Peano Axioms

Peano Axioms are axioms defining natural numbers set \(\mathbb N\) using set language. With \(+\) and \(\times\) defined by Peano Arithmetic, \((\mathbb N,+,0,\times,1)\) forms a commutative semiring. The goal of this analysis is to formalize arithmetic. As opposed to accepting arithmetic results as fact, arithmetic results are built through the Peano Axioms and the process of mathematical induction.

The successor function, denoted as \(S(n),\) takes a natural number as an argument, and returns the next successive natural number as a result. In the ordinary arithmetic notation, this is:

\[S(n)=n+1.\]

However, this notation is generally avoided when using the successor function. Typically, the goal is to define mathematical operations and results purely through the successor function, in particular, as a series of nested successor functions. For example,

\[\begin{align} 1 &= S(0) \\ 2 &= S(S(0)) \\ 3 &= S(S(S(0))) \\ &\vdots \end{align}\]

The axioms are as follows. \(0\) is a symbol standing for a constant and \(S(\cdot)\) is the unary successor function.

1. \(0 \in \mathbb N.\) (Note that 0 is included as a natural number in this analysis.)
2. \(\forall x \in \mathbb N, S(x) \in \mathbb N.\)
3. \(x=y \Leftrightarrow S(x)=S(y).\)
4. \(\not \exists x: S(x)=0.\)
5. \(\forall K \subseteq \mathbb N,(0 \in K) \wedge ((x \in K) \Rightarrow (S(x) \in K)) \Rightarrow K=\mathbb N.\)

These axioms will be used to "prove" arithmetic facts on the set of natural numbers with the process of induction.

Peano Arithmetic defines two binary functions in set \(\mathbb N\), namely addition \(+\) and multiplication \(\times\). They are defined recursively:

Addition
1. \(x+0=x\)
2. \(x+S(y)=S(x+y)\)

Find \(S(S(0))+S(S(S(0)))\).

\(\;\;\;\;\color{red}{S(S(0))}+S(\color{red}{S(S(0))})\)
\(=S(\color{red}{S(S(0))}+S(\color{red}{S(0)}))\) (Apply (2))
\(=S(S(\color{red}{S(S(0))}+S(\color{red}0)))\) (Again)
\(=S(S(S(S(\color{red}{S(0)}+\color{red}0))))\) (And again)
\(=S(S(S(S(S(0)))))\) (Apply (1))

And this is actually \(2+3=5\) in our normally used symbols.

From now on, for simplicity, we will use the normally used symbols to represent the natural numbers instead of a long series of \(S\).

Multiplication
1. \(x \times 0 = 0\)
2. \(x \times S(y) = x + (x \times y)\)

Find \(2 \times 3\).

\(\;\;\;\;2 \times 3\)
\(=2+(2 \times 2)\)
\(=2+(2+(2 \times 1))\)
\(=2+(2+(2+(2 \times 0)))\)
\(=2+(2+(2+(0)))\)
\(=2+(2+(2))\)
\(=2+(4)\)
\(=6\)