116c- Lecture 20

We showed that implies a weak version of choice, , namely, every countable family of non-empty sets of reals admits a choice function. This implies that 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 for any real and therefore is strongly inaccessible in for any real .

We closed the course by showing that, in fact, 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 of the Turing degrees and then “pulling back” this measure to .

Finally, given any measurable cardinal , let be a (-complete, non-principal) measure on . Then is a model of choice in which is measurable. In particular,

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

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