- 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.
- IC 342 (Wikipedia)
IC 342 (also known as Caldwell 5) is an intermediate spiral galaxy in the constellation Camelopardalis, located relatively close to the Milky Way. Despite its size and actual brightness, its location behind dusty areas near the galactic equator makes it difficult to observe, leading to the nickname “The Hidden Galaxy”, though it can readily be detected even with binoculars. If the galaxy were not obscured, it would be visible by naked eye. The dust makes it difficult to determine its precise distance; modern estimates range from about 7 million light-years (Mly) to about 11 Mly. The galaxy was discovered by William Frederick Denning in 1892. It is one of the brightest in the IC 342/Maffei Group, one of the closest galaxy groups to the Local Group. Edwin Hubble first thought it to be in the Local Group, but it was later determined not to be a member.