Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 16F / b

False. With $\alpha =\beta =1$ and $\gamma =\omega$,

True. Left distributivity over a sum in the right argument,follows by transfinite induction on $\gamma$ from the recursive definition of ordinal multiplication; the successor and limit steps are exactly its defining clauses.

False. The ordinal $\alpha =\omega \cdot 2$ is a limit ordinal, but

True. Every $\beta <{\omega }_1$ is a countable ordinal, andWriting any $\alpha \ge {\omega }_1$ as ${\omega }_1+\delta$ and using associativity gives

True. The hypothesis says that $\alpha +\alpha$ and $\beta$ commute under ordinal addition. By the commuting ordinal addition classification, comparison of their Cantor normal forms implies that $\alpha$ and $\beta$ themselves commute. Thereforeso both outer expressions equal the middle one.