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Aug 8, 2019 at 20:30 comment added Asaf Karagila Well, in the proof that $\lozenge$ implies that every Whitehead group is free there is a characterization of what and where things fail. To my understanding that was the origin of club guessing, too.
Aug 8, 2019 at 19:09 comment added Sean Cox I would be interested in knowing if Whitehead groups of size $\omega_1$ can be characterized by strong $\mathcal{C}$-filtering, for some $\mathcal{C}$ that satisfies the requirements of the question (assuming we're in a universe where there is a non-free Whitehead group, of course).
Aug 8, 2019 at 16:43 comment added Asaf Karagila Whitehead's Problem comes to mind...
Aug 8, 2019 at 13:22 history edited Sean Cox CC BY-SA 4.0
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Aug 7, 2019 at 22:54 history edited Sean Cox CC BY-SA 4.0
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Aug 7, 2019 at 19:59 history edited Sean Cox CC BY-SA 4.0
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Aug 7, 2019 at 18:51 history edited Sean Cox CC BY-SA 4.0
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Aug 7, 2019 at 18:46 history asked Sean Cox CC BY-SA 4.0