title: Craps

text: This bet can only be placed on the numbers 4, 6, 8, and 10. In order for this bet to win, the chosen number must be rolled the "hard way" (as doubles) before a 7 or any other non-double combination ("easy way") totaling that number is rolled. For example, a player who bets a hard 6 can only win by seeing a 3–3 roll come up before any 7 or any easy roll totaling 6 (4–2 or 5–1); otherwise, he/she loses.


title: Craps

text: In Las Vegas casinos, this bet is generally working, including when no point has been established, unless the player specifies otherwise. In other casinos such as those in Atlantic City, hard ways are not working when the point is off unless the player requests to have it working on the come out roll.


title: Craps

text: Like single-roll bets, hard way bets can be lower than the table minimum; however, the maximum bet allowed is also lower than the table maximum. The minimum hard way bet can be a minimum one unit. For example, lower stake table minimums of $5 or $10, generally allow minimum hard ways bets of $1. The maximum bet is based on the maximum allowed win from a single roll.


title: Craps

text: Easy way is not a specific bet offered in standard casinos, but a term used to define any number combination which has two ways to roll. For example, (6–4, 4–6) would be a "10 easy". The 4, 6, 8 or 10 can be made both hard and easy ways. Betting point numbers (which pays off on easy or hard rolls of that number) or single-roll ("hop") bets (e.g., "hop the 2–4" is a bet for the next roll to be an easy six rolled as a two and four) are methods of betting easy ways.


title: Craps

text: A player can choose either the 6 or 8 being rolled before the shooter throws a seven. These wagers are usually avoided by experienced craps players since they pay even money (1:1) while a player can make place bets on the 6 or the 8, which pay more (7:6). Some casinos (especially all those in Atlantic City) do not even offer the Big 6 & 8. The bets are located in the corners behind the Pass line, and bets may be placed directly by players.


title: Craps

text: The only real advantage offered by the Big 6 & 8 is that they can be bet for the table minimum, whereas a place bet minimum may sometimes be greater than the table minimum (e.g. $6 place bet on a $3 minimum game.) In addition place bets are usually not working, except by agreement, when the shooter is "coming out" i.e. shooting for a point, and Big 6 and 8 bets always work. Some modern layouts no longer show the Big 6/Big 8 bet.


title: Craps

text: Single-roll (proposition) bets are resolved in one dice roll by the shooter. Most of these are called "service bets", and they are located at the center of most craps tables. Only the stickman or a dealer can place a service bet. Single-roll bets can be lower than the table minimum, but the maximum bet allowed is also lower than the table maximum. The maximum bet is based on the maximum allowed win from a single roll. The lowest single-roll bet can be a minimum one unit bet. For example, tables with minimums of $5 or $10 generally allow minimum single-roll bets of $1. Single bets are always working by default unless the player specifies otherwise. The bets include:


title: Craps

text: 2 or 12 (hi-lo): Wins if shooter rolls a 2 or 12. The stickman places this bet on the line dividing the 2 and 12 bets.


title: Craps

text: C & E: A combined bet, a player is betting half their bet on craps (2,3,12) and the other half on 11 (yo). The combine payout is 3:1 on craps and 7:1 on 11 (yo). Another method of calculating the payout is to divide the total bet in half. The player would receive 7:1 minus half the total bet payout on half the total bet for craps and 15:1 minus half the total bet payout on half the total bet for 11 (yo). For example, using this method if a player were to bet $2 on C & E, $1 would receive 7:1 payout on craps minus $1 for the bet on 11 so the total profit would be $6. If an 11 was rolled the player would receive 15:1 minus $1 for the bet on craps so the player's total profit is $14. Both methods of calculation yield the same result so either method can be used. If a player wishes to take the bet down after a win the player would receive the whole bet not half even though only one of the two bets can win per roll. The minimum bet on C & E is double the lowest unit bet allowed at the table. So if the minimum single roll bet is $1 the lowest C & E bet allowed would be $2. Players are, however, able to make odd number bets larger than $2 if desired. One of the two bets will always lose, the other may win.


title: Craps

text: Any seven: A single roll bet which wins if the shooter rolls a 7 with 4:1 payout. This bet is also nicknamed Big Red, since the 7 on its betting space on the layout is usually large and red, and it is considered bad luck and a breach of etiquette among gamblers to speak the word "seven" at the table.


title: Craps

text: Horn: This is a bet that involves betting on 1 unit each for 2, 3, 11, and 12 at the same time for the next roll. The bet is actually four separate bets, and pays off depending on which number is actually rolled. The combined payout is 27:4 for 2, 12 and 3:1 for 3, 11. Each individual bet has the same payout as a single bet on the specific numbers, 30:1 for 2 and 12 minus the other three bets, 15:1 for 3 and 11 minus the other three bets. If a player wins the bet he can take down all four bets instead of a single bet even though only one bet can win per roll. Many players, in order to eliminate the confusion of tossing four chips to the center of the table or having change made while bets are being placed, will make a five-unit Horn High bet, which is a four-way bet with the extra unit going to one specific number. For example, if one tosses a $5 chip into the center and says "horn high yo", they are placing four $1 bets on each of the horn numbers and the extra dollar will go on the yo (11). Horn bets are generally required to be in multiples of 4 or 5 with the minimum bet being 4 times the minimum unit allowed. For example, if the single roll minimum at the table is $1 the Horn bet must be $4 or more.


title: Craps

text: Whirl or World: A five-unit bet that is a combination of a horn and any-seven bet, with the idea that if a seven is rolled the bet is a push, because the money won on the seven is lost on the horn portions of the bet. The combine odds are 26:5 on the 2, 12, 11:5 on the 3, 11, and a push on the 7. Like the C & E and Horn bet, if a player wishes to take down the bet after a win he or she would receive all five units back. The minimum bet is five of the minimum units. For example, if the minimum single roll bet is $1, the minimum World/Whirl bet is $5.


title: Craps

text: On the Hop, Hop or Hopping: A single roll bet on any particular combination of the two dice on the next roll including combinations whose sum is 7 (e.g. 4 and 3). For example, if someone bets on "5 and 1" on the hop, they are betting that the next roll will have a 5 on one die and a 1 on the other die. The bet pays 15:1 on easy ways (same as a bet on 3 or 11). Hard ways hop pays 30:1 (e.g., 3 and 3 on the hop, same as a bet on 2 or 12). The true odds are 17:1 and 35:1, resulting in a house edge of 11.11% and 13.89% respectively. When presented, hop bets are located at the center of the craps layout with the other proposition bets. If hop bets are not on the craps layout, they still may be bet on by players but they become the responsibility of the boxman to book the bet. Sometimes players may request to hop a whole number. In this case the money on the bet different combinations. For example, if a player says "hop the tens" (6–4, 5–5, 4–6) the player must give the dealer an even number bet so it can be divided among the hard and easy ways. If the player gives $10, $5 would be placed on the easy ways 10 with 15:1 odds and $5 would be placed on the hard way with 30:1 odds. If a player wishes to "hop the sevens" there would be three different combinations and six possible ways to roll a 7 (6–1, 5–2, 4–3, 3–4, 2–5, 1–6) therefore the player should bet in multiples of 3 so the bet can be divided among each combination with a 15:1 payout minus the other two bets, otherwise if players does not bet in multiples of 3, they would specific which combination has additional units.


title: Craps

text: Field: This bet is a wager that one of the numbers 2, 3, 4, 9, 10, 11, or 12 will appear on the next roll of the dice. This bet typically pays more (2:1 or 3:1) if 2 or 12 is rolled, and 1:1 if 3, 4, 9, 10, or 11 is rolled.


title: Craps

text: The Field bet is a "Self-Service" Bet. Unlike the other proposition bets which are handled by the dealers or stickman, the field bet is placed directly by the player.


title: Craps

text: Players identify their Field bets by placing them in the Field area directly in front of them or as close to their position as possible. The initial bet and/or any payouts can "ride" through several rolls until they lose, and are assumed to be "riding" by dealers. It is thus the player's responsibility to collect their bet and/or winnings immediately upon payout, before the next dice roll, if they do not wish to let it ride.


title: Craps

text: Fire Bet: Before the shooter begins, some casinos will allow a bet known as a fire bet to be placed. A fire bet is a bet of as little as $1 and generally up to a maximum of $5 to $10 sometimes higher, depending on casino, made in the hope that the next shooter will have a hot streak of setting and getting many points of different values. As different individual points are made by the shooter, they will be marked on the craps layout with a fire symbol.


