1.2 Axioms1

1.2.1 Field Axioms

As you might know from experience, natural numbers won’t get us very far in maths. So instead, we turn to studying the real numbers (\(\mathbb{R}\)). But before we do, we need to know how these numbers behave under certain operations. This is where the field axioms come in. There’s a long list of them in Section 3.1 of the lecture notes , so they’re not repeated here in full. However, we can summarise2 them as follows:

  • Addition: On \(\mathbb{R}\), addition is associative and commutative, an additive identity exists, and additive inverses exist.
  • Multiplication: On \(\mathbb{R}\setminus \lbrace 0 \rbrace\), multiplication is associative and commutative, a multiplicative identity exists, and multiplicative inverses exist.
  • Multiplication distributes over addition.

Try matching the properties here to the numbered axioms in the lecture notes!

1.2.2 Order Axioms

As if the first 9 field axioms weren’t enough, there are 5 order axioms you need to know. As these are useful for the problem sheet, they are presented below.
Order Axioms:

For \(x,y,z \in \mathbb{R}\):

  • \(x \leq y\) or \(y \leq x\).
  • If \(x\leq y\) and \(y \leq x\), then \(x = y\).
  • If \(x \leq y\) and \(y \leq z\), then \(x \leq z\).
  • If \(x \leq y\), then \(x + z \leq y + z\).
  • If \(x\leq y\) and \(z\geq0\), then \(xz\leq yz\).

  1. Or Axiomata, if you’re into your archaic English.↩︎

  2. Once you’ve learned some group theory, we can obscure everything behind more definitions. The first two bullet points can be stated as: \((\mathbb{R},+)\) and \((\mathbb{R}\setminus\lbrace0\rbrace, \cdot)\) are abelian groups. The third bullet point remains the same.↩︎