- In mathematics, the well-ordering theorem, also known as Zermelo’s theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering. The well-ordering theorem together with Zorn’s lemma are the most important mathematical statements that are equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents). Ernst Zermelo introduced the axiom of choice as an “unobjectionable logical principle” to prove the well-ordering theorem. One can conclude from the well-ordering theorem that every set is susceptible to transfinite induction, which is considered by mathematicians to be a powerful technique. One famous consequence of the theorem is the Banach–Tarski paradox.
- Lake of the Woods (Wikipedia)
Lake of the Woods (French: Lac des Bois; Ojibwe: Pikwedina Sagainan, lit. ‘“inland lake of the sand hills”’) is a lake occupying parts of the Canadian provinces of Ontario and Manitoba and the U.S. state of Minnesota. Lake of the Woods is over 70 miles (110 km) long and wide, containing more than 14,552 islands and 65,000 miles (105,000 km) of shoreline. It is fed by the Rainy River, Shoal Lake, Kakagi Lake and other smaller rivers. The lake drains into the Winnipeg River and then into Lake Winnipeg. Ultimately, its outflow goes north through the Nelson River to Hudson Bay.