title: Hunter-class frigate

text: Following a report by the RAND Corporation into options for Australia’s naval shipbuilding industry, the Government announced an $89 billion naval shipbuilding plan. This plan brought the schedule of the Future Frigate Program forward by three years and announced a "continuous onshore build program to commence in 2020" in South Australia. A competitive evaluation process was announced in April 2016, with Navantia, Fincantieri, and BAE Systems revealed as the contenders to design the ships.


title: Hunter-class frigate

text: The outlined "ambitious plans for the Navy’s surface fleet." At its centre was the requirement for twelve Future Submarines and "eight new Future Frigates, which will be larger than the "Anzac"-class vessels" with a focus on anti-submarine warfare. The accompanying Defence Capability Plan stated that a Government decision would be expected "beyond 2019."


title: Hunter-class frigate

text: The 2013 Defence White Paper reaffirmed the Future Frigate program and suggested that the replacement of the vessels could be brought forward. In the early 2010s, there was significant concern over the 'valley of death' in Australian shipbuilding following the conclusion of the destroyer program. With concerns both over the cost and management of the "Hobart"-class program and a union campaign calling for job security at government-owned shipyard ASC, the Abbott Government committed over $78 million to preliminary studies to determine whether the "Hobart"-class hull could be utilised for the Future Frigate.


title: Hunter-class frigate

text: The 2016 Defence White Paper increased the number of future frigates by one to a total of nine ships.


title: Hunter-class frigate

text: Against this backdrop, the Abbott Government commissioned a study by the RAND Corporation to determine options for the future of naval shipbuilding in Australia. The report found that:


title: Hunter-class frigate

text: In response to the RAND report, the Government announced a $89 billion shipbuilding program. This included bringing forward the Future Frigate program with a "continuous onshore build programme to commence in 2020." The budget for the program has been confirmed as "more than $35 billion" and the Government claims it will "directly create over 2,000 jobs." All nine vessels will be constructed in Adelaide, South Australia.


title: Hunter-class frigate

text: In April 2016 the government announced a competitive evaluation process between Navantia, Fincantieri and BAE Systems for the Future Frigate Program. Additionally, a tender for the combat system was also held between Saab and Lockheed Martin. In October 2017, the government announced that the Aegis combat system and a Saab tactical interface would be used for the class.


title: Hunter-class frigate

text: Navantia offered an evolution of its F-100 base design, which forms the basis for the "Hobart"-class destroyers currently being built in Adelaide for the RAN. In 2014, the Australian Government commissioned a study to use the "Hobart"-class hull which Navantia claims shows it could be adapted to meet the requirements of the Future Frigate program, including integration of the CEAFAR radar and Saab 9LV combat system. Based on this study, a Navantia-designed Future Frigate would have 75 per cent systems commonality with the "Hobart"-class destroyers. Systems on the "Hobart" class include a 48-cell Mk41 vertical launch system, five-inch Mark 45 naval gun, undersea warfare capabilities including a hull mounted sonar and active and passive towed variable depth sonar, as well as the capability to operate a MH-60R "Romeo" Seahawk.


title: Hunter-class frigate

text: The "Hobart"-class Air Warfare Destroyer program has attracted criticism for cost and schedule over-runs: by 2015 the program was three years behind schedule and $800 million over budget. In late 2015, Navantia was selected to bring a shipbuilding management team into government-owned shipyard ASC as part of the AWD reform program. Following the reform program, initiated by ASC prior to Navantia management integration, ASC has stated that "when we reach our budget on ship three...we will be as good as the other Aegis yards in the world."


title: Hunter-class frigate

text: Fincantieri offered the anti-submarine warfare (ASW) variant of its FREMM frigate ("Bergamini" class). Fincantieri originally said that the general hull configuration of the "Bergamini" design will require little or no modification to meet Australian requirements, including the incorporation of the CEAFAR radar, although it has confirmed that some redesign would have been required to incorporate the US Navy Mark 45 five inch naval gun. In 2016, then Australian Defence Minister Christopher Pyne stated that "one of the advantages for this company is that this vessel has been built, it is already in operation. One of the disadvantages is that the company doesn't operate here."


