 Solution

ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2008/b/5/3/solution

False. Use

α = 2

and

β = ω

2 ω = sup_{n < ω} 2 n = ω, ω \cdot 2 = ω + ω > ω .

The first product concatenates countably many finite two-point blocks. The second concatenates two whole copies of the order type

ω

, producing a strictly larger ordinal. Hence ordinal multiplication is not commutative.

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