Mathematical Analysis
Section 1.1 notes
2026-01-02
This section defines the algebraic or arithmetic properties of the real numbers \(\symbb{R}\). Even though the content of this section—Axiom 1.1 of Schröder (2008, 2) and Theorem 1.2 of Schröder (2008, 2–3)—is described in terms of \(\symbb{R}\), nothing in this section is unique to the real numbers. Rather, these are algebraic results applicable to any field \(F\), similar to the results in the Section 1.3.2 notes for Katznelson and Katznelson (2024).
It is a curiosity that the definition of a field given in Axiom 1.1 of Schröder (2008, 2) is slightly different from the definition given in Definition 1.3.2 of Katznelson and Katznelson (2024, 9). According to Schröder (2008) a field \(F\) has at least two elements but it’s not required axiomatically that \(0 \neq 1\), whereas according Katznelson and Katznelson (2024) it’s required axiomatically that \(F \neq \emptyset\) and \(0 \neq 1\).
Despite these superficial differences the two definitions are equivalent. If \(F\) is a field according to Schröder (2008), then \(F\) having at least two elements implies \(F \neq \emptyset\), and it’s proven in Theorem 1.2 part 2 from Schröder (2008) that \(0 \neq 1\). Therefore \(F\) is a field according Katznelson and Katznelson (2024). Conversely, if \(F\) is a field according to Katznelson and Katznelson (2024), then \(F \neq \emptyset\) and \(0 \neq 1\), so \(F\) has at least two elements. Therefore \(F\) is a field according to Schröder (2008).
Starting from the definition of a field according to Schröder (2008), the proof that \(0 \neq 1\) uses as a lemma that for all \(x \in \symbb{R}\), \(x \cdot 0 = 0\). This is proven in Theorem 1.2 part 1. The proof then proceeds by contradiction. If \(0 = 1\) then for \(x \in \symbb{R}\setminus \{ 0 \}\) and \(1 \in \symbb{R}\), \(1 \cdot x = x\) by definition of \(1\) from Axiom 1.1 part 7. On the other hand, by the lemma it follows that \(1 \cdot x = 0 \cdot x = 0 \neq x\), which is a contradiction. Therefore it must be that \(0 \neq 1\). This is essentially the proof given by Schröder (2008, 2–3).