For all cardinal numbers , , (a) + = + and =

For all cardinal numbers α, β̃, γ
(a)  α + β̃ = β̃ +α and αβ̃ = β̃α (commutative laws).
(b) (α +  β̃) + γ = β̃ + (α + γ) and (αβ̃ = α(β̃γ) (associative laws).
(c) α(β̃ + γ) = α β̃ + αγ and (α + β̃ = α, γ + {β̃γ (distributive laws).
(d) α + 0 = α and αl = α.
(e) If α ≠ 0, then there is no β̃ such that α + β̃ = 0 and if α ≠ 1, then there is no β̃ such that αβ̃ = 1. Therefore subtraction and division of cardinal numbers cannot be defined.