TY - JOUR
T1 - On the strong equality between supercompactness and strong compactness
AU - After, Arthur W.
AU - Shelah, Saharon
PY - 1997
Y1 - 1997
N2 - We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V = ZFC 4- GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V[G] = ZFC + GCH in which, (a) (preservation) for κ ≤ γ regular, if V = "κ is λ supercompact", then V[G] = "κ is λ supercompact" and so that, (b) (equivalence) for κ ≤ λ regular, V[G] = " κ is A strongly compact" iff V[G] = "κ is λ supercompact", except possibly if κ is a measurable limit of cardinals which are λ supercompact.
AB - We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V = ZFC 4- GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V[G] = ZFC + GCH in which, (a) (preservation) for κ ≤ γ regular, if V = "κ is λ supercompact", then V[G] = "κ is λ supercompact" and so that, (b) (equivalence) for κ ≤ λ regular, V[G] = " κ is A strongly compact" iff V[G] = "κ is λ supercompact", except possibly if κ is a measurable limit of cardinals which are λ supercompact.
KW - Strongly compact cardinal
KW - Supercompact cardinal
UR - http://www.scopus.com/inward/record.url?scp=21444455339&partnerID=8YFLogxK
U2 - 10.1090/s0002-9947-97-01531-6
DO - 10.1090/s0002-9947-97-01531-6
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AN - SCOPUS:21444455339
SN - 0002-9947
VL - 349
SP - 103
EP - 128
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 1
ER -