title: Craps

text: The first three points will not pay out on the fire bet, but the fourth, fifth, and sixth will pay out at increasing odds. The fourth point pays at 24-to-1, the fifth point pays at 249-to-1, and the 6th point pays at 999-to-1. (The points must all be different numbers for them to count toward the fire bet.) For example, a shooter who successfully hits a point of 10 twice will only garner credit for the first one on the fire bet. Players must hit the established point in order for it to count toward the fire bet. The payout is determine by the number of points which have been established and hit after the shooter sevens out.


title: Craps

text: Bonus Craps: Prior to the initial "come out roll", players may place an optional wager (usually a $1 minimum to a maximum $25) on one or more of the three Bonus Craps wagers, "All Small", "All Tall", or "All or Nothing at All." For players to win the "All Small" wager, the shooter must hit all five small numbers (2, 3, 4, 5, 6) before a seven is rolled; similarly, "All Tall" wins if all five high numbers (8, 9, 10, 11, 12) are hit before a seven is rolled.


title: Craps

text: These bets pay 35-for-1, for a house advantage of 7.76%. "All or Nothing at All" wins if the shooter hits all 10 numbers before a seven is rolled. This pays 176-for-1, for a house edge of 7.46%. For all three wagers, the order in which the numbers are hit does not matter. Whenever a seven is hit, including on the come out roll, all bonus bets lose, the bonus board is reset, and new bonus bets may be placed.


title: Craps

text: A player may wish to make multiple different bets. For example, a player may be wish to bet $1 on all hard ways and the horn. If one of the bets win the dealer may automatically replenish the losing bet with profits from the winning bet. In this example, if the shooter rolls a hard 8 (pays 9:1), the horn loses. The dealer may return $5 to the player and place the other $4 on the horn bet which lost. If the player does not want the bet replenished, he or she should request any or all bets be taken down.


title: Craps

text: A working bet is a live bet. Bets may also be on the board, but not in play and therefore not working. Pass line and come bets are always working meaning the chips are in play and the player is therefore wagering live money. Other bets may be working or not working depending whether a point has been established or player's choice. Place and buy bets are working by default when a point is established and not working when the point is off unless the player specifies otherwise. Lay bets are always working even if a point has not been established unless the player requests otherwise. At any time, a player may wish to take any bet or bets out of play. The dealer will put an "Off" button on the player's specific bet or bets; this allows the player to keep his chips on the board without a live wager. For example, if a player decides not to wager a place bet mid-roll but wishes to keep the chips on the number, he or she may request the bet be "not working" or "Off". The chips remain on the table, but the player cannot win from or lose chips which are not working.


title: Craps

text: The opposite is also allowed. By default place and buy bets are not working without an established point; a player may wish to wager chips before a point has been established. In this case, the player would request the bet be working in which the dealer will place an "On" button on the specified chips.


title: Craps

text: The probability of dice combinations determine the odds of the payout. The following chart shows the dice combinations needed to roll each number. The two and twelve are the hardest to roll since only one combination of dice is possible. The game of craps is built around the dice roll of seven, since it is the most easily rolled dice combination.


title: Craps

text: The expected value of all bets is usually negative, such that the average player will always lose money. This is because the house always sets the paid odds to below the actual odds. The only exception is the "odds" bet that the player is allowed to make after a point is established on a pass/come Don't Pass/Don't Come bet (the odds portion of the bet has a long-term expected value of 0). However, this "free odds" bet cannot be made independently, so the expected value of the entire bet, including odds, is still negative. Since there is no correlation between die rolls, there is normally no possible long-term winning strategy in craps.


title: Craps

text: There are occasional promotional variants that provide either no house edge or even a player edge. One example is a field bet that pays 3:1 on 12 and 2:1 on either 3 or 11. Overall, given the 5:4 true odds of this bet, and the weighted average paid odds of approximately 7:5, the player has a 5% advantage on this bet. This is sometimes seen at casinos running limited-time incentives, in jurisdictions or gaming houses that require the game to be fair, or in layouts for use in informal settings using play money. No casino currently runs a craps table with a bet that yields a player edge full-time.


title: Craps

text: Maximizing the size of the odds bet in relation to the line bet will reduce, but never eliminate the house edge, and will increase variance. Most casinos have a limit on how large the odds bet can be in relation to the line bet, with single, double, and five times odds common. Some casinos offer 3–4–5 odds, referring to the maximum multiple of the line bet a player can place in odds for the points of 4 and 10, 5 and 9, and 6 and 8, respectively. During promotional periods, a casino may even offer 100x odds bets, which reduces the house edge to almost nothing, but dramatically increases variance, as the player will be betting in large betting units.


title: Craps

text: Since several of the multiple roll bets pay off in ratios of fractions on the dollar, it is important that the player bets in multiples that will allow a correct payoff in complete dollars. Normally, payoffs will be rounded down to the nearest dollar, resulting in a higher house advantage. These bets include all place bets, taking odds, and buying on numbers 6, 8, 5, and 9, as well as laying all numbers.


title: Craps

text: These variants depend on the casino and the table, and sometimes a casino will have different tables that use or omit these variants and others.


title: Craps

text: When craps is played in a casino, all bets have a house advantage. That is, it can be shown mathematically that a player will (with 100% probability) lose all his or her money to the casino in the long run, while in the short run the player is more likely to lose money than make money. There may be players who are lucky and get ahead for a period of time, but in the long run these winning streaks are eroded away. One can slow, but not eliminate, one's average losses by only placing bets with the smallest house advantage.


title: Craps

text: The Pass/Don't Pass line, Come/Don't Come line, place 6, place 8, buy 4 and buy 10 (only under the casino rules where commission is charged only on wins) have the lowest house edge in the casino, and all other bets will, on average, lose money between three and twelve times faster because of the difference in house edges.


title: Craps

text: The place bets and buy bets differ from the Pass line and come line, in that place bets and buy bets can be removed at any time, since, while they are multi-roll bets, their odds of winning do not change from roll to roll, whereas Pass line bets and come line bets are a combination of different odds on their first roll and subsequent rolls. The first roll of a Pass line bet is 2:1 advantage for the player (8 wins, 4 losses), but it's "paid for" by subsequent rolls that are at the same disadvantage to the player as the Don't Pass bets were at an advantage. As such, they cannot profitably let the player take down the bet after the first roll. Players can bet or lay odds behind an established point depending on whether it was a Pass/Come or Don't Pass/Don't Come to lower house edge by receiving true odds on the point. Casinos which allow put betting allows players to increase or make new pass/come bets after the come-out roll. This bet generally has a higher house edge than place betting, unless the casino offers high odds.


title: Craps

text: Conversely, a player can take back (pick up) a Don't Pass or Don't Come bet after the first roll, but this cannot be recommended, because they already endured the disadvantaged part of the combination – the first roll. On that come-out roll, they win just 3 times (2 and 3), while losing 8 of them (7 and 11) and pushing one (12) out of the 36 possible rolls. On the other 24 rolls that become a point, their Don't Pass bet is now to their advantage by 6:3 (4 and 10), 6:4 (5 and 9) and 6:5 (6 and 8). If a player chooses to remove the initial Don't Come and/or Don't Pass line bet, he or she can no longer lay odds behind the bet and cannot re-bet the same Don't Pass and/or Don't Come number (players must make a new Don't Pass or come bets if desired). However, players can still make standard lay bets odds on any of the point numbers (4,5,6,8,9,10).


title: Craps

text: Among these, and the remaining numbers and possible bets, there are a myriad of systems and progressions that can be used with many combinations of numbers.


title: Craps

text: An important alternative metric is house advantage per roll (rather than per bet), which may be expressed in loss per hour. The typical pace of rolls varies depending on the number of players, but 102 rolls per hour is a cited rate for a nearly full table. This same reference states that only "29.6% of total rolls are come out rolls, on average", so for this alternative metric, needing extra rolls to resolve the Pass line bet, for example, is factored. This number then permits calculation of rate of loss per hour, and per the 4 day/5 hour per day gambling trip:


title: Craps

text: Besides the rules of the game itself, a number of formal and informal rules are commonly applied in the table form of Craps, especially when played in a casino.


title: Craps

text: To reduce the potential opportunity for switching dice by sleight-of-hand, players are not supposed to handle the dice with more than one hand (such as shaking them in cupped hands before rolling) nor take the dice past the edge of the table. If a player wishes to change shooting hands, they may set the dice on the table, let go, then take them with the other hand.


title: Craps

text: When throwing the dice, the player is expected to hit the farthest wall at the opposite end of the table (these walls are typically augmented with pyramidal structures to ensure highly unpredictable bouncing after impact). Casinos will sometimes allow a roll that does not hit the opposite wall as long as the dice are thrown past the middle of the table; a very short roll will be nullified as a "no roll". The dice may not be slid across the table and must be tossed. These rules are intended to prevent dexterous players from physically influencing the outcome of the roll.


