Zorn's Lemma and Equivalents of the Axiom of Choice

Intuition. The Axiom of Choice (AC) says: from any family of nonempty sets you can simultaneously pick one element from each. Several powerful, very different-looking statements turn out to be logically equivalent to AC, so accepting one forces accepting all.

Result (the equivalence circle).

• Zorn's Lemma: If $(A,\leq)$ is a poset in which every chain (totally ordered subset) has an upper bound in $A$, then $A$ has a maximal element.
• Hausdorff Maximal Principle: Every poset contains a maximal chain (maximal under inclusion).
• Well-Ordering Theorem: Every set can be well-ordered.
• Cardinal comparability (trichotomy): For cardinals $\alpha,\beta$, either $\alpha \leq \beta$ or $\beta \leq \alpha$.
• Surjection-to-injection: A surjection $f:X \twoheadrightarrow Y$ yields an injection $g:Y \hookrightarrow X$ (pick $g(y)\in f^{-1}(y)$).

Worked example. Why does a surjection $f:X\to Y$ give an injection back? Each fiber $f^{-1}(y)$ is nonempty; by AC choose $g(y)\in f^{-1}(y)$. Distinct $y$ have disjoint fibers, so $g$ is injective. This single step is exactly where AC is needed.

Pitfall: Zorn's Lemma uses an upper bound on every chain to produce a maximal element. The hypothesis pushes up, so the conclusion is maximal, never minimal. A statement that keeps the upper-bound hypothesis but concludes a minimal element is FALSE — e.g. the negative integers $\{-1,-2,-3,\dots\}$ are bounded above (by $-1$) yet have no minimal element.