Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 5

König's theorem for cardinal numbers states that, for a set-indexed family with ${\kappa }_i<{\lambda }_i$ at every index,Here the sum is the cardinality of a disjoint union and the product is the cardinality of a Cartesian product. Work in ZFC. Choose disjoint sets $A_i$ of sizes ${\kappa }_i$ and sets $B_i$ of sizes ${\lambda }_i$. For any function $f:{⨆}_i{A}_i\to {\prod }_i{B}_i$, its $i$th coordinate values on $A_i$ form a subset of $B_i$ of cardinality at most ${\kappa }_i$. Choose $b_i\in B_i$ outside that subset, using choice. Then $b=(b_i{)}_i$ cannot equal $f(a)$ for any $a$: if $a\in A_j$, its $j$th coordinate was specifically excluded. Thus there is no surjection from the disjoint union onto the product.

Cardinal comparability under choice now implies the displayed strict inequality. Empty index set gives $0<1$ directly; empty individual $A_i$ cause no difficulty, since the corresponding coordinate excludes an empty image. This is the cardinal diagonal theorem, rather than the finitely branching tree result also bearing König's name.

The generalized continuum hypothesis isEquivalently, for every ordinal $\alpha$, $2^{{\aleph }_{\alpha }}={\aleph }_{\alpha +1}$. Here $2^{\kappa }$ is the cardinality of the power set of a set of cardinality $\kappa$, and ${\kappa }^{+}$ is the least strictly larger cardinal. The hypothesis says that no intermediate cardinal lies between an infinite set and its power set.

For GCH arithmetic at aleph omega, let $\kappa ={\aleph }_{\omega }$. Use monotonicity of cardinal exponentiation, the law $(a^b{)}^c=a^{bc}$, and $\kappa \cdot \kappa =\kappa$ for infinite cardinals in ZFC. Since $2\le \kappa \le 2^{\kappa }$,Thus ${\kappa }^{\kappa }=2^{\kappa }$, and the generalized continuum hypothesis yieldsThe lower bound embeds binary-valued functions into $\kappa$-valued ones; the upper bound codes each value by a subset and combines the resulting pairs of indices. In particular the equality before applying the hypothesis is a ZFC equality.

Write again $\kappa ={\aleph }_{\omega }$. The countable cardinal sum ${\sum }_{n<\omega }{\aleph }_n$ equals $\kappa$: it is at least their supremum, and at most ${\aleph }_0\cdot \kappa =\kappa$. König's theorem givesThe latter product injects into $P={\prod }_{n<\omega }{\aleph }_n$ by fixing coordinate zero and putting its $n$th coordinate in coordinate $n+1$. Conversely $P$ injects into the set of all functions $\omega \to \kappa$. ThereforeNo cardinal lies strictly between $\kappa$ and its successor, soThe strict lower bound is essential: replacing the product by the supremum of its factors would incorrectly give ${\aleph }_{\omega }$.

The product from the preceding part injects into ${\kappa }^{{\aleph }_0}$, where $\kappa ={\aleph }_{\omega }$, because each coordinate value is an ordinal below $\kappa$. Combining its strict König lower bound with the exponentiation upper bound givesHenceOnly the instance $2^{{\aleph }_{\omega }}={\aleph }_{\omega +1}$ of the generalized continuum hypothesis was needed in these three computations, though the question assumes the full hypothesis. No assumption that countable exponentiation leaves this singular cardinal unchanged is valid: König's theorem rules that out.