title: Hunter-class frigate

text: BAE Systems offered an export variant of its Type 26. The Type 26 and the smaller, cheaper Type 31 will replace the Royal Navy's Type 22 and Type 23 frigate fleets.


title: Hunter-class frigate

text: The Type 26 in UK service will be equipped with an advanced anti-submarine warfare (ASW) capability, a 24-cell strike length Mk 41 VLS for long-range strike weapons such as the Tomahawk, a 48-cell vertical launch silo (VLS) for Sea Ceptor anti-air missiles, a 5-inch gun, and is capable of landing a Chinook helicopter on its flight deck.


title: Hunter-class frigate

text: On 29 June 2018 the Australian Government announced that the Type 26 had been selected to fulfil the Royal Australian Navy's Future Frigate requirement. The Government also announced that the ships were to be built in Australia with Australian specific modifications and would be named the "Hunter Class" in Australian service. The Government stated that,


title: Hunter-class frigate

text: "The Hunter class will provide the Australian Defence Force with the highest levels of lethality and deterrence our major surface combatants need in periods of global uncertainty. The Hunter class will have the capability to conduct a variety of missions independently, or as part of a task group, with sufficient range and endurance to operate effectively throughout the region. The frigates will also have the flexibility to support non-warfare roles such as humanitarian assistance and disaster relief. Incorporating the leading edge Australian-developed CEA Phased-Array Radar and the US Navy’s Aegis combat management system, with an Australian interface developed by Saab Australia, the Hunter class will be one of the most capable warships in the world."


title: Hunter-class frigate

text: The "Hunter"-class frigate will have a full load displacement and will be approximately in length. The vessel will be capable of sailing in excess of and will have a full complement of 180 crew.


title: Hunter-class frigate

text: A Saab tactical interface with the Aegis combat interface will be used. The vessel will be able to carry one MH-60R ASW helicopter, and has the ability to host other Australian aircraft such as the MRH90 helicopter.


title: Cara Seymour

text: Cara Seymour (born 6 January 1964) is a British actress from Essex, England. She has appeared in films such as "You've Got Mail", "American Psycho", "Adaptation", "Dancer in the Dark", "Gangs of New York", "Hotel Rwanda", "The Savages", and "An Education". She appeared on stage in the New York Shakespeare Festival production of Caryl Churchill's "The Skriker".


title: Irredeemable

text: Irredeemable is an American comic book series written by Mark Waid, drawn by Peter Krause and Diego Barreto, and published by Boom! Studios. The series follows the fall of the world's greatest superhero, the Plutonian, as he begins slaughtering the population of Earth. His former allies, the superhero group The Paradigm, attempt to find a way to stop his rampage while dealing with their own feelings of betrayal and hopelessness. "Irredeemable" #1 premiered in April 2009, and the series ended in May 2012, after thirty-seven issues and one special. Peter Krause was the artist for the first twenty-four issues, after which Diego Barretto took over as artist. A spin-off, "Incorruptible", was released in December 2009, which follows one of the Plutonian's greatest enemies, Max Damage, on his journey to become a superhero in the wake of Plutonian's fall. Waid brought both series to a conclusion in May 2012.


title: Irredeemable

text: Initial pre-publication publicity utilized the tagline "Mark Waid is Evil! Mark Waid is Irredeemable!", which culminated with the release of a limited edition "Mark Waid is Evil" tee-shirt at the 2009 New York Comic Con. On February 23, 2009, Boom! Studios released a trailer, by Craig Kennedy at CK Creative, for the series on YouTube and posted the first 7 pages of the first issue on the company website.


title: Irredeemable

text: The first issue, which included an afterword by Grant Morrison, featured a cover by John Cassaday, a 1-in-4 variant cover by Barry Kitson, and a 1-in-50 incentive cover signed by Mark Waid with artwork by Jeffrey Spokes. The incentive variants by Spokes for the first 12 issues of the series spelled out "Irredeemable", with one letter being featured on each cover. A silver holofoil edition, limited to 500 copies with a cover stating "Mark Waid is Evil", was released at the 2009 Emerald City Convention. The first issue sold out of Diamond Comics Distribution on the day of release, which caused the publisher to immediately solicit a second printing of the comic. The second printing's cover is a sketch version of Kitson variant and a 1-in-20 incentive reprinting of the Spokes cover, not signed by Waid.


