Jensen's covering theorem

Remarks

The proof by Jensen uses the fine structure theory of $L$; one takes an elementary substructure that contains $X$ and analyses its transitive collapse. A consequence: if $0^\sharp$ does not exist, then every singular cardinal $\kappa$ of $V$ is singular in $L$, and $(\kappa^+)^L = \kappa^+$. It also follows that the singular cardinals hypothesis holds when $0^\sharp$ does not exist. The hypothesis is necessary. If $0^\sharp$ exists, then $\aleph_\omega$ is a regular cardinal in $L$. Then the set $X = \omega_1 \cup \set{\aleph_n : n < \omega}$ of size $\aleph_1$ cannot be covered. Indeed, a set $Y \in L$ with $\abs{Y} = \aleph_1$ has order type less than $\omega_2$. So $Y \cap \aleph_\omega$ has $L$-cardinality less than $\aleph_\omega$, and it is bounded in $\aleph_\omega$.

Going deeper

For a student

The theorem is the model for many covering results for larger inner models, such as the Dodd–Jensen core model and the core model in general. It turns the nonexistence of a large cardinal object into strong combinatorial consequences, such as the singular cardinals hypothesis. Read the other way, it gives lower bounds for consistency strength. If the singular cardinals hypothesis fails, then $0^\sharp$ exists, and in fact much stronger large cardinal hypotheses hold in inner models.