Page 57 - Textos de Matemática Vol. 38

ORDINALS, COMPUTATIONS, AND MODELS OF SET THEORY 49
Definition 2.13. The ordinal sum α + β is defined by recursion on β ∈ Ord by − α+0=α,
− α+(β+1)=(α+β)+1,
− α+λ=β<λ(α+β).
Definition 2.14. The ordinal product α · β is defined by recursion on β ∈ Ord by
− α·0=0,
− α·(β+1)=(α·β)+α, − α·λ=β<λ(α·β).
These operations obviously extend the arithmetic on natural numbers.
Exercise 3. Exhibit explicit recursion rules for + and · as in the recursion Theorem 2.11.
Exercise 4. Prove the following arithmetic laws for ordinal arithmetic: a) (α+β)+γ=α+(β+γ).
b) (α·β)·γ=α·(β·γ).
c) α·(β+γ)=α·β+α·γ.
Show that the operations are not commutative, and that the distributive law (α + β) · γ = α · γ + α · γ fails.
Exercise 5. Prove a) ∀α∃βα+β=β. b) ∀α∃βα·β=β.
The operations + and · are continuous at limit ordinals with respect to ordinal limits:
Definition 2.15. Let (δi|i < λ) be a sequence of ordinals of limit length λ. Then a) limi<λ δi = i<λ δi is the limit of (δi|i < λ),
b) liminfi<λ δi = limi<λ min{δj|i j < λ} is the inferior limit of (δi|i < λ).
Exercise 6. Define a topology on the class Ord such that limit ordinals are limit points in the sense of the topology and such that the operations + and · are continuous in the sense of the topology.