Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 16F (Logic and Set Theory)

The inductive definition of ordinal addition isfor a limit ordinal $\lambda$.

The synthetic definition takes the order type of the disjoint union of a copy of $\alpha$ followed by a copy of $\beta$: every point of the first copy precedes every point of the second, and each copy retains its original total order.

Fix $\alpha$ and use transfinite induction on $\beta$. The synthetic sum with the empty order is $\alpha$. Appending a greatest element produces the successor ordinal rule. At a limit ordinal $\lambda$, the second copy is the union of its initial segments of types $\beta <\lambda$, so the whole order has type
${\sup }_{\beta <\lambda }(\alpha +\beta )$. Thus the synthetic operation satisfies the inductive recursion; uniqueness in transfinite recursion proves the definitions equivalent.

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.