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I am currently writing my master's thesis about Whitehead's problem in homological algebra. My main source is Eklof's article https://www.jstor.org/stable/2318684. Eklof does not name the Diamond principle in the article (though I know Shelah uses it in his original proof), but instead uses the following axiom, which he lists as a consequence of V=L.

Let C be a set which is the union of a strictly increasing smooth chain of countable sets $\{C_{\upsilon}\ |\ \upsilon<\omega_{1}\}$ (smoothness means that $C_\lambda = \bigcup_{\upsilon<\lambda} C_\upsilon$ for $\lambda$ a limit ordinal), and let $E$ be a stationary subset of $\omega_{1}$. Then there is a sequence $\{S_{\upsilon}\ |\ \upsilon\in E\}$ such that $S_{\upsilon}\subseteq C_{\upsilon}$ for all $\upsilon\in E$ and such that for any subset $X$ of $C$, the set of $\upsilon\in E$ with $X\cap C_{\upsilon} = S_{\upsilon}$ is stationary in $\omega_{1}$.

On my first read of the article, I assumed this was either equivalent to or a simple consequence of the Diamond Principle. However, after actually trying to prove it, I'm not so sure anymore. I have not been able to construct a proof, and I'm not even sure this is actually the Diamond Principle. Are there any users out there who might shed some light on this? If this axiom is a consequence of $\diamondsuit$, how do I prove this? If not, how is it related to $\diamondsuit$?

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    $\begingroup$ What does it mean for a chain of sets to be "smooth"? $\endgroup$ Commented Nov 11, 2021 at 20:01
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    $\begingroup$ Oh, I suppose that's not a common term. Let me update. $\endgroup$
    – Elswyyr
    Commented Nov 11, 2021 at 20:08

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To see that $\lozenge$ follows from this principle, simply set $C=\omega_1$ with $C_\nu=\nu$ and take $E=\rm Lim$, the club of limit ordinals. Then for every subset of $C$, that is $\omega_1$, the set of correct guesses is stationary.

In the opposite direction, it seems to me that Eklof defines $\lozenge_E$, which in general is stronger than $\lozenge$. Of course, under $V=L$ none of this matters, and ultimately, this is just a question of what is the most convenient tool for explaining Shelah's proof.

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  • $\begingroup$ I was really hoping that $\diamondsuit$ was sufficient, as I was trying to do a comparative analysis of the independence of Whitehead's problem and Suslin's problem. As such, I'd love to use the exact same extensions of ZFC to prove each, and I was hoping to avoid going through all of V = L. Do you have a reference on $\diamondsuit_E$, so I can read a little more about it? $\endgroup$
    – Elswyyr
    Commented Nov 11, 2021 at 22:03
  • $\begingroup$ You can find it mentioned in all the usual places like Jech and such (often written as $\lozenge(E)$. Assaf Rinot has a nice write up about diamond and its relatives. As for the proof that it is stronger, I couldn't track a reference, and had to ask a colleague to make sure I remembered correctly. $\endgroup$
    – Asaf Karagila
    Commented Nov 12, 2021 at 9:47

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