- 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.
- British Columbia (Wikipedia)
British Columbia is the westernmost province of Canada. Situated in the Pacific Northwest between the Pacific Ocean and the Rocky Mountains, the province has a diverse geography, with rugged landscapes that include rocky coastlines, sandy beaches, forests, lakes, mountains, inland deserts and grassy plains. British Columbia borders the province of Alberta to the east; the territories of Yukon and Northwest Territories to the north; the U.S. states of Washington, Idaho and Montana to the south, and Alaska to the northwest. With an estimated population of over 5.7 million as of 2025, it is Canada’s third-most populous province. The capital of British Columbia is Victoria, while the province’s largest city is Vancouver. Vancouver and its suburbs together make up the third-largest metropolitan area in Canada, with the 2021 census recording 2.6 million people in Metro Vancouver. British Columbia is Canada’s third-largest province in terms of total area, after Quebec and Ontario.