title: Irredeemable

text: In April 2011, Krause announced that he would be leaving the series to focus on opportunities outside of comics. Krause claimed that he made the decision in February 2011 after finding himself facing stress and deadlines from multiple commitments. These commitments resulted in art duties for multiple issues of the series being shared between Krause, who drew pages involving the Plutonian, and Diego Barreto, who drew pages relating to the Paradigm. Starting with "Irredeemable" #29 in September 2011, Krause officially left the series, being replaced by Diego and his father Eduardo Barreto.


title: Irredeemable

text: On February 3, 2012, Waid announced that he was bringing both "Irredeemable" and "Incorruptible" to an end with issues #37 and #30 respectively in May 2012. Waid stated that he was "stretched thin right now both personally and professionally", and that the cast of "Irredeemable" were moving naturally towards the series ending he had planned, and so Waid desired to "go out big and grand". Waid did not rule out revisiting the "Irredeemable" series in the future to explore some of the characters but intimated that this would not be possible for some of them, following the series' end.


title: Irredeemable

text: "Irredeemable" is author Mark Waid's third and "most complex" story concerning the "cost of superheroics" or the "path of villainy". "Kingdom Come" concerned the "ethical price of heroism" and "Empire" premised the ultimate failure of superheroes, but "Irredeemable" is "about how the lessons we learn about right and wrong as children can become warped and twisted when challenged by the realities of the adult world." Waid realized that the concept was one he could never properly explore at either DC or Marvel Comics, a "Twilight of the Superheroes"-style story revolving around the premise of "how does a man go from being the world’s greatest superhero to its greatest supervillain?"


title: Irredeemable

text: Waid's premise stems from the rejection of the idea that, in "superhero comics, pretty much everyone who’s called upon to put on a cape is, at heart, emotionally equipped for the job." He expounds of this by stating that:


title: Irredeemable

text: The beauty of Superman is that he can deal with that level of adulation without it going to his head, without it warping him, but he's a very special individual. We presume, whenever we write superheroes and we come up with superhero origins, that anybody who gets the powers of a superhero — even if they are like Spider-Man and they've got things they've got to work out that issue and responsibility and power and responsibility — we assume that they eventually have the emotional makeup it takes to overcome these things. Well, what if you gave that level of power to someone who, at heart, didn't have that emotional capability?


title: Irredeemable

text: Waid further notes that, "by the classic superhero rules," a hero can't concern themselves with what people think of them, but that if "you are so far removed as to not care what people think of you, it takes one less step to not care what people think."


title: Irredeemable

text: During the 2011 San Diego Comic-Con International, Waid stated that he had developed ideas for ending the story but had no end issue planned while sales of the book continued. During the same event, he added that he did not have any intention of rehabilitating the Plutonian character or redeeming him for his actions, saying "There's no hope for Plutonian...but that said, I never actually said the title "Irredeemable" refers to Plutonian."


title: Irredeemable

text: The Plutonian, a powerful being once thought to be the world's greatest superhero, has now become its greatest supervillain. He has destroyed Sky City — the metropolis he once protected — and murdered millions of people across the globe. The series starts with the Plutonian killing his former ally, the Hornet, and his entire family. The remaining superheroes, the Paradigm — Bette Noir, Scylla, Charybdis (Cary), Volt, Qubit, Gil, and Kaidan — search for the reason behind Plutonian's change by speaking to his former sidekick Samsara, whom Plutonian lobotomized with his heat vision.


title: Irredeemable

text: Former enemies of Plutonian attempt to work with him, but they immediately betray him when he offers a way to earn his trust, resulting in a self-destruct of the facility in which they are meeting. The resulting explosion kills Scylla and injures Cary, superhero brothers who are observing the villains.


title: Irredeemable

text: The United Nations offer Plutonian their subservience, hoping to avoid his wrath. The Plutonian takes offense when, after speaking to the Singaporean representative, he realizes that the decision is made out of fear. In response, Plutonian sinks Singapore into the ocean. Meanwhile, Qubit searches for Modeus, the Plutonian's nemesis, using multiple robots designed to look and think like the villain.