title: Craps

text: Players are generally asked not to throw the dice above a certain height (such as the eye level of the dealers). This is both for the safety of those around the table, and to eliminate the potential use of such a throw as a distraction device in order to cheat.


title: Craps

text: Dice are still considered "in play" if they land on players' bets on the table, the dealer's working stacks, on the marker puck, or with one die resting on top of the other. The roll is invalid if either or both dice land in the boxman's bank, the stickman's bowl (where the extra three dice are kept between rolls), or in the rails around the top of the table where players chips are kept. If one or both dice hits a player or dealer and rolls back onto the table, the roll counts as long as the person being hit did not intentionally interfere with either of the dice, though some casinos will rule "no roll" for this situation. If one or both leave the table, it is also a "no roll", and the dice may either be replaced or examined by the boxman and returned to play.


title: Craps

text: Shooters may wish to "set" the dice to a particular starting configuration before throwing (such as showing a particular number or combination, stacking the dice, or spacing them to be picked up between different fingers), but if they do, they are often asked to be quick about it so as not to delay the game. Some casinos disallow such rituals to speed up the pace of the game. Some may also discourage or disallow unsanitary practices such as kissing or spitting on the dice.


title: Craps

text: In most casinos, players are not allowed to hand anything directly to dealers, and vice versa. Items such as cash, checks, and chips are exchanged by laying them down on the table; for example, when "buying in" (paying cash for chips), players are expected to place the cash on the layout: the dealer will take it and then place the chips in front of the player. This rule is enforced in order to allow the casino to easily monitor and record all transfers via overhead surveillance cameras, and to reduce the opportunity for cheating via sleight-of-hand.


title: Craps

text: Most casinos prohibit "call bets", and may have a warning such as "No Call Bets" printed on the layout to make this clear. This means a player may not call out a bet without also placing the corresponding chips on the table. Such a rule reduces the potential for misunderstanding in loud environments, as well as disputes over the amount that the player intended to bet after the outcome has been decided. Some casinos choose to allow call bets once players have bought-in. When allowed, they are usually made when a player wishes to bet at the last second, immediately before the dice are thrown, to avoid the risk of obstructing the roll.


title: Craps

text: Craps is among the most social and most superstitious of all gambling games, which leads to an enormous variety of informal rules of etiquette that players may be expected to follow. An exhaustive list of these is beyond the scope of this article, but the guidelines below are most commonly given.


title: Craps

text: Tipping the dealers is universal and expected in Craps. As in most other casino games, a player may simply place (or toss) chips onto the table and say, "For the dealers", "For the crew", "etc." In craps, it is also common to place a bet for the dealers. This is usually done one of three ways: by placing an ordinary bet and simply declaring it for the dealers, as a "two-way", or "on top". A "Two-Way" is a bet for both parties: for example, a player may toss in two chips and say "Two Way Hard Eight", which will be understood to mean one chip for the player and one chip for the dealers. Players may also place a stack of chips for a bet as usual, but leave the top chip off-center and announce "on top for the dealers". The dealer's portion is often called a "toke" bet, which comes from the practice of using $1 slot machine tokens to place dealer bets in some casinos.


title: Craps

text: In some cases, players may also tip each other, for example as a show of gratitude to the thrower for a roll on which they win a substantial bet.


title: Craps

text: Craps players routinely practice a wide range of superstitious behaviors, and may expect or demand these from other players as well.


title: Craps

text: Most prominently, it is universally considered bad luck to say the word "seven" (after the "come-out", a roll of 7 is a loss for "pass" bets). Dealers themselves often make significant efforts to avoid calling out the number. When necessary, participants may refer to seven with a "nickname" such as "Big Red" (or just "Red"), "the S-word", etc.


title: Craps

text: Although no wagering system can consistently beat casino games based on independent trials such as craps, that does not stop gamblers from believing in them. One of the best known systems is the Martingale System. In this strategy, the gambler doubles his bet after every loss. After a win, the bet is reset to the original bet. The theory is that the first win would recover all previous losses plus win a profit equal to the original stake.


title: Craps

text: Other systems depend on the gambler's fallacy, which in craps terms is the belief that past dice rolls influence the probabilities of future dice rolls. For example, the gambler's fallacy indicates that a craps player should bet on eleven if an eleven has not appeared or has appeared too often in the last 20 rolls. In practice this can be observed as players respond to a roll such as a Hard Six with an immediate wager on the Hard Six.


title: Craps

text: In reality, each roll of the dice is an independent event, so the probability of rolling eleven is exactly 1/18 on every roll, regardless of the number of times eleven has come up in the last x rolls. Even if the dice are actually biased toward particular results ("loaded"), each roll is still independent of all the previous ones. The common term to describe this is "dice have no memory".


title: Craps

text: Another approach is to "set" the dice in a particular orientation, and then throw them in such a manner that they do not tumble randomly. The theory is that given exactly the same throw from exactly the same starting configuration, the dice will tumble in the same way and therefore show the same or similar values every time.


title: Craps

text: Casinos take steps to prevent this. The dice are usually required to hit the back wall of the table, which is normally faced with a jagged angular texture such as pyramids, making controlled spins more difficult. There has been no independent evidence that such methods can be successfully applied in a real casino.


title: Craps

text: Bank craps is a variation of the original craps game and is sometimes known as Las Vegas Craps. This variant is quite popular in Nevada gambling houses, and its availability online has now made it a globally played game. Bank craps uses a special table layout and all bets must be made against the house. In Bank Craps, the dice are thrown over a wire or a string that is normally stretched a few inches from the table's surface. The lowest house edge (for the Pass/Don't Pass) in this variation is around 1.4%. Generally, if the word "craps" is used without any modifier, it can be inferred to mean this version of the game, to which most of this article refers.


title: Craps

text: Crapless craps, also known as bastard craps, is a simple version of the original craps game, and is normally played as an online private game. The biggest difference between crapless craps and original craps is that the shooter (person throwing the dice) is at a far greater disadvantage and has a house edge of 5.38%. Another difference is that this is one of the craps games in which a player can bet on rolling a 2, 3, 11 or 12 before a 7 is thrown. In crapless craps, 2 and 12 have odds of 11:2 and have a house edge of 7.143% while 3 and 11 have odds of 11:4 with a house edge of 6.25%.


title: Craps

text: New York Craps is one of the variations of craps played mostly in the Eastern coast of the US, true to its name. History states that this game was actually found and played in casinos in Yugoslavia, the UK and the Bahamas. In this craps variant, the house edge is greater than Las Vegas Craps or Bank craps. The table layout is also different, and is called a double-end-dealer table. This variation is different from the original craps game in several ways, but the primary difference is that New York craps doesn't allow Come or Don't Come bets. New York Craps Players bet on box numbers like 4, 5, 6, 8, 9, or 10. The overall house edge in New York craps is 5%.


title: Craps

text: In order to get around California laws barring the payout of a game being directly related to the roll of dice, Indian reservations have adapted the game to substitute cards for dice.


title: Craps

text: To replicate the original dice odds exactly without dice or possibility of card-counting, one scheme uses two shuffle machines each with just one deck of Ace through 6 each. Each machine selects one of the 6 cards at random and this is the roll. The selected cards are replaced and the decks are reshuffled for the next roll.


title: Craps

text: In one variation, two shoes are used, each containing some number of regular card decks that have been stripped down to just the Aces and deuces through sixes. The boxman simply deals one card from each shoe and that is the roll on which bets are settled. Since a card-counting scheme is easily devised to make use of the information of cards that have already been dealt, a relatively small portion (less than 50%) of each shoe is usually dealt in order to protect the house.


title: Craps

text: In a similar variation, cards representing dice are dealt directly from a continuous shuffling machine (CSM). Typically, the CSM will hold approximately 264 cards, or 44 sets of 1 through 6 spot cards. Two cards are dealt from the CSM for each roll. The game is played exactly as regular craps, but the roll distribution of the remaining cards in the CSM is slightly skewed from the normal symmetric distribution of dice.


title: Craps

text: Even if the dealer were to shuffle each roll back into the CSM, the effect of buffering a number of cards in the chute of the CSM provides information about the skew of the next roll. Analysis shows this type of game is biased towards the Don't Pass and Don't Come bets. A player betting Don't Pass and Don't Come every roll and laying 10x odds receives a 2% profit on the initial Don't Pass / Don't Come bet each roll. Using a counting system allows the player to attain a similar return at lower variance.


