Cohen M42: Bull's Model | Back to this models page
Description: Let M be a countable transitive model of ZFC+ "There are uncountable regular cardinals ℵα<ℵβ<ℵγ such that ℵα is ℵγ-supercompact; ℵβ is the first measurable cardinal greater than ℵα; and ℵγ=|2ℵβ|." Using backward Easton forcing (which is due to Silver), Bull constructs a generic extension of M
When the book was first being written, only the following form classes were known to be true in this model:
Form Howard-Rubin Number | Statement |
---|---|
39 | C(ℵ1,∞): Every set A of non-empty sets such that |A|=ℵ1 has a choice function. Moore, G. [1982], p. 202. |
86-alpha | C(ℵα,∞): If X is a set of non-empty sets such that |X|=ℵα, then X has a choice function. |
When the book was first being written, only the following form classes were known to be false in this model:
Form Howard-Rubin Number | Statement |
---|---|
91 | PW: The power set of a well ordered set can be well ordered. |
163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
321 | There does not exist an ordinal α such that ℵα is weakly compact and ℵα+1 is measurable. |
Historical background: Then usingcollapsing functions, he constructs an inner model, M42, in whichℵβ is collapsed to ℵα+1. He proves thatℵα is an uncountable, weakly compact cardinal andℵα+1 is measurable (321 is false). Bull also proves thatC(ℵα,∞) (86) is true in this model. Since α≥1,form 39 (C(ℵ1,∞)) is also true. Form 163 (Every set iseither well orderable or has an infinite Dedekind finite subset.) is falsebecause 39 implies 8 and 8 + 163 implies AC (Brunner [1982a]).
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