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Stream: learning: questions

Topic: ✔ $$AC_{\mathbb{N}, 2}$$ and the fan theorem

Madeleine Birchfield (Dec 16 2024 at 15:33):

Does $\mathrm{AC}_{\mathbb{N}, 2}$ (i.e. countable choice for subsets of the boolean domain) imply the fan theorem?

Ryota Kuroki (Dec 17 2024 at 02:09):

Without LEM, I think even countable choice doesn't imply the fan theorem.

Madeleine Birchfield (Dec 17 2024 at 04:21):

You know what, I think you're right. What countable choice and $AC_{\mathbb{N}, 2}$ allows is for the fan theorem to imply that the locale of Dedekind real numbers be a spatial locale. But the fan theorem is still needed in the first place.

Notification Bot (Dec 17 2024 at 04:22):

Madeleine Birchfield has marked this topic as resolved.