title: Craps

text: In this game variation, one red deck and one blue deck of six cards each (A through 6), and a red die and a blue die are used. Each deck is shuffled separately, usually by machine. Each card is then dealt onto the layout, into the 6 red and 6 blue numbered boxes. The shooter then shoots the dice. The red card in the red-numbered box corresponding to the red die, and the blue card in the blue-numbered box corresponding to the blue die are then turned over to form the roll on which bets are settled.


title: Craps

text: Another variation uses a red and a blue deck of 36 custom playing cards each. Each card has a picture of a two-die roll on it – from 1–1 to 6–6. The shooter shoots what looks like a red and a blue die, called "cubes". They are numbered such that they can never throw a pair, and that the blue one will show a higher value than the red one exactly half the time. One such scheme could be 222555 on the red die and 333444 on the blue die.


title: Craps

text: One card is dealt from the red deck and one is dealt from the blue deck. The shooter throws the "cubes" and the color of the cube that is higher selects the color of the card to be used to settle bets. On one such table, an additional one-roll prop bet was offered: If the card that was turned over for the "roll" was either 1–1 or 6–6, the other card was also turned over. If the other card was the "opposite" (6–6 or 1–1, respectively) of the first card, the bet paid 500:1 for this 647:1 proposition.


title: Craps

text: And additional variation uses a single set of 6 cards, and regular dice. The roll of the dice maps to the card in that position, and if a pair is rolled, then the mapped card is used twice, as a pair.


title: Craps

text: Recreational or informal playing of craps outside of a casino is referred to as street craps or private craps. The most notable difference between playing street craps and bank craps is that there is no bank or house to cover bets in street craps. Players must bet against each other by covering or fading each other's bets for the game to be played. If money is used instead of chips and depending on the laws of where it is being played, street craps can be an illegal form of gambling.


title: Craps

text: There are many variations of street craps. The simplest way is to either agree on or roll a number as the point, then roll the point again before rolling a seven. Unlike more complex proposition bets offered by casinos, street craps has more simplified betting options. The shooter is required to make either a Pass or a Don't Pass bet if he wants to roll the dice. Another player must choose to cover the shooter to create a stake for the game to continue.


title: Craps

text: If there are several players, the rotation of the player who must cover the shooter may change with the shooter (comparable to a blind in poker). The person covering the shooter will always bet against the shooter. For example, if the shooter made a "Pass" bet, the person covering the shooter would make a "Don't Pass" bet to win. Once the shooter is covered, other players may make Pass/Don't Pass bets, or any other proposition bets, as long as there is another player willing to cover.


title: Craps

text: Due to the random nature of the game, in popular culture a "crapshoot" is often used to describe an action with an unpredictable outcome.


title: Craps

text: Floating craps is an illegal operation of craps. The term "floating" refers to the practice of the game's operators using portable tables and equipment to quickly move the game from location to location to stay ahead of the law enforcement authorities. The term may have originated in the 1930s when Benny Binion (later known for founding the downtown Las Vegas hotel Binion's) set up an illegal craps game utilizing tables created from portable crates for the Texas Centennial Exposition.


title: Craps

text: The 1950 Broadway musical "Guys and Dolls" features a major plot point revolving around a floating craps game.


title: Craps

text: In the 1950s and 1960s The Sands Hotel in Las Vegas had a craps table that floated in the swimming pool, as a joke reference to the notoriety of the term.


title: Craps

text: A Golden Arm is a craps player who rolls the dice for longer than one hour without losing. The first Golden Arm was Oahu native Stanley Fujitake, who rolled 118 times without sevening out in 3 hours and 6 minutes at the California Hotel and Casino on May 28, 1989.


title: Craps

text: The current record for length of a "hand" (successive rounds won by the same shooter) is 154 rolls including 25 passes by Patricia DeMauro of New Jersey, lasting 4 hours and 18 minutes, at the Borgata in Atlantic City, New Jersey, on May 23–24, 2009. She bested by over an hour the record held for almost 20 years – that of Fujitake.


title: Rhineland-Palatinate

text: Rhineland-Palatinate ( , ; ; ; ) is a western state of Germany. It covers and has about 4.05 million residents. It is the ninth largest and sixth most populous of the sixteen states. Mainz is the capital and largest city. Other cities are Ludwigshafen am Rhein, Koblenz, Trier, Kaiserslautern, Worms and Neuwied. It is bordered by North Rhine-Westphalia, Saarland, Baden-Württemberg and Hesse and by the countries France, Luxembourg and Belgium.


title: Rhineland-Palatinate

text: Rhineland-Palatinate was established in 1946 after World War II, from parts of the former states of Prussia (part of its Rhineland and Nassau provinces), Hesse (Rhenish Hesse) and Bavaria (its former outlying Palatinate kreis or district), by the French military administration in Allied-occupied Germany. Rhineland-Palatinate became part of the Federal Republic of Germany in 1949 and shared the country's only border with the Saar Protectorate until the latter was returned to German control in 1957. Rhineland-Palatinate's natural and cultural heritage includes the extensive Palatinate winegrowing region, picturesque landscapes, and many castles and palaces.


title: Rhineland-Palatinate

text: Rhineland-Palatinate is currently the only federal state in Germany where nuclear weapons are stored extraterritorially under the responsibility and supervision of US forces.


title: Rhineland-Palatinate

text: The state of Rhineland-Palatinate was founded shortly after the Second World War, on 30 August 1946. It was formed mainly from the southern part of the Prussian Rhine Province (the of Koblenz and Trier), from Rhenish Hesse, from the western part of Nassau and the Bavarian Rhenish Palatinate minus the county of Saarpfalz. The Joint German-Luxembourg Sovereign Region () is the only unincorporated area of the state of Rhineland-Palatinate. This condominium is formed by the rivers Moselle, Sauer, and Our, where they run along the border between Luxembourg and Rhineland-Palatinate or the Saarland.


title: Rhineland-Palatinate

text: The present state of Rhineland-Palatinate formed part of the French Zone of Occupation (1945–1949) after the Second World War. It comprised the former Bavarian Palatinate, the ("government districts") of Koblenz and Trier (which formed the southern part of the Prussian Rhine Province), the parts of the Province of Rhenish Hesse () west of the River Rhine and belonged to the People's State of Hesse (), parts of the Prussian province of Hesse-Nassau (Montabaur), and the former Oldenburg region around Birkenfeld (Principality of Birkenfeld).


title: Rhineland-Palatinate

text: On 10 July 1945, the occupation authority on the soil of the present-day Rhineland-Palatinate transferred from the Americans to the French. To begin with, the French divided the region provisionally into two "upper presidiums" (), Rhineland-Hesse-Nassau (for the hitherto Prussian government districts and regions of Koblenz, Trier, and Montabaur) and Hesse-Palatinate (for the hitherto Bavarian Palatinate and old Hessian-Darmstadt province of Rhenish Hesse). The formation of the state was ordained on 30 August 1946, the last state in the Western Zone of Occupation to be established, by Regulation No. 57 of the French military government under General Marie-Pierre Kœnig. It was initially called Rhenish-Palatinate ( or ); the name Rhineland-Palatinate () was first confirmed in the constitution of 18 May 1947.


title: Rhineland-Palatinate

text: The provisional French government at that time wanted originally to leave the option open of annexing further areas west of the Rhine after the Saarland was turned into a protectorate. When the Americans and British, however, had led the way with the establishment of German states, the French came under increasing pressure and eventually followed their example by setting up the states of Baden, Württemberg-Hohenzollern, and Rhineland-Palatinate. However, the French military government forbade the Saarland from joining Rhineland-Palatinate. Mainz was named as the state capital in the regulation; the "Mixed Commission" (), named as the highest organ of state charged with the administration of the new state and with the preparation of an advisory state assembly, started its work in Mainz. However, war damage and destruction meant that Mainz did not have enough administrative buildings, so the headquarters of the state government and parliament was provisionally established in Koblenz. On 22 November 1946, the constituent meeting of the Advisory State Assembly () took place there, and a draft constitution was drawn up. Previously, local elections had been held. Wilhelm Boden was (after a short term of office as the of Rhineland-Hesse-Nassau) nominated on 2 December as the minister president of the new state by the French military government.


title: Rhineland-Palatinate

text: Adolf Süsterhenn submitted to the Advisory State Assembly a draft constitution, which was passed after several rounds of negotiation on 25 April 1947 in a final vote, with the absolute majority of the CDU voting for and the SPD and KPD voting against. A point of contention involved the draft constitution providing for separate schools based on Christian denomination. On 18 May 1947 53% of the electorate adopted the Constitution for Rhineland-Palatinate in a referendum. While the Catholic north and west of the new state adopted the constitution by a majority, the majority in Rhenish Hesse and the Palatinate voted against. On the same date the first elections took place for the state parliament, the Landtag of Rhineland-Palatinate. The inaugural assembly of parliament took place on 4 June 1947 in the large city hall at Koblenz. Wilhelm Boden was elected the first minister-president of Rhineland-Palatinate. Just one month later, Peter Altmeier succeeded him.


