- ℕ
- Positional notation
- +
- (0) + n = n if n ∈ ℕ
- (m+1) + n = ((m + n)+1) if m, n ∈ ℕ
- (n+1) = n + 1 if n ∈ ℕ
- (l + m) + n = l + (m + n) if l, m, n ∈ ℕ
- + is associative
- m + n = n + m if m, n ∈ ℕ
- + is commutative
- m + n = 0 ⇒ m = n = 0 if m, n ∈ ℕ
- a + c = b + c ⇒ a = b if a, b, c ∈ ℕ
- m + n = m ⇒ n = 0 if m, n ∈ ℕ
- ≤
- <
- ≤ is a total order
- a < b ≤ c ⇒ a < c if a, b, c ∈ ℕ
- a ≤ b < c ⇒ a < c if a, b, c ∈ ℕ
- n ≥ 0 if n ∈ ℕ
- n ≤ 0 ⇒ n = 0 if n ∈ ℕ
- n + 1 > n if n ∈ ℕ
- Subsets
- ∃ m ∈ M : m is a ≤-greatest element of M if M ⊆ ℕ, k ∈ ℕ, f : ℕ≤k ↔ M is a bijection
- M is finite ⇔ M is empty or ∃ m ∈ M : m is a ≤-greatest element of M ⇔ M is empty or ∃! m ∈ M : m is a ≤-greatest element of M ⇔ ∃ a ∈ ℕ : M ⊆ ℕ≤a ⇔ ∃ b ∈ ℕ : M ⊆ ℕ<b if M ⊆ ℕ
- −
- ⋅
- n ⋅ 1 = n if n ∈ ℕ
- ⋅ is associative
- ⋅ is commutative
- ⋅ is distributive over +
- a ⋅ c = b ⋅ c ⇒ a = b if a, b ∈ ℕ, c ∈ ℕ+
- ∣
- Divℕ(n)
- m ∣ n ⇒ m ≤ n if m, n ∈ ℕ+
- a ∣ b ⇒ [a ∣ c ⇔ a ∣ (b + c)] if a ∈ ℕ+, b ∈ ℕ, c ∈ ℕ
- even/odd
- ∕
- mn
- p(0), [∀ m ∈ ℕ s.t. p(m) : p(m + 1)] ⇒ p(n) if p is a property on ℕ, n ∈ ℕ
- 0 ∈ M and ∀ m ∈ M : m + 1 ∈ M ⇔ ∀ n ∈ ℕ s.t. ℕ<n ⊆ M : n ∈ M ⇔ M = ℕ if M ⊆ ℕ
- ≤ is a well-order
- min
- min
- max
- max
- Sequences
- n!
- (n!)n ∈ ℕ starts with (1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000)
- (nk)
- (nk) = n!∕(k! ⋅ (n − k)!) if n ∈ ℕ, k ∈ ℕ≤n
- Iterated operators
- Prime numbers
- coprime
- m and m ⋅ n are not coprime if m, n ∈ ℕ>1