Zermelo's Well-Ordering Theorem/Proof 2
Theorem
Let the Axiom of Choice be accepted.
Then every set is well-orderable.
Proof
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
By the Axiom of Choice, there is a choice function $c$ defined on $\powerset S \setminus \set \O$.
We will use $c$ and the Principle of Transfinite Induction to define a bijection between $S$ and some ordinal.
Intuitively, we start by pairing $\map c S$ with $0$, and then keep extending the bijection by pairing $\map c {S \setminus X}$ with $\alpha$, where $X$ is the set of elements already dealt with.
Basis for the Induction
$\alpha = 0$
Let $s_0 = \map c S$.
Inductive Step
Suppose $s_\beta$ has been defined for all $\beta < \alpha$.
If $S \setminus \set {s_\beta: \beta < \alpha}$ is empty, we stop.
Otherwise, define:
- $s_\alpha := \map c {S \setminus \set {s_\beta: \beta < \alpha} }$
The process eventually stops, else we have defined bijections between subsets of $S$ and arbitrarily large ordinals.
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Now, we can impose a well-ordering on $S$ by embedding it via $s_\alpha \to \alpha$ into the ordinal $\beta = \ds {\bigcup_{s_\alpha \mathop \in S} \alpha}$ and using the well-ordering of $\beta$.
$\blacksquare$
Axiom of Choice
This theorem depends on the Axiom of Choice.
Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.
Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.
However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.
Also known as
Zermelo's Well-Ordering Theorem is also known just as the well-ordering theorem.
Some sources omit the hyphen: (Zermelo's) well ordering theorem.
It is also known just as Zermelo's Theorem.
Under this name it can often be seen worded:
- Every set of cardinals is well-ordered with respect to $\le$.
This is called by some authors the Trichotomy Problem.
It is also referred to as the well-ordering principle, but this causes confusion with the result that states that the natural numbers are well-ordered.
Source of Name
This entry was named for Ernst Friedrich Ferdinand Zermelo.