title: Rhineland-Palatinate

text: The constitutional bodies -the Government (), the Parliament () and the Constitutional Court () – established their provisional seat in Koblenz. In the following period, Koblenz and Mainz each emphasized their suitability as the state capital in a public debate. From the beginning, Minister-President Altmeier pressed for Mainz as the capital because he knew that the south of the country, especially the Palatinate, would not accept Koblenz, which was far to the north and formerly Prussian. On 16 May 1950, the decided to relocate itself and the from Koblenz to Mainz.


title: Rhineland-Palatinate

text: After the government and parliament moved to Mainz, many state authorities and courts remained in Koblenz, including the Constitutional Court and the State Archives. In addition, the German Federal Archives and Federal Office of Hydrology were established in Koblenz in 1952.


title: Rhineland-Palatinate

text: A sense of community developed only very gradually in the "land of the retort", which had been established largely without regard to the historical affiliations of its inhabitants. It was given little chance of survival, especially as it had very few large industrial centres. However, the establishment of numerous military bases, both Allied and Bundeswehr, helped to some extent to boost the economy. In 1956, under Article 29 of the Basic Law for the Federal Republic of Germany, petitions were made in the regions of Koblenz, Trier, Montabaur, Rhenish Hesse, and Palatinate for their separation from the state and incorporation into the respective states of North Rhine-Westphalia, Hesse, Bavaria, and Baden-Württemberg. All petitions for a referendum except those in the administrative district of Palatinate won the necessary majority; however, almost 20 years passed before the referendums finally took place. On 19 January 1975, none of the regions concerned returned a majority for being transferred to another state. This put an end to decades of discussion. Only the AKK conflict, a dispute over the districts of Mainz-Amöneburg, Mainz-Kastel, and Mainz-Kostheim, has continued to exercise politicians up to the present day.


title: Rhineland-Palatinate

text: Rhineland-Palatinate shares international borders with France (Grand Est), Luxembourg (Clervaux, Diekirch, Echternach, Grevenmacher, Remich, and Vianden), and Belgium (Wallonia). Within Germany, its neighbours are Baden-Württemberg, Hesse, North Rhine-Westphalia, and the Saarland. It is the ninth-largest state by area. Rhineland-Palatinate is part of the SaarLorLux euregion.


title: Rhineland-Palatinate

text: With 42% of its area covered by forests, it is the most forested state along with Hesse. The state's major rivers are the Rhine, including the UNESCO World Heritage Site Middle Rhine, and the Moselle. Several crater lakes of volcanic origin are in the Eifel, the largest of which is the Laacher See.


title: Rhineland-Palatinate

text: The Rhenish Massif with a typically humid continental climate (Dfb). Summers here are usually warm with much precipitation and winters tend to be dry and cold. Snowfall is a common occasion.


title: Rhineland-Palatinate

text: The deep Valleys of Rhine and Moselle with an oceanic climate (Köppen climate classification: "Cfb"), very closely bordering a humid subtropical climate (Köppen climate classification: "Cfa"). Summers are hot and moist, while winters are mild. Freezing temperatures are rare and snow usually melts within hours. The mild climate allows palm trees, fig trees, oleander shrubs, olive trees and other exotic plants to grow in this area.


title: Rhineland-Palatinate

text: The Rhenish Massif forms roughly the northern half of the state, including the regions Eifel, Moselle Valley, Hunsrück, Westerwald, and parts of the Taunus. The Palatinate forms the biggest part of the southern half along with Rhenish Hesse. The Nahe Valley separates both parts.


title: Rhineland-Palatinate

text: Rhineland-Palatinate is divided into 24 districts ("Landkreise"), 12 independent cities ("Kreisfreie Städte").


title: Rhineland-Palatinate

text: , 40.3% of the population of the state adhered to the Roman Catholic Church and 26.8% to the Evangelical Church in Germany; 32.9% of the population is irreligious or adheres to other religions. Muslims made up 5.0% of the total.


title: Rhineland-Palatinate

text: The league of ShUM-cities in the later Rhineland-Palatinate comprised the Jewish communities of Mainz, Speyer, and Worms, which became the center of Jewish life during medieval times. The "Takkanot Shum" (), or Enactments of ShU"M were a set of decrees formulated and agreed upon over a period of decades by their Jewish community leaders. Today, there are approximately 20,000 Jews (0.5% of the population) living in the state.


title: Set theory

text: Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.


title: Set theory

text: The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of "naive set theory". After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.


title: Set theory

text: Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.


title: Set theory

text: Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers".


title: Set theory

text: Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874, and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking, and culminated in Cantor's 1874 paper.


title: Set theory

text: Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre", contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.


title: Set theory

text: The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899, Cantor had himself posed the question "What is the cardinal number of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his "The Principles of Mathematics". Rather than the term "set", Russell used the term "class", which has subsequently been used more technically.


title: Set theory

text: In 1906, the term "set" appeared in the book "Theory of Sets of Points" by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.


title: Set theory

text: The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of analysts, such as that of Henri Lebesgue, demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation.


title: Set theory

text: Set theory begins with a fundamental binary relation between an object and a set . If is a "member" (or "element") of , the notation is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. Since sets are objects, the membership relation can relate sets as well.


title: Set theory

text: A derived binary relation between two sets is the subset relation, also called "set inclusion". If all the members of set are also members of set , then is a "subset" of , denoted . For example, is a subset of , and so is but is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term "proper subset" is defined. is called a "proper subset" of if and only if is a subset of , but is not equal to . Also, 1, 2, and 3 are members (elements) of the set , but are not subsets of it; and in turn, the subsets, such as , are not members of the set .


title: Set theory

text: Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The following is a partial list of them:


title: Set theory

text: Some basic sets of central importance are the set of natural numbers, the set of real numbers and the empty set—the unique set containing no elements. The empty set is also occasionally called the "null set", though this name is ambiguous and can lead to several interpretations.


title: Set theory

text: A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the "von Neumann universe" of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number formula_2, known as its "rank." The rank of a pure set formula_3 is defined to be the least ordinal that is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set containing only the empty set is assigned rank 1. For each ordinal formula_2, the set formula_5 is defined to consist of all pure sets with rank less than formula_2. The entire von Neumann universe is denoted formula_7.


title: Set theory

text: Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.


title: Set theory

text: The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:


title: Set theory

text: The above systems can be modified to allow "urelements", objects that can be members of sets but that are not themselves sets and do not have any members.


title: Set theory

text: The "New Foundations" systems of NFU (allowing urelements) and NF (lacking them), associate with Willard Van Orman Quine, are not based on a cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, Thomas Forster has argued that it does reflect an iterative conception of set.


title: Set theory

text: Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic instead of classical logic. Yet other systems accept classical logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject.


title: Set theory

text: Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, vector spaces, and relational algebras can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.


title: Set theory

text: Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of "Principia Mathematica", it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second-order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.


title: Set theory

text: Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes human-written, computer-verified derivations of more than 12,000 theorems starting from ZFC set theory, first-order logic and propositional logic.


title: Set theory

text: "Combinatorial set theory" concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdős–Rado theorem.


title: Set theory

text: "Descriptive set theory" is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.


title: Set theory

text: The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.


title: Set theory

text: A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.


title: Set theory

text: In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In "fuzzy set theory" this condition was relaxed by Lotfi A. Zadeh so an object has a "degree of membership" in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.


title: Set theory

text: An "inner model" of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe "L" developed by Gödel.


title: Set theory

text: One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model "V" of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model "L" constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.


title: Set theory

text: The study of inner models is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).


title: Set theory

text: A "large cardinal" is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo–Fraenkel set theory.


title: Set theory

text: "Determinacy" refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure.


title: Set theory

text: Paul Cohen invented the method of "forcing" while searching for a model of ZFC in which the continuum hypothesis fails, or a model of ZF in which the axiom of choice fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models.


title: Set theory

text: A "cardinal invariant" is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.


title: Set theory

text: "Set-theoretic topology" studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.


title: Set theory

text: From set theory's inception, some mathematicians have objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop's influential book "Foundations of Constructive Analysis".


title: Set theory

text: A different objection put forth by Henri Poincaré is that defining sets using the axiom schemas of specification and replacement, as well as the axiom of power set, introduces impredicativity, a type of circularity, into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that Solomon Feferman has said that "all of scientifically applicable analysis can be developed [using predicative methods]".


