Definitions of the naturals and their arithmetic: 1. The [[Peano Axioms|Peano axioms]] formulate the natural numbers directly in $L_{2}.$ 2. The [[Set-theoretic Construction of the Naturals|set-theoretic construction]] defines them inductively using the empty set. # Properties **Algebra:** addition & multiplication of natural numbers satisfy Peano axioms. **Relations on $\mathbb{N}$:** (Order) We can define a [[Ordering the natural numbers|total strict order]] on $\mathbb{N}.$ The [[Well-Ordering Principle|well-ordering principle]] is that every non-empty subset of $\mathbb{N}$ has a unique least element. (Factor) Take $a,b\in\mathbb{N}.$ We say $a$ [[Divisibility in Integers|divides]] $b$ iff $a = b\times c$ for some $c\in\mathbb{N}.$ (Congruence) $a,b\in\mathbb{N}$ are said to [[Congruence Modulo n|congruent modulo]] $n\in \mathbb{N}$ iff $n\mid(b-a)$ or equivalently, $a$ and $b$ have the same remainder after [[Division with remainder for integers|division with remainder]]. (Coprime) $a$ is [[Coprime Integers|relatively prime]] to $b$ iff their [[Greatest Common Divisor (GCD)|gcd]] is $1.$ **Prime factorisation:** A [[Prime numbers|prime]] is natural number that is divisible by exactly two positive natural numbers. The [[Fundamental theorem of arithmetic|FTA]] asserts that every natural number gt;1$ can be uniquely expressed as a product of prime powers. # Applications **Integers**: The [[Integers|integers]] can be formally constructed as the set of equivalence classes of ordered pairs of naturals $(a,b)$ where two pairs are equivalent if they have the same difference (i.e. $(a,b)\sim(c,d)\iff a+d=b+c$). **Induction**: [[Induction Principle]].