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I was reading Wilansky's book Modern methods in Topological Vector Spaces and came across this problem on set theory (p. 7):

"The maximal axiom for countable posets is equivalent to induction."

He gives no clue as to what he refers to as induction, but I believe it is something along the lines: if $1\in A \subseteq \mathbb{N}$, and $n+1 \in A$ whenever $n \in A$, then $A=\mathbb{N}$. His maximal axiom reads: if $(X,\geq)$ is a partially ordered set, then it has a $\supseteq$-maximal chain $Y$ (that is, if $Z$ is any other chain in $X$ and satisfies $Y \subseteq Z$, then $Z=Y$). This is Hausdorff maximal principle. The maximal axiom for countable posets is when $X$ is countable.

I am not really interested in this part of the book, but I am puzzled about this problem. Specially because I have no idea on how to approach it. Any hint, reference, or comment is appreciated!

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    @Andreas: My memory is worse (sometimes) than the alleged three-seconds memory o$f$ a goldfish. One post from over a $y$ear ago is a tough cookie. I do recall approaching this question with a foundation approach of much less than ZF (i.e. some theory which cannot really talk about the enumeration internally).2012-12-24

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