Explain what is meant by a chain-complete poset. State the Bourbaki-Witt fixedpoint theorem.

We call a poset $(P, \leqslant)$ Bourbakian if every order-preserving map $f: P \rightarrow P$ has a least fixed point $\mu(f)$. Suppose $P$ is Bourbakian, and let $f, g: P \rightrightarrows P$ be order-preserving maps with $f(x) \leqslant g(x)$ for all $x \in P$; show that $\mu(f) \leqslant \mu(g)$. [Hint: Consider the function $h: P \rightarrow P$ defined by $h(x)=f(x)$ if $x \leqslant \mu(g), h(x)=\mu(g)$ otherwise.]

Suppose $P$ is Bourbakian and $f: \alpha \rightarrow P$ is an order-preserving map from an ordinal to $P$. Show that there is an order-preserving map $g: P \rightarrow P$ whose fixed points are exactly the upper bounds of the set $\{f(\beta) \mid \beta<\alpha\}$, and deduce that this set has a least upper bound.

Let $C$ be a chain with no greatest member. Using the Axiom of Choice and Hartogs' Lemma, show that there is an order-preserving map $f: \alpha \rightarrow C$, for some ordinal $\alpha$, whose image has no upper bound in $C$. Deduce that any Bourbakian poset is chain-complete.