Zermelo's Well-Ordering Theorem/Proof 1
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 exists a choice function on $S$.
The result follows from Set with Choice Function is Well-Orderable.
$\blacksquare$
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.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 4$ Well ordering and choice: Theorem $4.8$