title: Set theory

text: Ludwig Wittgenstein condemned set theory philosophically for its connotations of mathematical platonism. He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein identified mathematics with algorithmic human deduction; the need for a secure foundation for mathematics seemed, to him, nonsensical. Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical constructivism and finitism. Meta-mathematical statements — which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory — are not mathematics. Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in "Remarks on the Foundations of Mathematics": Wittgenstein attempted to refute Gödel's incompleteness theorems after having only read the abstract. As reviewers Kreisel, Bernays, Dummett, and Goodstein all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as Crispin Wright begun to rehabilitate Wittgenstein's arguments.


title: Set theory

text: Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory. Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for pointless topology and Stone spaces.


title: Set theory

text: An active area of research is the univalent foundations and related to it homotopy type theory. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with universal properties of sets arising from the inductive and recursive properties of higher inductive types. Principles such as the axiom of choice and the law of the excluded middle can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results.


title: Set theory

text: As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics of naive set theory early in mathematics education.


title: Set theory

text: In the US in the 1960s, the New Math experiment aimed to teach basic set theory, among other abstract concepts, to primary school students, but was met with much criticism. The math syllabus in European schools followed this trend, and currently includes the subject at different levels in all grades. Venn diagrams are widely employed to explain basic set-theoretic relationships to primary school students (even though John Venn originally devised them as part of a procedure to assess the validity of inferences in term logic).


title: Set theory

text: Set theory is used to introduce students to logical operators (NOT, AND, OR), and semantic or rule description (technically intensional definition) of sets (e.g. "months starting with the letter "A""), which may be useful when learning computer programming, since boolean logic is used in various programming languages. Likewise, sets and other collection-like objects, such as multisets and lists, are common datatypes in computer science and programming.


title: Set theory

text: In addition to that, sets are commonly referred to in mathematical teaching when talking about different types of numbers (the sets formula_8 of natural numbers, formula_9 of integers, formula_10 of real numbers, etc.), and when defining a mathematical function as a relation from one set (the domain) to another set (the range).


title: Gay men

text: Gay men are male homosexuals. Some bisexual and homoromantic men may also dually identify as gay, and a number of young gay men also identify as queer. Historically, gay men have been referred to by a number of different terms, including "inverts" and "uranians".


title: Gay men

text: Gay men continue to face significant discrimination in large parts of the world, particularly in most of Asia and Africa. In the United States, many gay men still face discrimination in their daily lives, though some openly gay men have reached national success and prominence. In Europe, Xavier Bettel currently serves as the prime minister of Luxembourg; Leo Varadkar serves as the deputy head of the Government of Ireland (he was Taoiseach (Prime Minister) from June 2017 to June 2020); and from 2011 to 2014, Elio Di Rupo served as Prime Minister of Belgium.


title: Gay men

text: For a time, the term "gay" was used as a synonym for anything related to homosexual men. For example, the term "gay bar" still often refers to a bar which caters primarily to a homosexual male clientele or is otherwise part of gay men's culture. By the end of the 20th century, however, the word "gay" was recommended by LGBT groups and style guides to describe all people exclusively attracted to members of the same sex, while "lesbian" referred specifically to female homosexuals, and "gay men" referred exclusively to male homosexuals.


title: Gay men

text: Some scholars argue that the terms "homosexual" and "gay" are problematic when applied to men in ancient cultures since, for example, neither Greeks or Romans possessed any one word covering the same semantic range as the modern concept of "homosexuality". Furthermore, there were diverse sexual practices that varied in acceptance depending on time and place. Other scholars argue that there are significant similarities between ancient and modern male homosexuals.


title: Gay men

text: In cultures influenced by Abrahamic religions, the law and the church established sodomy as a transgression against divine law or a crime against nature. The condemnation of anal sex between males, however, predates Christian belief. Throughout the majority of Christian history, most Christian theologians and denominations have considered homosexual behavior as immoral or sinful. Many historical figures, including Socrates, Lord Byron, Edward II, and Hadrian, have had terms such as "gay" or "bisexual" applied to them. Some scholars, such as Michel Foucault, have regarded this as risking the anachronistic introduction of a contemporary construction of sexuality foreign to their times, though other scholars challenge this.


title: Gay men

text: The first record of a possible homosexual male couple in history is commonly regarded as Khnumhotep and Niankhkhnum, an ancient Egyptian couple, who lived around 2400 BCE. The pair are portrayed in a nose-kissing position, the most intimate pose in Egyptian art, surrounded by what appear to be their heirs. The 6th- or 7th-century Ashmolean Parchment AN 1981.940 provides the only example in Coptic language of a love spell between men. At around of 1240, the Coptic Egyptian Christian writer Abul Fada'il Ibn al-'Assal compiled a legal code known as Fetha Nagast. Written in Ge'ez language, Ibn al-'Assal referred his laws from apostolic writer and former laws of Byzantine Empire. Fetha Nagast was written in two parts: the first dealt with the Church hierarchy sacraments and connected to religious rites. The second concerned laity, civil administration such as family laws. In 1960, when the government enacted the civil code of Ethiopia, it cited the Fetha Nagast as an inspiration to the codification commission. More recently, the European colonization of Africa resulted in the introduction of anti-sodomy laws, and is generally regarded as the central reason why African nations have such stringent laws against gay men today. Three countries or jurisdictions have imposed the death penalty for gay men in Africa. These include Mauritania and several regions in Nigeria and Jubaland.


title: Gay men

text: As is true of many other non-Western cultures, it is difficult to determine the extent to which Western notions of sexual orientation apply to Pre-Columbian cultures. Evidence of homoerotic sexual acts between men has been found in many pre-conquest civilizations in Latin America, such as the Aztecs, Mayas, Quechuas, Moches, Zapotecs, the Incas, and the Tupinambá of Brazil. The Spanish "conquistadors" expressed horror at discovering sodomy openly practiced among native men and used it as evidence of their supposed inferiority. The "conquistadors" talked extensively of sodomy among the natives to depict them as savages and hence justify their conquest and forced conversion to Christianity. As a result of the growing influence and power of the conquistadors, many Native leaders started condemning homosexual acts themselves. During the period following European colonization, homosexuality was prosecuted by the Inquisition, sometimes leading to death sentences on the charges of sodomy, and the practices became clandestine. Many homosexual men went into heterosexual marriages to keep appearances, and some turned to the clergy to escape public scrutiny.


title: Gay men

text: During the Mexican Inquisition, after a series of denunciations, authorities arrested 123 men in 1658 on suspicion of homosexuality. Although many escaped, the Royal Criminal Court sentenced fourteen men from different social and ethnic backgrounds to death by public burning, in accordance to the law passed by Isabella the Catholic in 1497. The sentences were carried out together on one day, 6 November 1658. The records of these trials and those that occurred in 1660, 1673 and 1687, suggest that Mexico City, like many other large cities at the time had an active underworld.


title: Gay men

text: In East Asia, same-sex relations between men has been noted since the earliest recorded history. Homosexuality in China, known as the "passions of the cut peach" and various other euphemisms, has been recorded since approximately 600 BCE. Male homosexuality was mentioned in many famous works of Chinese literature. The instances of same-sex affection and sexual interactions described in the classical novel "Dream of the Red Chamber" seem as familiar to observers in the present as do equivalent stories of romances between heterosexual people during the same period. Confucianism, being primarily a social and political philosophy, focused little on sexuality, whether homosexual or heterosexual. Ming dynasty literature, such as "Bian Er Chai" (弁而釵/弁而钗), portray homosexual relationships between men as more enjoyable and more "harmonious" than heterosexual relationships. Writings from the Liu Song Dynasty by Wang Shunu claimed that homosexuality was as common as heterosexuality in the late 3rd century China. Opposition to male homosexuality in China originates in the medieval Tang Dynasty (618–907), attributed to the rising influence of Christian and Islamic values, but did not become fully established until the Westernization efforts of the late Qing Dynasty and the Republic of China.


title: Gay men

text: The earliest Western documents (in the form of literary works, art objects, and mythographic materials) concerning same-sex male relationships are derived from ancient Greece. These relationships were constrained between "normal" men and their young male lovers. Relationships between adult men, however, were still largely considered taboo in Ancient Greek culture. Given the importance in Greek society of cultivating the masculinity of the adult male and the perceived feminizing effect of being the passive partner, relations between adult men of comparable social status were considered highly problematic, and usually associated with social stigma.


