TY - JOUR
T1 - Strongly compact cardinals and the continuum function
AU - Apter, Arthur W.
AU - Dimopoulos, Stamatis
AU - Usuba, Toshimichi
N1 - Funding Information:
The research of the first author was partially supported by PSC-CUNY Grant 63505-00-51 . The research of the third author was supported by JSPS KAKENHI Grant Nos. 18K03403 and 18K03404 . The authors wish to thank the referee for suggestions which have been incorporated into the current version of the paper.
Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2021/10/1
Y1 - 2021/10/1
N2 - We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals. We begin by showing that it is possible to force violations of GCH at an arbitrary strongly compact cardinal using only strong compactness as our initial assumption. This result is due to the third author. We then investigate realising Easton functions at and above the least measurable limit of supercompact cardinals starting from an initial assumption of the existence of a measurable limit of supercompact cardinals. By results due to Menas, assuming 2κ=κ+, the least measurable limit of supercompact cardinals κ is provably in ZFC a non-supercompact strongly compact cardinal which is not κ+-supercompact. We also consider generalisations of our earlier theorems in the presence of more than one strongly compact cardinal. We conclude with some open questions.
AB - We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals. We begin by showing that it is possible to force violations of GCH at an arbitrary strongly compact cardinal using only strong compactness as our initial assumption. This result is due to the third author. We then investigate realising Easton functions at and above the least measurable limit of supercompact cardinals starting from an initial assumption of the existence of a measurable limit of supercompact cardinals. By results due to Menas, assuming 2κ=κ+, the least measurable limit of supercompact cardinals κ is provably in ZFC a non-supercompact strongly compact cardinal which is not κ+-supercompact. We also consider generalisations of our earlier theorems in the presence of more than one strongly compact cardinal. We conclude with some open questions.
KW - Easton function
KW - Strongly compact cardinal
KW - Supercompact cardinal
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U2 - 10.1016/j.apal.2021.103013
DO - 10.1016/j.apal.2021.103013
M3 - Article
AN - SCOPUS:85108313639
SN - 0168-0072
VL - 172
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 9
M1 - 103013
ER -