User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Theorem
Well-Ordering Theorem

Proof
Let $X$ be a non-empty set.

If $X$ considered are empty or singletons, the theorem holds vacuously or trivially.

So suppose $X$ contains at least two elements.

We note that there exists at least one tower in $X$ by considering any doubleton $\{x,y\}$ and defining $x \preccurlyeq y$.

Define:


 * $\mathcal T = \left\{ { \left({T_k,\preccurlyeq_k}\right) } \right\}_{k \mathop \in K}$

as the indexed family of all towers in $X$, where $K$ is some indexing set.

Define:


 * $T = \displaystyle \bigcup_{k \mathop \in K} T_k$


 * $\preccurlyeq \, = \displaystyle \bigcup_{k \mathop \in K} \preccurlyeq_k$

Define a relation on the towers in $X$:


 * $\left({T_j,\preccurlyeq_k}\right) < \left({T_\ell,\preccurlyeq_\ell}\right)$


 * $\left({T_j,\preccurlyeq_k}\right)$ equals an initial segment of $\left({T_j,\preccurlyeq_k}\right)$.

By Equality to Initial Segment Imposes Well-Ordering, $\left({\mathcal T, <}\right)$ is a strictly well-ordered set.

We claim that $T = X$.