Now, consider stronger forms of the negation of AC. For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets. Note that strengthened negations may be compatible with weakened forms of AC. For example, ZF + DC + BP is consistent, if ZF is.
It is also consistent with ZF + DC that every set of reals is Lebesgue measurable; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption (the existence of an inaccessible cardinal). The much stronger axiom of determinacy, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the perfect set property (all three of these results are refuted by AC itself). ZF + DC + AD is consistent provided that a sufficiently strong large cardinal axiom is consistent (the existence of infinitely many Woodin cardinals).
The sea was traditionally known as the Archipelago (in Greek, , meaning "chief sea"), but in English the meaning of Archipelago has changed to refer to the Aegean Islands and, generally, to any island group.
Amsterdam's name derives from Amstelredamme, indicative of the city's origin around a dam in the river Amstel. Originating as a small fishing village in the late 12th century, Amsterdam became one of the most important ports in the world during the Dutch Golden Age (17th century), a result of its innovative developments in trade. During that time, the city was the leading centre for finance and diamonds. In the 19th and 20th centuries the city expanded, and many new neighborhoods and suburbs were planned and built. The 17th-century canals of Amsterdam and the 1920th century Defence Line of Amsterdam are on the UNESCO World Heritage List. Since the annexation of municipality Sloten in 1921 by the municipality of Amsterdam, the oldest historic part of the city lies in Sloten (9th century).
As the commercial capital of the Netherlands and one of the top financial centres in Europe, Amsterdam is considered an alpha world city by the Globalization and World Cities (GaWC) study group. The city is also the cultural capital of the Netherlands. Many large Dutch institutions have their headquarters there, and seven of the world's 500 largest companies, including Philips and ING, are based in the city. In 2012, Amsterdam was ranked the second best city to live in by the Economist Intelligence Unit (EIU) and 12th globally on quality of living for environment and infrastructure by Mercer. The city was ranked 3rd in innovation by Australian innovation agency 2thinknow in their Innovation Cities Index 2009. The Amsterdam seaport to this day remains the second in the country, and the fifth largest seaport in Europe.
Famous Amsterdam residents include the diarist Anne Frank, artists Rembrandt van Rijn and Vincent van Gogh, and philosopher Baruch Spinoza.
The Amsterdam Stock Exchange, the oldest stock exchange in the world, is located in the city center. Amsterdam's main attractions, including its historic canals, the Rijksmuseum, the Van Gogh Museum, Stedelijk Museum, Hermitage Amsterdam, Anne Frank House, Amsterdam Museum, its red-light district, and its many cannabis coffee shops draw more than 5 million international visitors annually. It is also the world's most multicultural city with at least 177 nationalities represented.