Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 19 / 2 / i

In the usual tail-normalized version, an Ulam matrix on omega-one is a family $A_{\alpha ,n}$ such that, for each $\alpha <{\omega }_1$, the sets indexed by $n<\omega$ partition the tail $(\alpha ,{\omega }_1)$, while for each fixed $n$ the sets indexed by $\alpha <{\omega }_1$ are pairwise disjoint. Empty cells are allowed; harmless bounded-tail variants give the same applications.

For a concrete realization, choose injections $e_{\beta }:\beta \to \omega$ for every countable ordinal $\beta$, and setFor fixed $\alpha$ each $\beta >\alpha$ chooses exactly one $n$; for fixed $n$, injectivity of $e_{\beta }$ prevents one $\beta$ from belonging to two cells. This verifies the two matrix conditions explicitly.

König theorem for cardinal numbers states that if ${\kappa }_i<{\lambda }_i$ for each $i\in I$, then ${\sum }_i{\kappa }_i<{\prod }_i{\lambda }_i$. Its cofinality formulation gives, for every infinite cardinal $\kappa$,Taking a cofinal sequence of smaller cardinals in $\kappa$ and comparing their sum with the product of their successors gives this inequality. Another standard consequence, for infinite $\lambda$ and $\kappa \ge 2$, isIndeed, if $\mu =\mathrm{cf}({\kappa }^{\lambda })\le \lambda$, the preceding inequality at ${\kappa }^{\lambda }$ would contradict $({\kappa }^{\lambda }{)}^{\mu }={\kappa }^{\lambda \mu }={\kappa }^{\lambda }$.

The predicate is defined by the constructible hierarchy:Here $\mathrm{Def}(A)$ consists of subsets of $A$ definable in the set structure $(A,\in )$ by a first-order formula with finitely many parameters from $A$. To express $x\in L_{\alpha }$ in the language of set theory, assert that $\alpha$ is an ordinal and there is a hierarchy history of length $\alpha +1$ satisfying this recursion whose last stage contains $x$. Formulas are coded by natural numbers and truth is the definable satisfaction for a set structure. Transfinite recursion gives a unique history. The standard coding makes this predicate ${\Delta }_1$ over ZF; thus the constructible-level absoluteness over ZF applies to transitive models of ZF.