### Peano Arithmetic

**Peano Arithmetic** is an attempt to axiomatize arithmetic and number theory in first-order logic . It consists of the following axioms:

1. $(\forall x)(0\not\equiv Sx)$

2. $Sx\equiv Sy \rightarrow x\equiv y$

3. $x+0\equiv x$

4. $x+Sy\equiv S(x+y)$

5. $x\cdot 0\equiv 0$

6. $x\cdot Sy\equiv (x\cdot y)+x$

7. For each formula $\varphi(v_0...v_n)$ of $\mathscr{L}$, where $v_0$ is free in $\varphi$, we have $\big(\varphi(0v_1...v_n)\land (\forall v_0)(\varphi(v_0v_1...v_n)\rightarrow \varphi(Sv_0v_1...v_n))\big)\rightarrow (\forall v_0)\varphi(v_0...v_n)$ This is the **axiom schema of induction** and it consists of infinitely many axioms, one for each formula $\varphi$ satisfying the aforementioned stipulations. The axiom corresponding to $\varphi$ may be denoted $(7_\varphi)$.

These axioms can be interpreted intuitively as the following informal observations about our "mental picture" of how the arithmetic on the natural numbers should look:

1. $0$ is the first number, and has no predecessor.

2. No two distinct numbers have the same successor.

3. $0$ is the additive identity, i.e. adding $0$ to a number does not change it.

4. The successor of a sum of numbers is equal to the sum of the first number with the successor of the second.

5. Any number times $0$ equals $0$.

6. Multiplication is repeated addition (roughly speaking).

7. If a statement is true about $0$, and is true for the successor of any other number for which it is true, then it is true for all numbers.

The **standard model of number theory** is the model $\langle\omega,+,\cdot,S,0\rangle$, where each symbol has its usual meaning, and all other nonisomorphic models are called **nonstandard**. **Complete number theory** is the set of all sentences of $\mathscr{L}$ that hold in the standard model.

# model-theory

# first-order-logic

# number-theory

# axioms

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