The size of the class of countable sequences of ordinals

Chan, William; Jackson, Stephen; Trang, Nam

doi:10.1090/tran/8573

The size of the class of countable sequences of ordinals
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by William Chan, Stephen Jackson and Nam Trang PDF

Trans. Amer. Math. Soc. 375 (2022), 1725-1743 Request permission

Abstract:

Assume $\mathsf {ZF} + \mathsf {AD} + \mathsf {DC}_\mathbb {R}$. There is no injection of ${}^{<\omega _{1}}{\omega _{1}}$ (the set of countable length sequences of countable ordinals) into ${}^\omega \mathrm {ON}$ (the class of $\omega$ length sequences of ordinals). There is no injection of $[{\omega _{1}}]^{{\omega _{1}}}$ (the powerset of ${\omega _{1}}$) into ${}^{<{\omega _{1}}}\mathrm {ON}$ (the class of countable length sequences of ordinals).

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