The size of the class of countable sequences of ordinals

Abstract

Assume Z F + A D + D C R \mathsf {ZF} + \mathsf {AD} + \mathsf {DC}_\mathbb {R} . There is no injection of > ω 1 ω 1 {}^{>\omega _{1}}{\omega _{1}} (the set of countable length sequences of countable ordinals) into ω O N {}^\omega \mathrm {ON} (the class of ω \omega length sequences of ordinals). There is no injection of [ ω 1 ] ω 1 [{\omega _{1}}]^{{\omega _{1}}} (the powerset of ω 1 {\omega _{1}} ) into > ω 1 O N {}^{>{\omega _{1}}}\mathrm {ON} (the class of countable length sequences of ordinals).