Peano arithmetic

Definition

Let’s add another function: addition. Defined by:

$$\forall a \in \mathbb{N} (a+0=a)$$

$$\forall a b \in \mathbb{N} (a+s(b)=s(a+b))$$

That is, adding zero to a number doesn’t change it, and $$(a+b)+1=a+(b+1)$$.

Example

Let’s use this to solve $$1+2$$:

$$1+2=1+s(1)$$

$$1+s(1)=s(1+1)$$

$$s(1+1)=s(1+s(0))$$

$$s(1+s(0))=s(s(1+0))$$

$$s(s(1+0))=s(s(1))$$

$$s(s(1))=s(2)$$

$$s(2)=3$$

$$1+2=3$$

All addition can be done iteratively like this.

Commutative property of addition

$$x+y=y+x$$

Associative property of addition

$$x+(y+z)=(x+y)+z$$

Multiplication

Definition

Multiplication can be defined by:

$$\forall a \in \mathbb{N} (a.0=0)$$

$$\forall a b \in \mathbb{N} (a.s(b)=a.b+a)$$

Example

Let’s calculate $$2.2$$.

$$2.2=2.s(1)$$

$$2.s(1)=2.1+2$$

$$2.1+2=2.s(0)+2$$

$$2.s(0)+2=2.0+2+2$$

$$2.0+2+2=2+2$$

$$2+2=4$$

Commutative property of multiplication

Multiplication is commutative:

$$xy=yx$$

Associative property of multiplication

Multiplication is associative:

$$x(yz)=(xy)z$$

Distributive property of multiplication

Multiplication is distributive over addition:

$$a(b+c)=ab+ac$$

$$(a+b)c=ac+bc$$