Axiom of Choice Implies Zorn's Lemma
Theorem
Acceptance of the Axiom of Choice implies the truth of Zorn's Lemma.
Statement of Zorn's Lemma
Let $\left({X, \preceq}\right), X \ne \varnothing$ be a non-empty ordered set such that every non-empty chain in $X$ has an upper bound in $X$.
Then $X$ has at least one maximal element.
Proof 1
For each $x \in X$, consider the lower closure $x^\preceq$:
- $x^\preceq = \set {y \in X: y \preceq x}$
Let $\mathbb S \subseteq \powerset X$ be the image of $\cdot^\preceq$ considered as a mapping from $X$ to $\powerset X$, where $\powerset X$ is the power set of $X$.
From Ordering is Equivalent to Subset Relation:
- $\forall x, y \in X: x^\preceq \subseteq y^\preceq \iff x \preceq y$
Thus the task of finding a maximal element of $X$ is equivalent to finding a maximal set in $\mathbb S$.
Thus the statement of the result is equivalent to a statement about chains in $\mathbb S$:
- Let $\mathbb S$ be a non-empty subset of $\powerset X, X \ne \O$ such that every non-empty chain in $\mathbb S$, ordered by $\subseteq$, has an upper bound in $\mathbb S$.
- Then $\mathbb S$ has at least one maximal set.
Let $\mathbb X$ be the set of all chains in $\struct {X, \preceq}$.
Every element of $X$ is included in $\map {s^\preceq} x$ for some $x \in X$.
$\mathbb X$ is a non-empty set of sets which are ordered (perhaps partially) by subset relation.
If $\mathcal C$ is a chain in $\mathbb X$, then:
- $\displaystyle \bigcup_{A \mathop \in \mathcal C} A \in \mathbb X$
Since each set in $\mathbb X$ is dominated by some set in $\mathbb S$, going from $\mathbb S$ to $\mathbb X$ can not introduce any new maximal elements.
The main advantage of using $\mathbb X$ is that the chain hypothesis is in a slightly more specific form.
Instead of saying that each chain in $\mathcal C$ has an upper bound in $\mathbb S$, we can explicitly state that the union of the sets of $\mathcal C$ is an element of $\mathbb X$.
This union of the sets of $\mathcal C$ is clearly an upper bound of $\mathcal C$.
Another advantage of $\mathbb X$ is that, from Subset of Toset is Toset, it contains all the subsets of each of its sets.
Thus we can enlarge non-maximal sets in $\mathbb X$ one element at a time.
So, from now on, we need consider only this non-empty collection $\mathbb X$ of subsets of a non-empty set $X$.
$\mathbb X$ is subject to two conditions:
- $(1): \quad$ Every subset of each set in $\mathbb X$ is in $\mathbb X$.
- $(2): \quad$ The union of each chain of sets in $\mathbb X$ is in $\mathbb X$.
It follows from $(1)$ that $\O \in \mathbb X$.
We need to show that there exists a maximal set in $\mathbb X$.
Let $f$ be a choice function for $X$:
- $\forall A \in \powerset X \setminus \O: \map f A \in A$
For each $A \in \mathbb X$, let $\hat A$ be defined as:
- $\hat A := \set {x \in X: A \cup \set x \in \mathbb X}$
That is, $\hat A$ consists of all the elements of $X$ which, when added to $A$, make a set which is also in $\mathbb X$.
From its definition:
- $\displaystyle \hat A = \bigcup_{x \mathop \in \hat A} \paren {A \cup \set x}$
where each of $A \cup \set x$ are chains in $X$ and so elements of $\mathbb X$.
Suppose there exists a maximal set $M$ in $\mathbb X$.
Then, by definition of maximal, there are no elements in $x \in X \setminus M$ which can be added to $M$ to make $M \cup \set x$ another element of $\mathbb X$.
Thus it follows that $\hat M = M$.
From Set Difference with Self is Empty Set it then follows that if $A$ is maximal, then $\hat A \setminus A = \O$.
The mapping $g: \mathbb X \to \mathbb X$ can now be defined as:
- $\forall A \in \mathbb X: \map g A = \begin{cases} A \cup \set {\map f {\hat A \setminus A} } & : \hat A \setminus A \ne \O \\ A & : \text{otherwise} \end{cases}$
Thus what we now have to prove is that:
- $\exists A \in \mathbb X: \map g A = A$
Note that from the definition of $g$:
- $\forall A \in \mathbb X: A \subseteq \map g A$
Also note that $\map f {\hat A \setminus A}$ is a single element of $\hat A \setminus A$.
Thus we obtain the crucial fact that $\map g A$ contains at most one more element than $A$.
We (temporarily) define a tower as being a subset $\mathcal T$ of $\mathbb X$ such that:
- $(1): \quad \O \in \mathcal T$
- $(2): \quad A \in \mathcal T \implies \map g A \in \mathcal T$
- $(3): \quad $ If $\mathcal C$ is a chain in $\mathcal T$, then $\displaystyle \bigcup_{A \mathop \in \mathcal C} A \in \mathcal T$
There is of course at least one tower in $\mathbb X$, as $\mathbb X$ itself is one.
It follows from its definition that the intersection of a collection of towers is itself a tower.
It follows in particular that if $\mathcal T_0$ is the intersection of all towers in $\mathbb X$, then $\mathcal T_0$ is the smallest tower in $\mathbb X$.
Next we demonstrate that $\mathcal T_0$ is a chain.
