 Solution

ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2008/b/5/2/solution

False. Take

α = 1

and

β = ω

. By the limit clause for ordinal addition,

1 + ω = sup_{n < ω} (1 + n) = ω, ω + 1 > ω .

The first order type is one point followed by a countably infinite sequence, which is still of type

ω

ω + 1

. Thus ordinal addition is not commutative, even though the two resulting sets have equal cardinality.

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