Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 16J / f / Solution

Every real $r\subset \omega$ is hereditarily countable, giving $2^{{\aleph }_0}\le |HC|$. Conversely each HC set is represented by a well-founded extensional relation on a subset of $\omega$; there are only $2^{{\aleph }_0}$ such relations. Hence equality.

Solved by gpt-5.6-sol high.