title: Irredeemable

text: Qubit and the other heroes discover Modeus' location within Plutonian's citadel. Plutonian is alerted to the heroes' hidden location and attacks them. Cary waits to confront Plutonian while the others teleport into his lair. Cary reveals that he is aware that Plutonian, when he was still a hero, used alien technology recovered from an invasion to buy the silence of a scientist who was a vociferous Plutonian critic due to the latter's tight control of said technology. Plutonian thought the technology was safe and believed the scientist to be sincere in his criticism, but subsequent tampering released a virus that resulted in the deaths of many children. Plutonian tried to conceal his involvement in the event, but when Samsara learned the truth, Plutonian snapped, enraged that no matter what he attempts to do, people are terrified of him.


title: Irredeemable

text: Plutonian attempts to kill Cary, but Cary reveals that his brother's death has made him significantly more powerful. The rest of the Paradigm returns, having failed to locate Modeus, but they have recovered Plutonian's captive Encanta. The group witness Cary beat Plutonian into submission. Plutonian flees and hides in Samsara's grave.


title: Irredeemable

text: Qubit teleports to his base with Encanta, believing she may know how to find Modeus. Encanta details how Modeus had her cast a spell that would put him in the "safest place on Earth". Samsara's restorative powers revive him in his grave to Plutonian's gratitude, but it is revealed that Modeus secretly possesses Samsara's body. Meanwhile, in response to Cary's new power, the US military summons the demon Orian to destroy the heroes entirely. Orian kidnaps Bette to use as bait and learns of her secret past by threatening her husband Gil's life. Bette reveals that she had an affair with Plutonian before his fall, using a magical candle that renders Plutonian mortal to allow them to have sex. Bette stole a piece of the candle wax, but she did not tell the other Paradigm members about it, fearing that her affair would be exposed. The Paradigm confront Orian, but even Cary is unable to compete with his power. The military captures the team, except for a fleeing Bette, and Cary, who is dragged away by Orian into his own dimension.


title: Irredeemable

text: Encanta is teleported from Qubit's lab by a rogue Modeus robot that is revealed to be in possession of Scylla, who is still alive. Cary and Orian return to Earth, agreeing to work together to kill Plutonian. Bette liberates Qubit, Kaidan, and Volt from prison, but she finds that Gil has already escaped. The group teleport to Bette's home to retrieve the wax fragment that she has crafted into a bullet, but Gil has already taken it. Cary, Orian, Gil, and Bette confront Plutonian. While they fight, Modeus/Samsara pushes Volt over a cliff to his death. Bette shoots at Plutonian with the wax bullet, but Qubit uses his portals to redirect the bullet and kill Orian, having correctly surmised that Orian intended to invade Earth after Plutonian's death. Plutonian and Samsara retreat, and the Paradigm shun Qubit for wasting the opportunity to stop Plutonian. It is revealed that Qubit retrieved the wax bullet after it killed Orian.


title: Irredeemable

text: At Volt's funeral, Kaidan discovers she can summon the spirits of her fallen comrades. For their alliance with Orian, Cary destroys the controls for the US nuclear arsenal, making them dependent on the Paradigm's power. Elsewhere, Modeus/Samsara offers Plutonian a chance to undo his actions. Kaidan realizes that Scylla is alive when she fails to summon his spirit. Modeus/Samsara and Plutonian travel to Sky City with a magical gem. Plutonian laments that only Modeus could figure out how to use the gem to restore Sky City, and he reminisces about the time he realized that Modeus was in love with him. Plutonian reveals that he is aware of Modeus' possession of Samsara and burns off Samsara's face. Qubit shows Kaidan and Cary a recording made by Hornet prior to his death, detailing a deal he made with the aliens the Vespan when they invaded Earth; the locations of habitable alien worlds in exchange for them leaving Earth alone, but returning to subdue the Plutonian if he ever turned evil. Hornet activated the signal to the Vespan before he was killed. The Vespan arrive and subdue Plutonian with advanced technology. The rogue Modeus robot and Encanta find the injured Modeus/Samsara and transfer Modeus into the robot's body. Plutonian is taken off world, kept passive in an artificial reality where he is still a hero.


