116c- Lecture 20

We showed that \mathsf{AD} implies a weak version of choice, \mathsf{AC}_\omega({}^\omega\omega), namely, every countable family of non-empty sets of reals admits a choice function. This implies that \omega_1 is regular and suffices to develop classical analysis in a straightforward fashion (in particular, to construct Lebesgue measure and to prove its basic properties).

Coupled with the fact that all sets of reals have the perfect set property, this implies that \omega_1>\omega_1^{L[x]} for any real x and therefore \omega_1^V is strongly inaccessible in L[x] for any real x.

We closed the course by showing that, in fact, \omega_1 is a measurable cardinal. We proved this result of Solovay by showing Martin’s result that the “cone measure” is indeed a non-atomic measure on the structure {\mathcal D} of the Turing degrees and then “pulling back” this measure to \omega_1.

Finally, given any measurable cardinal \kappa, let \mu be a (\kappa-complete, non-principal) measure on \kappa. Then L[\mu] is a model of choice in which \kappa is measurable. In particular, {\rm Con}({\sf ZF}+{\sf AD})\Rightarrow{\rm Con}({\sf ZFC}+\mbox{There is a measurable cardinal}).

Since, under choice, any measurable cardinal is strongly inaccessible and the limit of strongly inaccessible cardinals, this shows that {\sf AD} has significant consistency strength.

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