We (temporarily) define a set $C \in \mathcal T_0$ as comparable if it is comparable with every element of $\mathcal T_0$.
That is, if $A \in \mathcal T_0$ then $C \subseteq A$ or $A \subseteq C$.
To say that $\mathcal T_0$ is a chain means that all sets of $\mathcal T_0$ are comparable.
There is at least one comparable set in $\mathcal T_0$, as $\O$ is one of them.
So, suppose $C \in \mathcal T_0$ is comparable.
Let $A \in \mathcal T_0$ such that $A \subsetneq C$.
Consider $\map g A$.
Because $C$ is comparable, either $C \subsetneq \map g A$ or $\map g A \subseteq C$.
In the former case $A$ is a proper subset of a proper subset of $\map g A$.
This contradicts the fact that $\map g A \setminus A$ can be no more than a singleton.
Thus if such an $A$ exists, we have that:
- $(A): \quad \map g A \subseteq C$
Now let $\mathcal U$ be the set defined as:
- $\mathcal U := \set {A \in \mathcal T_0: A \subseteq C \lor \map g C \subseteq A}$
Let $\mathcal U'$ be the set defined as:
- $\mathcal U' := \set {A \in \mathcal T_0: A \subseteq \map g C \lor \map g C \subseteq A}$
That is, $\mathcal U'$ is the set of all sets in $\mathcal T_0$ which are comparable with $\map g C$.
If $A \in \mathcal U$, then as $C \subseteq \map g C$, either $A \subseteq \map g C \lor \map g C \subseteq A$
So $\mathcal U \subseteq \mathcal U'$.
The aim now is to demonstrate that $\mathcal U$ is a tower.
From Empty Set is Subset of All Sets, $\O \subseteq C$.
Hence condition $(1)$ is satisfied.
Now let $A \in \mathcal U$.
As $C$ is comparable, there are three possibilities:
- $(1'): \quad A \subsetneq C$
Then from $(A)$ above, $\map g A \subseteq C$.
Therefore $\map g A \in \mathcal U$.
- $(2'): \quad A = C$
Then $\map g A = \map g C$ and so $\map g C \subseteq \map g A$.
Therefore $\map g A \in \mathcal U$.
- $(3'): \quad \map g C \subseteq A$
Then $\map g C \subseteq \map g A$
Therefore $\map g A \in \mathcal U$.
Hence condition $(2)$ is satisfied.
From the definition of $\mathcal U$, it follows immediately that the union of a chain in $\mathcal U$ is also in $\mathcal U$.
Hence condition $(3)$ is satisfied.
The conclusion is that $\mathcal U$ is a tower such that $\mathcal U \subseteq \mathcal T_0$.
But as $\mathcal T_0$ is the smallest tower, $\mathcal T_0 \subseteq \mathcal U$.
It follows that $\mathcal U = \mathcal T_0$.
Consider some comparable set $C$, then.
From that $C$ we can form $\mathcal U$, as above.
But as $\mathcal U = \mathcal T_0$:
- $A \in \mathcal T_0 \implies \paren {A \subseteq C \implies A \subseteq \map g C} \lor \map g C \subseteq A$
and so $g \left({C}\right)$ is also comparable.
We now know that:
- $\O$ is comparable
- the mapping $g$ maps comparable sets to comparable sets.
Since the union of a chain of comparable sets is itself comparable, it follows that the comparable sets all form a tower $\mathcal T_C$.
But by the nature of $\mathcal T_0$ it follows that $\mathcal T_0 \subseteq \mathcal T_C$.
So the elements of $\mathcal T_0$ must all be comparable.
Since $\mathcal T_0$ is a chain, the union $M$ of all the sets in $\mathcal T_0$ is itself a set in $\mathcal T_0$.
Since the union includes all the sets of $\mathcal T_0$, it follows that $\map g M \subseteq M$.
Since it is always the case that $M \subseteq \map g M$, it follows that $M = \map g M$.
The result follows.
$\blacksquare$
Proof 2
Aiming for a contradiction, suppose that for each $x \in X$ there is a $y \in X$ such that $x \prec y$.
By the Axiom of Choice, there is a mapping $f: X \to X$ such that:
- $\forall x \in X: x \prec \map f x$
Let $\mathcal C$ be the set of all chains in $X$.
By the premise, each element of $\mathcal C$ has an upper bound in $X$.
Thus by the Axiom of Choice, there is a mapping $g: \mathcal C \to X$ such that for each $C \in \mathcal C$, $\map g C$ is an upper bound of $C$.
Let $p$ be an arbitrary element of $X$.
Define a mapping $h: \operatorname {Ord} \to X$ by transfinite recursion thus:
\(\ds \map h 0\) | \(=\) | \(\ds p\) | ||||||||||||
\(\ds \map h {\alpha^+}\) | \(=\) | \(\ds \map f {\map h \alpha }\) | ||||||||||||
\(\ds \map h \lambda\) | \(=\) | \(\ds \map f {\map g {f \sqbrk \lambda} }\) | if $\lambda$ is a limit ordinal |
Then $h$ is strictly increasing, and thus injective.
Let $h'$ be the restriction of $h$ to $\operatorname {On} \times \map h {\operatorname{On} }$.
Then ${h'}^{-1}$ is a surjection from $\map h {\operatorname{On} } \subseteq X$ onto $\operatorname{On}$.
By the Axiom of Replacement, $\operatorname{On}$ is a set.
By Burali-Forti Paradox, this is a contradiction.
Thus we conclude that some element of $X$ has no strict successor, and is thus maximal.
$\blacksquare$