title: Irredeemable

text: Cary takes credit for Plutonian's defeat and offers amnesty to supervillains that join the Paradigm to help rebuild the Earth. Plutonian is fitted with an exoskeleton to put his unconscious body to work mining on an alien world; the Vespan find they cannot awaken him from his fantasy. Qubit confronts Cary, accusing him of not searching for Scylla out of fear of losing his new power. Cary threatens to tell the world that Qubit spared Plutonian's life. Modeus plots to return Plutonian to Earth. In space, the Vespan realize that they cannot contain the Plutonian, and send him to Gehnom, an insane asylum in the heart of a sun.


title: Irredeemable

text: On Earth, Cary uses the telepathic villain Burrows to interrogate the villains seeking amnesty to determine if their motives are pure. Qubit has Burrows secretly scan Cary and learns enough to deem him as big a threat as Plutonian. Modeus kidnaps Kaidan to force Qubit into an alliance to recover Plutonian. On Gehnom, Plutonian wakes from his fantasy to find the Auroran, a heroic figure from his dream, is present in the waking world.


title: Irredeemable

text: Qubit and Modeus travel to the Vespan homeworld and coerce Plutonian's location from the Vespan ambassador by threatening to disable the portal technology they employ that is based on Qubit's designs. On Gehnom, Plutonian and Auroran work together to escape the asylum, fighting their way through the inmates until they face Mordanse, a bestial alien. They negotiate an alliance with Mordanse to escape, and they are later joined by another alien, Kurne.


title: Irredeemable

text: On Earth, Cary tricks Burrows into revealing that he has been reading Cary's mind, learning of his increasing instability and anger. On Gehnom, Plutonian's group are joined by the aliens Cutter and Mallus, the latter of whom can punch a person through time. Before they escape, Plutonian confronts Auroran about why he resembles the first person Plutonian ever saved. Auroran is unveiled as a shape-shifting symbiote feeding on Plutonian's dreams. Plutonian kills Auroran and travels with his group to the portal that will lead them from Gehnom. There, they find Qubit and Modeus waiting. Qubit remotely destroys all of his teleportation technology to prevent Plutonian's escape. Plutonian notices a blur and realizes that it is himself from the immediate future, sent back in time by Mallus' power. While Qubit believed he had teleported Plutonian's allies away, the future Plutonian had gathered his allies and traveled through the Gehnom portal to Earth. The present Plutonian fades away and Modeus' robot body falls apart, leaving Qubit and Mallus trapped on Gehnom.


title: Irredeemable

text: Witnessing Plutonian's return, the public turn against the Paradigm for promising Earth was safe. Plutonian's group assaults the new Paradigm, but stands aside as their villain recruits attack Cary for bringing them to Plutonian's attention. After peering into Plutonian's mind, Burrows hangs himself. Plutonian leaves, taking some of the villain recruits with him, whom he later kills. Chinese and Japanese representatives meet with the US President to reveal their plan to kill Plutonian, which will result in the death of two billion people. Plutonian locates Bette and brings her into his group.


title: Irredeemable

text: Elsewhere, Kaidan escapes Modeus' imprisonment, summoning Scylla's spirit to fight his mindless body. Scylla's body is killed during the battle. Kaidan and Scylla's freed spirit find Gil and recruit him to locate Elliott, Scylla and Cary's triplet brother, to stop Cary. Cary meets with Elliott, who has become a priest and abandoned his powers. Bette repeatedly attempts suicide but is saved by the Plutonian. China, Japan, and the US launch their plan, releasing two towering creatures from a hidden facility, causing widespread destruction and billions of deaths. Scylla, Gil, and Kaidan find Elliott with Cary, and Gil kills Cary to transfer his power into Elliott, not knowing that Cary is the source of the power. Once Cary has died, the power is lost.


title: Irredeemable

text: Cutter confronts Bette and is revealed to be possessed by Modeus. He takes possession of Bette and discards Cutter's body. Elsewhere, the two towering alien creatures confront Plutonian, claiming to be his parents.


