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"Hausdorff's Maximal Principle" says that any partial order P has a maximal chain (chain = linear suborder). It is equivalent to the axiom of choice.

If we restrict Hausdorff's Maximal Principle to trees, i.e. partial orders, where a minimum element exists, and the sets $\{x \mid x < y\}$ form linear orders, what exactly do we get? Is it still equivalent to AC?

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  • $\begingroup$ Clearly $AC \implies HMP \implies HMP$ restricted. So the question is does $HMP $restricted$ \implies AC$ ? Probably not, or it would likely have been stated so with the minimum of dependencies. $\endgroup$ Commented Apr 15, 2016 at 14:06
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    $\begingroup$ I disagree. The restricted form is more complicated to state. And besides, Zorn's lemma which is both (a priori) weaker and more complicated is way more commonly used than HMP (because it is more convenient to use, I guess), so this does not amount to much. $\endgroup$
    – tomasz
    Commented Apr 15, 2016 at 14:15

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Even restricting this to well founded trees is enough to get $\sf DC_\kappa$ for every $\kappa$, which is enough to prove the axiom of choice.

So the answer is indeed positive.

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