An extract on #livefolk
There are several weaker statements that are not equivalent to the axiom of choice, but are closely related. One example is the axiom of dependent choice (DC). A still weaker example is the axiom of countable choice (AC or CC), which states that a choice function exists for any countable set of nonempty sets. These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.
Other choice axioms weaker than axiom of choice include the Boolean prime ideal theorem and the axiom of uniformization. The former is equivalent in ZF to the existence of an ultrafilter containing each given filter, proved by Tarski in 1930.
Frazee, Charles A. (November 2002). Two Thousand Years Ago: the World at the Time of Jesus. William B. Eerdmans Publishing Company. ISBN 978-0-8028-4805-5.
Heather, Peter (2010). Empires and Barbarians: The Fall of Rome and the Birth of Europe. Oxford University Press. ISBN 978-0-19-975272-0.
Heather, Peter (2007). The Fall of the Roman Empire: A New History of Rome and the Barbarians. Oxford University Press. ISBN 978-0-19-532541-6.
Many of the islands in the Aegean have safe harbours and bays. In ancient times, navigation through the sea was easier than travelling across the rough terrain of the Greek mainland (and to some extent the coastal areas of Anatolia). Many of the islands are volcanic, and marble and iron are mined on other islands. The larger islands have some fertile valleys and plains. Of the main islands in the Aegean Sea, two belong to Turkey Bozcaada (Tenedos ) and Gkeada (Imbros ); the rest belong to Greece. Between the two countries, there are political disputes over several aspects of political control over the Aegean space, including the size of territorial waters, air control and the delimitation of economic rights to the continental shelf.
Amsterdam has an oceanic climate (Kppen climate classification Cfb) strongly influenced by its proximity to the North Sea to the west, with prevailing westerly winds. Both winters and summers are considered mild, although winters can get quite cold, while summers are quite warm occasionally. Amsterdam, as well as most of the North Holland province, lies in USDA Hardiness zone 8b. Frosts mainly occur during spells of easterly or northeasterly winds from the inner European continent. Even then, because Amsterdam is surrounded on three sides by large bodies of water, as well as having a significant heat-island effect, nights rarely fall below 5 C (23 F), while it could easily be 12 C (10 F) in Hilversum, 25 kilometres (16 miles) southeast. Summers are moderately warm with a number of hot days every month. The average daily high in August is 22.1 C (71.8 F), and 30 C (86 F) or higher is only measured on average on 2.5 days, placing Amsterdam in AHS Heat Zone 2. The record extremes range from 15.4 C (4.3 F) to 34.5 C (94.1 F). Days with more than 1 millimetre (0.04 in) of precipitation are common, on average 133 days per year. Amsterdam's average annual precipitation is 838 millimetres (33 in), more than what is measured at Amsterdam Schiphol Airport. A large part of this precipitation falls as light rain or brief showers. Cloudy and damp days are common during the cooler months of October through March.