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A model for a very good scale and a bad scale. (English) Zbl 1156.03045

Summary: Given a supercompact cardinal \(\kappa \) and a regular cardinal \(\lambda < \kappa \), we describe a type of forcing such that in the generic extension the cofinality of \(\kappa \) is \(\lambda \), there is a very good scale at \(\kappa \), a bad scale at \(\kappa \), and SCH at \(\kappa \) fails. When creating our model we have great freedom in assigning the value of \(2^{\kappa }\), and so we can make SCH hold or fail arbitrarily badly.

MSC:

03E04 Ordered sets and their cofinalities; pcf theory
03E35 Consistency and independence results
03E55 Large cardinals
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[1] DOI: 10.1016/0003-4843(78)90031-1 · Zbl 0376.02055 · doi:10.1016/0003-4843(78)90031-1
[2] Cardinal arithmetic 29 (1994) · Zbl 0848.03025
[3] Fundamenta Mathematical 99 pp 61–
[4] DOI: 10.1142/S021906130100003X · Zbl 0988.03075 · doi:10.1142/S021906130100003X
[5] Set theory (2003)
[6] DOI: 10.1090/S0002-9939-07-08716-3 · Zbl 1140.03033 · doi:10.1090/S0002-9939-07-08716-3
[7] DOI: 10.1007/BF02761175 · Zbl 0381.03039 · doi:10.1007/BF02761175
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