title: Gay men

text: This stigma, however, was reserved for only the passive partner in the relationship. According to contemporary opinion, Greek men who took on a passive sexual role after reaching adulthood – at which point they were expected to take the reverse role in pederastic relationships and become the active and dominant member – thereby were feminized or "made a woman" of themselves. There is ample evidence in the theater of Aristophanes that derides these passive men and gives a glimpse of the type of biting social opprobrium and shame ("atimia") heaped upon them by their society.


title: Gay men

text: Some scholars argue that there are examples of male homosexual love in ancient literature, such as Achilles and Patroclus in the "Iliad". In Ancient Rome, young men's bodies remained a focus of male sexual attention, but relationships were between older free men and slaves or freed youths who took the receptive role in sex. The Hellenophile emperor Hadrian is renowned for his relationship with Antinous, but the Christian emperor Theodosius I decreed a law on 6 August 390, condemning passive males to be burned at the stake.


title: Gay men

text: During the Renaissance, wealthy cities in northern Italy—Florence and Venice in particular—were renowned for their widespread practice of same-sex love, engaged in by a considerable part of the male population and constructed along the classical pattern of Greece and Rome. But even as many of the male population were engaging in same-sex relationships, the authorities, under the aegis of the Officers of the Night court, were prosecuting, fining, and imprisoning a good portion of that population.


title: Gay men

text: From the second half of the 13th century, death was the punishment for male homosexuality in most of Europe.


title: Gay men

text: The relationships of socially prominent figures, such as King James I and the Duke of Buckingham, served to highlight the issue, including in anonymously authored street pamphlets: "The world is chang'd I know not how, For men Kiss Men, not Women now; ... Of J. the First and Buckingham: He, true it is, his Wives Embraces fled, To slabber his lov'd Ganimede" ("Mundus Foppensis, or The Fop Display'd", 1691).


title: Gay men

text: In ancient Sumer, a set of priests known as "gala" worked in the temples of the goddess Inanna, where they performed elegies and lamentations. "Gala" took female names, spoke in the "eme-sal" dialect, which was traditionally reserved for women, and appear to have engaged in homosexual intercourse. The Sumerian sign for "gala" was a ligature of the signs for "penis" and "anus". One Sumerian proverb reads: "When the "gala" wiped off his ass [he said], 'I must not arouse that which belongs to my mistress [i.e., Inanna]. In later Mesopotamian cultures, "kurgarrū" and "assinnu" were male servants of the goddess Ishtar (Inanna's East Semitic equivalent), who dressed in female clothing and performed war dances in Ishtar's temples. Several Akkadian proverbs seem to suggest that they may have also engaged in homosexual intercourse. In ancient Assyria, male homosexuality was present and common; it was also not prohibited, condemned, nor looked upon as immoral or disordered. Some religious texts contain prayers for divine blessings on homosexual relationships. The "Almanac of Incantations" contained prayers favoring on an equal basis the love of a man for a woman, of a woman for a man, and of a man for man.


title: Gay men

text: The use of "gay" to mean a "homosexual" man was first used as an extension of its application to prostitution: a "gay boy" was a young man or adolescent serving male clients. Similarly, a "gay cat" was a young man apprenticed to an older hobo and commonly exchanging sex and other services for protection and tutelage. The application to homosexuality was also an extension of the word's sexualized connotation of "uninhibited", which implied a willingness to disregard conventional sexual mores. In court in 1889, the prostitute John Saul stated: "I occasionally do odd-jobs for different gay people."


title: Gay men

text: "Bringing Up Baby" (1938) was the first film to use the word "gay" in an apparent reference to homosexuality. In a scene in which Cary Grant's character's clothes have been sent to the cleaners, he is forced to wear a woman's feather-trimmed robe. When another character asks about his robe, he responds, "Because I just went "gay" all of a sudden!" Since this was a mainstream film at a time, when the use of the word to refer to cross-dressing (and, by extension, homosexuality) would still be unfamiliar to most film-goers, the line can also be interpreted to mean, "I just decided to do something frivolous."


title: Gay men

text: In 1950, the earliest reference found to date for the word gay as a self-described name for male homosexuals came from Alfred A. Gross, executive secretary for the George W. Henry Foundation, who said in the June 1950 issue of "Sir" magazine: "I have yet to meet a happy homosexual. They have a way of describing themselves as gay but the term is a misnomer. Those who are habitues of the bars frequented by others of the kind, are about the saddest people I've ever seen."


title: Gay men

text: Gay men were one of the primary victims of the Nazi Holocaust. Historically, the earliest legal step towards the Nazi persecution of male homosexuality was 1871's Paragraph 175, a law passed after the unification of the German Empire. Paragraph 175 read: "An unnatural sex act committed between persons of male sex ... is punishable by imprisonment; the loss of civil rights might also be imposed." The law was interpreted differently in Germany until 23 April 1880, when the "Reichsgericht" ruled that criminal homosexual acts involved either anal, oral, or intercrural sex between two men. Anything less (such as kissing and cuddling) were deemed harmless play.


title: Gay men

text: Franz Gürtner, the "Reich" Justice Minister amended Paragraph 175 to address "loopholes" in the law after the Night of the Long Knives. The 1935 version of Paragraph 175 declared "expressions" of homosexuality as prosecutable crimes. The most important change to the law was the definitional shift of male homosexuality from "An unnatural sex act committed between persons of male sex" to instead "A male who commits a sex offense with another male." This expanded the reach of the law to persecute gay men as a people group, rather than male homosexuality as a sexual act. Kissing, mutual masturbation and love-letters between men were now seen as legitimate reasons for the police to make arrests. The law never defined a "sex offence", leaving it to interpretation.


title: Gay men

text: Between 1933 and 1945, an estimated 100,000 men were arrested as homosexuals under the Nazi regime, of whom some 50,000 were officially sentenced. Most of these men served time in prison, while an estimated 5,000 to 15,000 were incarcerated in Nazi concentration camps. Rüdiger Lautmann argued that the death rate of homosexuals in concentration camps may have been as high as 60%. Gay men in the camps suffered an unusual degree of cruelty by their captors and were regularly used as the subjects for Nazi medical experiments as scientists tried to find a "cure" for homosexuality.


title: Gay men

text: The HIV/AIDS epidemic is considered the deadliest period in modern history for gay men, and the generation of young gay men who died in the crisis is known as the "lost generation". At its start, the epidemic was particularly severe in the United States. In 1980, San Francisco resident Ken Horne was reported to the CDC with Kaposi's sarcoma (KS). He was retroactively identified as the first patient of the AIDS epidemic in the US. In 1981, Lawrence Mass became the first journalist in the world to write about the epidemic in the "New York Native". Later that year, the CDC reported a cluster of "Pneumocystis" pneumonia in five gay men in Los Angeles. The next month, "The New York Times" ran the headline: "Rare Cancer Seen in 41 Homosexuals". The illness was soon termed "Gay Related Immunodeficiency" (G.R.I.D.), because it was believed to only affect gay men. In June 1982, Larry Kramer founded the Gay Men's Health Crisis to provide food and support to gay men dying in New York City. During the early years of the AIDS crisis, gay men were treated pitilessly in hospital quarantine wards, left alone without contact for weeks at a time.


title: Gay men

text: During the early years of the epidemic, there was significant misinformation surrounding the illness. Rumors swirled that being in the same room or being touched by a gay man could lead one to contract HIV. It was not until April 1984 that the U.S. Health and Human Services Secretary Margaret Heckler announced in a press conference that the American scientist Robert Gallo had discovered the "probable cause" of AIDS, the retrovirus which would be named "human immunodeficiency virus" or HIV. In September 1985, during his second term in office, US President Ronald Reagan publicly mentioned AIDS for the first time after being asked about his administration's lack of medical research funding for the crisis. Four months later, Anthony Fauci, the director of the National Institute of Allergy and Infectious Diseases, stated, "One million Americans have already been infected with the virus and that this number will jump to at least 2 million or 3 million within 5 to 10 years." Gay men, trans women, and bisexual men faced the brunt of deaths during the first decade of the crisis. Activists claimed the government was responding to the epidemic with apathy because of the perceived "social undesirability" of these groups. To address this perceived apathy, activists such as Vito Russo, Larry Kramer, and others, took more militant approaches to AIDS activism, organizing direct action through organizations like ACT UP in order to force pharmaceutical corporations and government agencies to respond to the epidemic with more urgency. ACT UP eventually grew into a transnational organization, with 140 chapters around the world, while the AIDS crisis ultimately became a global epidemic. By 2019, complications related to AIDS had taken 32.7 million lives worldwide.


