Observations about Scott and Karp trees Original Research Article

Hyttinen and Väänänen (1990) study extensively the so-called Scott and Karp trees. Their paper leaves some open interesting questions: 1. (1) Are Scott trees closed under infimums? 2. (2) Are Karp trees closed under infimums? 3. (3) Does every Karp tree contain a subtree of small cardinality which is itself also a Karp tree? The present article addresses these questions. It turns out that there are counterexamples dictating a negative answer to (1) and (2). The answer to question (3), however, is independent of the standard ZFC axioms of set theory provided that the existence of a strongly compact cardinal is consistent.