Identity crises and strong compactness III: Woodin cardinals

We show that it is consistent, relative to n ∈ ω supercompact cardinals, for the strongly compact and measurable Woodin cardinals to coincide precisely. In particular, it is consistent for the first n strongly compact cardinals to be the first n measurable Woodin cardinals, with no cardinal above the nth strongly compact cardinal being measurable. In addition, we show that it is consistent, relative to a proper class of supercompact cardinals, for the strongly compact cardinals and the cardinals which are both strong cardinals and Woodin cardinals to coincide precisely. We also show how the techniques employed can be used to prove additional theorems about possible relationships between Woodin cardinals and strongly compact cardinals.

[1]  Arthur W. Apter.  Characterizing strong compactness via strongness , 2003, Math. Log. Q..

[2]  Arthur W. Apter.  A new proof of a theorem of Magidor , 2000, Arch. Math. Log..

[3]  Joel David Hamkins,et al.  Extensions with the approximation and cover properties have no new large cardinals , 2003, math/0307229.

[4]  Joel David Hamkins.  Gap forcing: Generalizing the Lévy-Solovay theorem , 1999, Bull. Symb. Log..

[5]  Joel David Hamkins.  The Lottery Preparation , 2000, Ann. Pure Appl. Log..

[6]  Richard Laver,et al.  Making the supercompactness of κ indestructible under κ-directed closed forcing , 1978 .

[7]  Akihiro Kanamori,et al.  Strong axioms of infinity and elementary embeddings , 1978 .

[8]  James Cummings,et al.  Squares, scales and stationary Reflection , 2001, J. Math. Log..

[9]  Saharon Shelah,et al.  On certain indestructibility of strong cardinals and a question of Hajnal , 1989, Arch. Math. Log..

[10]  A. Levy,et al.  Measurable cardinals and the continuum hypothesis , 1967 .

[11]  Arthur W. Apter.  On the least strongly compact cardinal , 1980 .

[12]  Arthur W. Apter.  On the first $n$ strongly compact cardinals , 1995 .

[13]  Joel David Hamkins,et al.  Small forcing creates neither strong nor Woodin cardinals , 1998 .

[14]  Menachem Magidor.  How large is the first strongly compact cardinal? or a study on identity crises , 1976 .

[15]  Arthur W. Apter,et al.  Identity crises and strong compactness , 2001, Arch. Math. Log..

[16]  Arthur W. Apter.  Some remarks on indestructibility and Hamkins’ lottery preparation , 2003, Arch. Math. Log..

[17]  A. Kanamori.  The Higher Infinite , 1994 .

[18]  Joel David Hamkins.  Gap forcing , 2001 .