title: Gay men

text: There are 54 nations in Africa recognized by the United Nations and/or African Union. In 34 of these states, male homosexuality is explicitly outlawed. In a 2015 report, Human Rights Watch noted that in Benin and the Central African Republic, male homosexuality is not explicitly outlawed, but both have laws which are applied differently for gay men than for straight men. In Mauritania, northern Nigeria, Somaliland, and Somalia, male homosexuality is punishable by death. In Sierra Leone, Tanzania, and Uganda, gay men receive life imprisonment for homosexual acts, though the law is not regularly enforced in Sierra Leone. In Nigeria, legislation has also made it illegal for family members, allies, and friends of gay men to openly express support for homosexuality, and the country is generally recognized for its "cold-blooded" attitudes toward gay men. Nigerian law states that any heterosexual person "who administers, witnesses, abets or aids" male homosexual activity should receive a 10-year jail sentence. In Uganda, Christian fundamentalist organizations from the United States funded the introduction of "Kill the Gays" legislation to impose the death penalty for gay men. The bill was ruled unconstitutional by the Ugandan Supreme Court in 2014, but retains support in the country and has been reconsidered for implementation. Of all countries in Sub-Saharan Africa, South Africa has the most liberal attitudes toward gay men. In 2006, South Africa became the fifth country in the world to legalize same-sex marriage, and the Constitution of South Africa guarantees gay men and lesbians full equal rights and protections. South Africa is the only country in Africa where LGBT discrimination is constitutionally forbidden; however, social discrimination against South African gay men persists in rural parts of the country, where high levels of religious tradition continue to fuel prejudice and violence.


title: Gay men

text: In the Americas (both North and South), male homosexuality is legal in every country (except Guyana). In the Caribbean, however, nine nations have criminal punishment for "buggery" on their statute books. These countries include Barbados, Saint Vincent and the Grenadines, Dominica, Saint Kitts and Nevis, Grenada, Saint Lucia, Antigua and Barbuda, Guyana, and Jamaica. In Jamaica, sexual intercourse between men is legally punishable by imprisonment, though the law's repeal is pending. As in Singapore, sexual intercourse between women is already legal, though lesbians in Jamaica still experience a high level of social stigma.


title: Gay men

text: In Jamaica, reports of vigilante violence and torture against gay men have been reported by the Jamaican police. In 2013, Amnesty International reported that "Gay men and lesbian women have been beaten, cut, burned, raped and shot on account of their sexuality. ... We are concerned that these reports are just the tip of the iceberg. Many gay men and women in Jamaica are too afraid to go to the authorities and seek help". As a result of this violence, hundreds of gay men from Jamaica sought to emigrate to countries with better human rights records. A 2016 poll from J-Flag showed that 88 percent of those polled disapprove of homosexuality, though since 2018, discriminatory attitudes have decreased slightly.


title: Gay men

text: In the Caribbean, like in other developing countries around the world, homosexual identity is often associated with Westernization, and as a result, homophobia is believed to be an anti-colonial tool. Wayne Marshall wrote that gay men are believed to be "decadent products of the West" and "are thus to be resisted alongside other forms of colonization, cultural or political". Wayne cites the example of the Jamaican dancehall hit "Dem Bow" by Shabba Ranks, which calls for the violent murder of gay men alongside a call for the "freedom for Black people". Marshall notes the irony of this ideological position, considering the historical evidence that homophobia was introduced to colonies by European colonists. Nevertheless, Caribbean scholars have noted the importance of opposition to gay men for Jamaican male gender construction. Kingsley Ragashanti Stewart, a professor of anthropology at the University of the West Indies, writes, "A lot of Jamaican men, if you call them a homosexual, ... will immediately get violent. It's the worst insult you could give to a Jamaican man." Stewart writes that homophobia influences Caribbean society even at the micro level of language. He writes of urban youth vernacular, "It's like if you say, 'Come back here,' they will say, 'No, no, no don't say "come back".' You have to say 'come forward,' because come back is implying that you're 'coming in the back,' which is how gay men have sex."


title: Gay men

text: In Eastern Europe, there has been a steady erosion of rights for gay men over the course of the last decade. In the Russian republic of Chechnya, gay men have been subjected to forced disappearances—secret abductions, imprisonment, torture—and extrajudicial killing by authorities. An unknown number of men, detained due to suspicion of them being gay or bisexual, have died while held in concentration camps. Independent media and human rights groups have reported that gay men are being sent to clandestine camps in Chechnya, described by one eyewitness as "closed prison, the existence of which no one officially knows". Some gay men have attempted to flee the region, but have been detained by Russian police and sent back to Chechnya. Reports have emerged of prison officials releasing accused gay men from the camps after securing assurances from their families that their families will kill them (at least one man was reported by a witness as having died after returning to his family). These imprisoned men are kept in extremely cramped conditions, with 30 to 40 people detained in one room (two to three metres big), and few are afforded a trial. Witnesses have also reported that the gay men are regularly beaten (with polypropylene pipes below the waist), tortured with electricity, and spat in the face by prison guards. In some cases the process of torture has resulted in the death of the person being tortured. As of 2021, the situation in Chechnya continues to worsen for gay men. In other countries in Eastern Europe, rights for gay men continue to deteriorate. Polish President Andrzej Duda has pledged to ban teaching about gay men in schools, forbid same-sex marriage and adoption, and establish "LGBT-free zones".


title: Gay men

text: A 2017 Pew Research Center poll found that the majority of Orthodox Christians in the Eastern European and former USSR states surveyed believe that homosexuality "should not be accepted by society"; 45% of Orthodox Christians in Greece and 31% in the United States answered the same way.


title: Gay men

text: In Southwest Asia and North Africa, gay men face some of the harshest and most hostile laws anywhere in the world. Sex between men is explicitly outlawed in 10 of the 18 "Middle Eastern" countries and is punishable by death in six. According to scholars, recent popular turns toward Islamic fundamentalism has strongly influenced the extreme violence against gay men. While all same-sex activity is legal in Bahrain, Cyprus, the West Bank, Turkey, Lebanon, Israel, Jordan, and Iraq, male homosexuality is illegal and punishable by imprisonment in Syria, Oman, Qatar, Kuwait, and Egypt. Israel is the most progressive concerning LGBT rights and recognizing unregistered cohabitation; however, it has faced criticism from academics such as Sarah Schulman and Jason Ritchie for "pinkwashing." And although same-sex marriage is not legal in the country, there is public support for recognizing and registering same-sex marriages performed in other countries.


title: Gay men

text: Male same-sex activity is also punishable by death in the UAE, Saudi Arabia, Iran, and Qatar. In the Gaza Strip and Yemen, punishment for male homosexuality varies between death and imprisonment depending on the act committed. In 2018, a transnational survey conducted in the region by Pew Research Center found that 80% of people polled believed homosexuality was "morally unacceptable", though others argue that the true number of people who support rights for gay men is unclear due to fear of backlash and punishment.


title: Gay men

text: Since the early 20th century, gay men, both out and closeted, have worked as some of the world's most influential fashion designers and started some of the most important fashion houses. Cristóbal Balenciaga (b. 1895), who started the Balenciaga brand, was gay but remained private about his sexuality for his entire life. His long-time partner, Franco-Polish millionaire Władzio Jaworowski d'Attainville, set up the fund for him to start the house. After d'Attainville died, Balenciaga's following collection was designed entirely in black to mourn his loss. The French fashion designer Christian Dior (b. 1905) was another hugely influential fashion designer of the 20th century. Dior never married nor openly identified as a gay man, however, he was known to frequent the gay cultural scene in Paris and was infamously described by Coco Chanel as never having intimately "known" a woman. Yves Saint Laurent (b. 1935), also regarded as being among the foremost fashion designers in the twentieth century, was openly gay; his long-time partner was Pierre Berge. Gianni Versace (b. 1946), an Italian fashion designer and founder of Versace, publicly came out in an interview with "The Advocate" in July 1995. Versace was murdered in 1997.


title: Gay men

text: A younger generation of gay men gained prominence in the fashion world during the late twentieth and early twenty-first centuries. Tom Ford (b. 1961) founded his eponymous brand Tom Ford in 2005 and has stated that he realized he was gay as a young man visiting Studio 54 in the 1980s. Alexander McQueen (b. 1969), who was also openly gay, founded his own label Alexander McQueen in 1992 and was recognized as a fashion prodigy for his revolutionary designs. McQueen said he realized his sexual orientation when he was six years old. In 2000, McQueen had a marriage ceremony with his partner George Forsyth, a documentary filmmaker, on a yacht in Ibiza. McQueen died by suicide in 2010, shortly after the death of his mother. In recent years, gay men have continued to produce some of the most influential fashion in the world, including by designers such as Jeremy Scott (b. 1975), Jason Wu (b. 1982), and Alexander Wang (b. 1983).