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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 α<β<γ such that α is γ-supercompact; β is the first measurable cardinal greater than α; and γ=|2β|." 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(1,): Every set A of non-empty sets such that |A|=1 has a choice function. Moore, G. [1982], p. 202.

86-alpha

C(α,): If X is a set of non-empty sets such that |X|=α, 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 α is weakly compact and α+1 is measurable.

Historical background: Then usingcollapsing functions, he constructs an inner model, M42, in whichβ is collapsed to α+1. He proves thatα is an uncountable, weakly compact cardinal andα+1 is measurable (321 is false). Bull also proves thatC(α,) (86) is true in this model. Since α1,form 39 (C(1,)) 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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