Axiom Of Choice

Cohen $\cal M42$ : Bull's Model | Back to this models page

Description: Let $\cal M$ be a countable transitive model of $ZFC +$ "There are uncountable regular cardinals $\aleph_\alpha <\aleph_\beta < \aleph_\gamma$ such that $\aleph_\alpha$ is $\aleph_\gamma$ -supercompact; $\aleph_\beta$ is the first measurable cardinal greater than $\aleph_\alpha$ ; and $\aleph_\gamma =|2^{\aleph_\beta}|$ ." Using backward Easton forcing (which is due to Silver), Bull constructs a generic extension of $\cal M$

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	$C(\aleph_{1},\infty)$ : Every set $A$ of non-empty sets such that $\vert A\vert = \aleph_{1}$ has a choice function. Moore, G. [1982], p. 202.
86-alpha	$C(\aleph_{\alpha},\infty)$ : If $X$ is a set of non-empty sets such that $\|X\| = \aleph_{\alpha }$ , 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 $\alpha$ such that $\aleph_{\alpha}$ is weakly compact and $\aleph_{\alpha+1}$ is measurable.

Historical background: Then usingcollapsing functions, he constructs an inner model, $\cal M42$ , in which $\aleph_\beta$ is collapsed to $\aleph_{\alpha+1}$ . He proves that $\aleph_\alpha$ is an uncountable, weakly compact cardinal and $\aleph_{\alpha+1}$ is measurable (321 is false). Bull also proves that $C(\aleph_\alpha,\infty)$ (86) is true in this model. Since $\alpha\ge 1$ ,form 39 ( $C(\aleph_1,\infty)$ ) 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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