As I understand there are two "axioms" that should be satisfied in floating-point arithmetic:
$$\forall x\in \mathbb R,\ \exists |\varepsilon|\leq\varepsilon_{\text{machine}},\ \mbox{fl} (x) = x(1+\varepsilon) \tag{1}$$
$$\forall x, y \in \mathbb F,\ x\circledast y = \mbox{fl} (x * y) \tag{2}$$
And from these two we can derive the Fundamental Axiom of Floating Point Arithmetic:
$$\forall x, y \in \mathbb F,\ \exists |\varepsilon|\leq\varepsilon_{\text{machine}},\ x \circledast y = (x*y)(1+\varepsilon) \tag{3}$$
In the stability section of Trefethen & Bau's Numerical Linear Algebra, the authors consider only axioms 1 and 3 (fundamental axiom), but not 2. Why is this morally? Can someone give an example of a floating-point system that satisfies only axiom 3 but not 2?
The first axiom defines an $\varepsilon_x>0$ for every $x\in \mathbb{R}$--this declares its associated floating point $fl(x)$ by the RHS. The second axiom defines a binary operation, $\circledast$, on the set of floating point numbers. The third defines the "$\varepsilon_{x\circledast y}$" for the product by declaration. This encodes the two together. Similar to what happens in the Subgroup Criterion in algebra.
That being said, if the system satisfies (3) it automatically satisfies (2) as well. Since: $$\color{blue}{\forall x,y\in \mathbb{F}},\text{ }\exists |\varepsilon_{x\circledast y}|\leq \varepsilon_{machine}:\bigg( \text{ }\text{ }\color{blue}{x\circledast y} = (x*y)(1+\epsilon) \color{blue}{=: fl(x*y)}\bigg)$$ We get: $$\color{blue}{\forall x,y\in \mathbb{F},\text{ }x\circledast y = fl(x*y)}.$$