 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ii/paper-1/16h/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

False. Take

α = ω^{2} + 1

, which satisfies the proposed lower bound. Associativity of ordinal addition and

1 + ω^{2} = ω^{2}

give

(ω^{2} + 1) + ω^{2} = ω^{2} \cdot 2, ω^{2} + (ω^{2} + 1) = ω^{2} \cdot 2 + 1.

The final successor survives only in the second expression.

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