact be derived from a more rudimentary set of assumptions called the Peano axioms. After stating these axioms and developing some understanding thereof, the derivation of all the properties taken by Katznelson and Katznelson (2024, sec 1.1.1) as axiomatic is carried out in the Section 1.1.1 notes for Katznelson and Katznelson (2024).
Let \(\symbb{N}\) be a non-empty set, \(1 \in \symbb{N}\) a distinguished element of \(\symbb{N}\), and \(\sigma \colon \symbb{N}\to \symbb{N}\) a function on \(\symbb{N}\). The Peano axioms are as follows:
The function \(\sigma\) is called the successor function, and if \(a \in \symbb{N}\), then \(\sigma(a)\) is called the successor of \(a\).
The first Peano axiom states that \(1\) is not the successor of any natural number, or more formally, \(1 \notin \symup{img}(\sigma)\). The second Peano axiom states that the successor function is injective. The third Peano axiom, called the principle of induction or a similar name, captures the idea that \(\symbb{N}\) contains only the natural numbers.
A triplet \((\symbb{N}, \sigma, 1)\) satisfying the three Peano axioms is called a Peano system.
The following examples of a non-empty set, distinguished element and function do not satisfy the Peano axioms.
Example 1 Let \(N \coloneq \{ a \}\) and \(s \colon N \to N\) be the function such that \(a \mapsto a\). The triplet \((N, s, a)\) satisfies the second and third Peano axioms but not the first. \(\blacksquare\)
Example 2 Let \(N \coloneq \{ a, b \}\) and \(s \colon N \to N\) be the function such that \(x \mapsto b\) for all \(x \in N\). The triplet \((N, s, a)\) satisfies the first and third Peano axioms but not the second. \(\blacksquare\)
Example 3 Suppose \((\symbb{N}, \sigma, 1)\) is a Peano system. Introduce a new symbol \(\vartheta \notin \symbb{N}\) and consider the set \(\symbb{N}\uplus \{ \vartheta \}\) with distinguished element \(1\). Consider the function \(\tau \colon \symbb{N}\uplus \{ \vartheta \} \to \symbb{N}\uplus \{ \vartheta \}\) such that \(\tau\vert_{\symbb{N}} = \sigma\) and \(\tau(\vartheta) = \vartheta\). Then \((\symbb{N}\uplus \{ \vartheta \}, \tau, 1)\) satisfies the first and second Peano axioms, but not the third, as evidenced by taking \(M = \symbb{N}\subseteq \symbb{N}\uplus \{ \vartheta \}\).1 \(\blacksquare\)
Before developing addition and multiplication in \(\symbb{N}\) from a Peano system \((\symbb{N}, \sigma, 1)\), the following propositions provide some evidence that a Peano system behaves in accordance with intuition.
Proposition 1 The successor function \(\sigma\) is a bijection when viewed as a function \(\sigma \colon \symbb{N}\to \symbb{N}\setminus \{ 1 \}\).
Proof. By definition \(\sigma \colon \symbb{N}\to \symbb{N}\) is injective, which implies that \(\sigma \colon \symbb{N}\to \symbb{N}\setminus \{ 1 \}\) is injective. Hence it suffices to show that \(\sigma \colon \symbb{N}\to \symbb{N}\setminus \{ 1 \}\) is surjective. Apply the third Peano axiom, i.e., the principle of induction, with \(M = \{ 1 \} \uplus \symup{img}(\sigma) \subseteq \symbb{N}\). (By the first Peano axiom, \(1 \notin \symup{img}(\sigma)\).) Then \(1 \in \{ 1 \} \uplus \symup{img}(\sigma)\), and \(a \in \{ 1 \} \uplus \symup{img}(\sigma)\) implies that \(a \in \symbb{N}\), so \(\sigma(a) \in \symup{img}(\sigma) \subseteq \{ 1 \} \uplus \symup{img}(\sigma)\). Hence by the third Peano axiom \(\{ 1 \} \uplus \symup{img}(\sigma) = \symbb{N}\). Therefore \(\symup{img}(\sigma) = \symbb{N}\setminus \{ 1 \}\), so \(\sigma \colon \symbb{N}\to \symbb{N}\setminus \{ 1 \}\) is surjective and therefore bijective. \(\blacksquare\)
Proposition 2 The successor function has no fixed points.
