Logic Seminar

This talk will compute the relations between the cardinality of some sets under determinacy. Woodin showed under ZFZF, dependent choice, and real determinacy that the set of countable sequences of countable ordinals does not inject into the set of ωω-sequences of countable ordinals and in fact does not even inject into the class of ωω-sequences of ordinals. This argument passes through a set called S1S1 and uses AD+AD+ techniques involving ∞∞-Borel codes, inner models of choice, and forcing arguments. In this talk, we will show a continuity result for functions from the set of sequences of countable ordinals of a fixed countable length into ω1ω1. This continuity result will be used to show in just ADAD that the set of ωω-sequences of countable ordinals has strictly smaller cardinality than the set of countable-length sequences of countable ordinals. Then under ADAD and dependent choice for the reals, this result along with category arguments, generic coding, and a bounding result of Steel will be used to show that the set of countable sequences of countable ordinals does not inject into the class of ωω-sequences of ordinals. These arguments are combinatorial and are more adaptable to the analogous questions for ω2ω2 and the odd projective ordinals. This is joint work with Stephen Jackson and Nam Tran.