## Franklin's Notes

### 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.