Let be strongly inaccessible cardinal in the ground model and perform the finite Lévy collapse to omega-one. The ground full binary set-theoretic tree of height has all levels of size less than , so these levels become countable. Its at least ground branches remain distinct. The chain in a partial order condition preserves , which is the extension . The unchanged ground set-theoretic tree therefore witnesses the Kurepa hypothesis.
There is of size such that is countable for every . Its initial-segment tree, pruned and leveled to be normal, is a Kurepa tree.
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A Kurepa tree is a type of mathematical structure that arises in set theory and combinatorial set theory, named after the mathematician E. Kurepa. It is a special kind of tree that is used to study the properties of certain kinds of sets and their cardinalities. Specifically, a Kurepa tree is an infinite tree that satisfies two primary conditions: 1. **Uncountably many branches**: Every branch (i.e.