Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 16I (Logic and Set Theory)

The inductive definition of ordinal addition is the transfinite recursionfor every nonzero limit ordinal $\lambda$. The synthetic definition takes disjoint well-orders $A$ and $B$ of order types $\alpha$ and $\beta$, puts every member of $A$ before every member of $B$, and defines $\alpha +\beta$ to be the order type of this ordered sum.

These definitions agree by transfinite induction on $\beta$. The empty second order contributes nothing; adjoining a greatest element gives the successor clause; and a limit well-order is the union of all of its proper initial segments, so its ordered sum with $A$ has order type ${\sup }_{\gamma <\lambda }(\alpha +\gamma )$.

Always true. The synthetic definition identifies both sides with the order type obtained by concatenating well-orders of types $\alpha$, $\beta$, and $\gamma$ in that order. Thus ordinal addition is associative:

False. With $\alpha ={\omega }_1$ and $\beta ={\omega }_1+1$, both ordinals are uncountable, butOrdinal addition is therefore not commutative even for uncountable ordinals.

Always true. Under the synthetic definition of ordinal multiplication, $\alpha \beta$ is the order type of $\beta$ consecutive copies of $\alpha$. Grouping the copies first into $\gamma$ blocks of $\beta$ copies, or first into single blocks of type $\alpha \beta$, gives the same lexicographic well-order. Hence

False. Take $\alpha =\omega$ and $\beta =\omega \cdot 2$. Thenbut associativity and the limit clause for ordinal multiplication give