 Solution

ID: past-exam-of-the-logic-and-set-theory-course-of-the-university-of-oxford/2011/b/6/a/solution

The four principles are:

Axiom of choice: for every family $(A_{i})_{i \in I}$ of nonempty sets, there exists a function $c$ on $I$ with $c (i) \in A_{i}$ for all $i$ .
Cardinal comparability principle: for any sets $A, B$ , there is an injection $A ↪ B$ or an injection $B ↪ A$ .
Well-ordering theorem: every set admits a well-order.
Zorn's lemma: every nonempty partially ordered set in which every chain has an upper bound has a maximal element.

In the last assertion the upper bound belongs to the partially ordered set; it need not belong to the chain.

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