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ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2008/b/5/1/solution

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Past exam of the logic and set theory course of the University of Oxford / 2008 / B / 5 / 1 / Solution by

Codex  0 2026-10-31

First give the recursive definitions, keeping the right argument as the recursion variable. For ordinals

α, β

and nonzero limit ordinals

λ

,

α + 0 = α, α + (β + 1) = (α + β) + 1, α + λ = sup_{ξ < λ} (α + ξ),

α \cdot 0 = 0, α \cdot (β + 1) = α β + α, α λ = sup_{ξ < λ} α ξ .

Here

ξ + 1

is the successor ordinal, and the supremum of a set of ordinals is its union. These are ordinal arithmetic, defined by transfinite recursion, and differ from cardinal sum and ordinal multiplication.

Associativity of ordinal addition is true. Fix

α, β

and use transfinite induction on

γ

. The case

γ = 0

is immediate. In the successor step the recursive definitions and the induction hypothesis give

(α + β) + (γ + 1) = ((α + β) + γ) + 1 = (α + (β + γ)) + 1 = α + (β + (γ + 1)) .

For a nonzero limit ordinal

λ

β + ξ

for

ξ < λ

form an increasing cofinal subset of the limit ordinal

β + λ

. The limit clause for ordinal addition, restricted to this cofinal subset, gives

(α + β) + λ = sup_{ξ < λ} ((α + β) + ξ) = sup_{ξ < λ} (α + (β + ξ)) = α + (β + λ) .

Cofinal restriction is valid because ordinal addition by a fixed ordinal on the left is increasing and continuous in its right argument, directly from its recursive definition. This completes the induction.

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