The Axioms for the Integers

Branko Ćurgus

In the axioms below we use the standard logical operators: the conjunction $\wedge$, the disjunction $\vee$, the exclusive disjunction $\oplus$, the implication $\Rightarrow$, universal quantifier $\forall,$ existential quantifier $\exists.$

We also use the standard set notation: the set membership $\in$, the subset $\subseteq$, the equality $=$, the set difference $\setminus$ and the Cartesian product $\times.$ For singleton sets instead of writing $\{a\} = \{b\}$ we write $a = b.$

The notation $f:A\to B$ stands for a function $f$ which is defined on a set $A$ with the values in $B.$

Axiom 2 (AE) below establishes the existence of the addition function defined on $\mathbb{Z}\!\times\!\mathbb{Z}$ with the values in $\mathbb{Z}.$ It is common to denote the value of $+$ at a pair $(a, b) \in \mathbb{Z}\!\times \!\mathbb{Z}$ by $a+b.$

Axiom 6 (ME) establishes the existence of the multiplication function defined on $\nZ\!\times\!\nZ$ with the values in $\nZ.$ It is common to denote the value of this function at a pair $(a,b) \in \nZ\!\times\!\nZ$ by $a\cdot b$ which is almost always abbreviated as $ab.$

Axiom 12 introduces the set of positive integers $\mathbb{P}.$

Definition. The set $\mathbb{Z}$ of integers satisfies the following 16 axioms.

Axiom 1. (ZE) $\nZ\neq\emptyset$
Axiom 2. (AE) There exists a function $+: \nZ\!\times\!\nZ \to \nZ.$
Axiom 3. (AA) $\forall a \in \nZ \ \ \forall b \in \nZ \ \ \forall c \in \nZ \quad a+(b+c) = (a+b)+c$
Axiom 4. (AC) $\forall a \in \nZ \ \ \forall b \in \nZ \quad a+b = b+a$
Axiom 5. (AZ) $\exists\, 0 \in \nZ \ \ \forall a \in \nZ \quad 0+a = a$
Axiom 6. (AO) $\forall a \in \nZ \ \ \exists (-a) \in \nZ \quad a + (-a) = 0$
Axiom 7. (ME) There exists a function $\cdot : \nZ \times \nZ \to \nZ.$
Axiom 8. (MA) $\forall a \in \nZ \ \ \forall b \in \nZ \ \ \forall c \in \nZ \quad a(bc) = (ab)c$
Axiom 9. (MC) $\forall a \in \nZ \ \ \forall b \in \nZ \quad ab = ba$
Axiom 10. (MO) $\exists\, 1 \in \nZ\!\setminus\!\{0\} \quad \forall a \in \nZ \quad 1a = a$
Axiom 11. (DL) $\forall a \in \nZ \ \ \forall b \in \nZ \ \forall c \in \nZ \quad a(b+c) = ab + ac$
Axiom 12. (PE) There exists a nonempty subset $\mathbb{P}$ of $\nZ\!\setminus\!\{0\}$.
Axiom 13. (PD) $\forall a \in \mathbb Z\!\setminus\!\{0\} \quad (a \in \mathbb P) \oplus (-a \in \mathbb P)$
Axiom 14. (PA) $\forall a \in \mathbb{P} \ \ \forall b \in \mathbb{P} \quad a + b \in \mathbb{P}$
Axiom 15. (PM) $\forall a \in \mathbb{P} \ \ \forall b \in \mathbb{P} \quad ab \in \mathbb{P}$
Axiom 16. (WO) $\bigl( S \subseteq \mathbb{P} \bigr)\!\wedge\!\bigl( S \neq \emptyset \bigr) \Rightarrow \bigl( \exists \, m \in S \ \forall x \in S\!\setminus\!\{m\} \quad x\!+\!(- m)\!\in\! \mathbb{P}\bigr)$
The end of the definition.

Explanation of the abbreviations: ZE--integers exist, AE--addition exists, AA--addition is associative, AC--addition is commutative, AZ--addition has zero, AO--addition has opposites, ME--multiplication exists, MA--multiplication is associative, MC--multiplication is commutative, MO--multiplication has one, DL--distributive law, PE--positive integers exist, PD--dichotomy involving positive integers, PA--positive integers respect addition, PM--positive integers respect multiplication, WO--the well-ordering axiom.