Zermelo's Well-Ordering Theorem/Proof 3
Theorem
Let the Axiom of Choice be accepted.
Then every set is well-orderable.
Proof
Let $S$ be a non-empty set.
Let $C$ be a choice function for $S$.
Let $A$ be an arbitrary set.
Let the mapping $h$ be defined as:
- $\map h A = \begin {cases}
\map C {S \setminus A} & : A \subsetneqq S \\ x & : \text {otherwise} \end {cases}$
where $x$ is an arbitrary element such that $x \notin S$.
The latter is known to exist from Exists Element Not in Set.
From the Transfinite Recursion Theorem: Formulation $5$, there exists a mapping $F$ on the class of all ordinals $\On$ such that:
- $\forall \alpha \in \On: \map F \alpha = \map h {F \sqbrk \alpha}$
It remains to show the following:
- $(1): \quad \exists \delta \in \On: \map F \delta \notin S$
- $(2): \quad$ Let $\beta$ be the smallest ordinal such that $\map F \beta \notin S$. Then:
- $F \sqbrk \beta = S$
- and:
- $F \restriction \beta$ is injective
- and so $\beta$ can be put into a one-to-one correspondence with $S$.
 | This needs considerable tedious hard slog to complete it. In particular: Proof needed for both $1$ and $2$ above To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Hence $S$ can be well-ordered.
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.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems: Exercise $5.3$