Proof. To show that for all \(a \in \symbb{N}\), \(a \neq \sigma(a)\), apply the third Peano axiom to the set \(M \coloneq \{ a \in \symbb{N}\colon a \neq \sigma(a) \} \subseteq \symbb{N}\). By the first Peano axiom, \(1 \neq \sigma(a)\) for all \(a \in \symbb{N}\), so taking \(a = 1\) implies that \(1 \in M\). If \(a \in M\), then \(a \neq \sigma(a)\). By the contrapositive of the second Peano axiom, \(\sigma(a) \neq \sigma(\sigma(a))\), so \(\sigma(a) \in M\) and therefore \(M = \symbb{N}\). \(\blacksquare\)
If the Peano axioms are more rudimentary than natural numbers, one might ask what is more rudimentary than the Peano axioms. The standard answer is the Zermelo–Fraenkel axioms with the axiom of choice, commonly abbreviated as ZFC. ZFC is a formalization of set theory that was developed in the early 20th century to rectify the earlier, less formalized set theory of the 19th century. In the 19th century a naïve form of set theory was used in which it was assumed that there exists a set of objects satisfying any property at all. Around the turn of the 20th century it was realized that this formulation of set theory is so permissive that it permits contradictions. For example, using this permissive form of set theory, let \(R \coloneq \{ X \colon X \notin X \}\) be the set of all sets \(X\) which do not contain themselves. The question arises whether \(R \in R\). If \(R \in R\), then by definition \(R \notin R\). On the other hand, if \(R \notin R\), then \(R \in R\). This is known as Russell’s paradox, and its discovery stimulated the search for a formalized set theory which led to ZFC.
ZFC is a list of nine axioms that stipulate, among other things, that two sets \(A\) and \(B\) are equal if and only if they have the same elements, and for any logical statement \(P(a)\) depending on \(a \in A\), the set \(\{ a \in A \colon P(a) \text{ is true} \} \subseteq A\) is well defined. Informal statements of the nine axioms can be found in Katznelson and Katznelson (2024, app. A) and Schröder (2008, app. B).
The ZFC axioms imply the existence of the empty set \(\emptyset\) and an infinite set \(S\). The set \(S\) has the properties that \(\emptyset \in S\) and if \(A \in S\), where \(A\) is a set, then \(A \cup \{ A \} \in S\). Using these ingredients, a Peano system can be constructed.
To construct the Peano system, \(S\) is taken as the non-empty set and \(\emptyset \in S\) as the distinguished element. The successor function is given by \(f \colon S \to S\) where \(f(A) = A \cup \{ A \}\). The natural numbers will then be as follows: \[\begin{align*} \emptyset\\[0.75em] f(\emptyset) &= \emptyset \cup \{ \emptyset \} = \{ \emptyset \}\\[0.75em] f(\{ \emptyset \}) &= \{ \emptyset \} \cup \{ \{ \emptyset \} \} = \{ \emptyset, \{ \emptyset \} \}\\[0.75em] f(\{ \emptyset, \{ \emptyset \} \}) &= \{ \emptyset, \{ \emptyset \} \} \cup \{ \{ \emptyset, \{ \emptyset \} \} \} = \{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \}\\[0.75em] &\vdotswithin{=} . \end{align*}\]
Given the Peano system \((S, f, \emptyset)\), the question arises whether there exist other Peano systems that are meaningfully different. The Peano system \((S, f, \emptyset)\) is in fact unique up to isomorphism, which means that any other Peano system \((\symbb{N}, \sigma, 1)\) is not meaningfully different from \((S, f, \emptyset)\). However, the theory to prove this is most naturally developed when constructing the integers \(\symbb{Z}\) from the natural numbers \(\symbb{N}\). Isomorphisms between Peano systems become relevant at that point because the construction of \(\symbb{Z}\) from \(\symbb{N}\) shows that \(\symbb{N}\) is not a subset of \(\symbb{Z}\), yet it is necessary to locate a subset of \(\symbb{Z}\) that is essentially the same as \(\symbb{N}\). This is done in the Section 1.2 notes for Katznelson and Katznelson (2024).
Note that \(1 \in \symbb{N}\) by assumption. If \(a \in \symbb{N}\), then \(\tau(a) = \sigma(a) \in \symbb{N}\) because \(\tau\vert_{\symbb{N}} = \sigma\), and yet \(\symbb{N}\neq \symbb{N}\uplus \{ \vartheta \}\).↩︎