Subtle cardinals were first introduced in a paper by Jensen and Kunen [JK]. They show that if κ is subtle then ◇κ holds. Subtle cardinals also play an important role in [B1], where Baumgartner proposed that certain large cardinal properties should be considered as properties of their associated normal ideals. He shows that in the case of ineffables, the ideals are particularly useful, as can be seen by the following theorem, κ is ineffable if and only if κ is subtle and Π½-indescribable and the subtle and Π½-indescribable ideals cohere, i.e. they generate a proper, normal ideal (which in fact turns out to be the ineffable ideal).
In this paper we examine properties of subtle cardinals and consider methods of forcing that destroy the property of subtlety while maintaining other properties. The following is a list of results.
1) We relativize the following two facts about subtle cardinals:
i) if κ is n-subtle then {α < κ: α is not n-subtle} is n-subtle, and
ii) if κ is (n + 1)-subtle then {α < κ: α is n-subtle} is in the (n + 1)-subtle filter to subsets of κ:
i′) if A is an n-subtle subset of κ then {α ϵ A: A ∩ α is not n-subtle} is n-subtle, and
ii′) if A is an (n + 1)-subtle subset of κ then {α ϵ A: A ∩ α is n-subtle} is (n + 1)-subtle.
2) We show that although a stationary limit of subtles is subtle, a subtle limit of subtles is not necessarily 2-subtle.
3) In §3 we use the technique of forcing to turn a subtle cardinal into a κ-Mahlo cardinal that is no longer subtle.
4) In §4 we extend the results of §3 by showing how to turn an (n + 1)-subtle cardinal into an n-subtle cardinal that is no longer (n + 1)-subtle.