Can a Large Cardinal Be Forced from a Condition Implying Its Negation?

In this note, we provide an affirmative answer to the title question by giving two examples of cardinals satisfying conditions implying they are non-Rowbottom which can be turned into Rowbottom cardinals via forcing. In our second example, our cardinal is also non-Jonsson. A well-known phenomenon is that under certain circumstances, it is possible to force over a given model V of ZFC containing a cardinal κ satisfying a large cardinal property φ(κ) to create a universe V in which φ(κ) no longer holds, yet over which φ(κ) can be resurrected via forcing. A folklore example of this is provided by supposing that V “ZFC + GCH + κ is measurable”. If P is defined as the reverse Easton iteration of length κ which adds a Cohen subset to each inaccessible cardinal below κ, then in V , κ is no longer measurable, by, e.g., a simpler version of the argument given in Lemma 2.4 of [1]. However, if one forces over V P by adding a Cohen subset to κ, then the standard reverse Easton arguments show that κ’s measurability has been resurrected, since the entire forcing can now be viewed as the length κ+ 1 reverse Easton iteration over V that adds a Cohen subset to each inaccessible cardinal less than or equal to κ. In this note, we consider a different, stronger phenomenon. Specifically, we examine the following. Question. Suppose φ is a given large cardinal property. Is it possible to find a formula ψ in one free variable in the language of set theory and a cardinal κ such that ψ(κ) holds, ZFC “∀λ[ψ(λ) =⇒ ¬φ(λ)]”, yet there is a partial ordering P such that P φ(κ)? We show that the answer to our Question is yes for φ the large cardinal property of Rowbottomness by proving the following theorem. Received by the editors August 30, 2003 and, in revised form, February 14, 2004 and June 15, 2004. 2000 Mathematics Subject Classification. Primary 03E02, 03E35, 03E55.