title: Irredeemable

text: The creatures take Plutonian to the end of time and existence, intending to leave him there. Plutonian asks them to explain how he came to be. The creatures reveal themselves as the Eleos, a race of interdimensional beings that explore and record universes. During their mission to Earth years prior, they find humans to be interesting and sacrifice a part of themselves to create a probe to explore humanity. The probe is inadvertently transformed into an infant human in response to the strong emotions of a mother who had killed her child. He is later raised by Bill Hartigan, the only foster parent who is not afraid of Plutonian's powers and trains him to be a selfless superhero. After learning of his wife's inoperable cancer, Bill and his wife are killed in an accident that Plutonian believes was suicide.


title: Irredeemable

text: Plutonian reveals that he hates his enemy Max Damage more than anyone because Max saw Plutonian as a child when he lived in the wilderness as the "wolf boy" and inspired his first act of unrestrained anger. The Eleos accept that Plutonian now feels remorse, but they abandon him at the end of time, refusing to absolve him of his crimes. Qubit and Max Damage teleport to Plutonian and return him to Earth.


title: Irredeemable

text: Qubit amplifies Mallus' powers to allow him to rapidly recreate his teleportation technology and escape Gehnom. Returning to find Earth devastated by radiation, and believing humanity will become extinct within three generations, Qubit recruits Max as muscle and seeks out Plutonian. When Qubit offers to make Plutonian even stronger in order to remove the radiation, Max abandons the plan and leaves.


title: Irredeemable

text: Qubit's efforts to remove the radiation fail. During a conversation about how Modeus helped create the radiation as a child, Plutonian realizes that he has possessed Cutter and flies after Bette, believing Modeus will hurt her. Modeus/Bette sends Kurne after Gil and Kaidan, and captures Mordanse. Plutonian returns, but Modeus uses Bette's gravity manipulation to overpower Plutonian. Modeus uses his power to brutally beat Plutonian when his advances are rejected, but Plutonian finds himself unable to harm Bette's body. Gil leads Kaidan and Scylla to the Garden of Eden. Kaidan and Scylla debate planting the seeds of the Tree of Life, granting immortality to humanity and immunity from the radiation but condemning them to an eternity of torture by Plutonian and preventing Kaidan from summoning the dead. Qubit confronts Modeus/Bette and convinces Plutonian to kill her body, admitting the hypocrisy of his no-killing rule. Modeus transfers himself into Qubit but is left trapped by Qubit's mental safeguards. Qubit uses Modeus' memories to discover how to stop the radiation, and Plutonian retrieves the knowledge of Bette's powers, necessary to stop the radiation. Plutonian confronts Qubit about his promise of redemption after using his time-observing power to discover that Mallus was killed sending Qubit to Earth.


title: Irredeemable

text: Qubit teleports the magic wax bullet into Plutonian's heart as a final threat, forcing Plutonian to use the last of Bette's gravity warping energy to absorb the deadly radiation by racing around the planet as fast as possible.


title: Irredeemable

text: The plan succeeds, and the radiation is removed, but the combined energy is too much, and it fatally ravages Plutonian's body. Qubit reveals his true plan to redeem Plutonian: using his portals to transport Plutonian's original essence, as created by the Eleos, into parallel universes to give it a chance to be remade into something good. In one universe, the essence inspires two young boys, implied to be Jerry Siegel and Joe Shuster, to create the ultimate comic superhero, drawing a character similar in appearance to Superman.


title: Irredeemable

text: A group of superheroes and the Plutonian's former teammates. Established four years before the beginning of the series by the Plutonian, Bette Noir, Qubit, Charybdis, Scylla, Metalman, Hornet, and Kaidan. The surviving members work together to discover a means to stop the Plutonian's rampage.


title: Irredeemable

text: Initial critical reception for the series was positive, the first issue receiving 3.5 out of 5 stars from Comic Book Resources, complimenting the way the art worked with the story, and 4 out of 5 stars from Major Spoilers. John Hardick of "The Express-Times" describes the series as "on its way to becoming one of the best books of the year". Stephen Joyce at Comics Bulletin praised the writing, characters and art (in particular the coloring) in the first issue, concluding "This is an amazing book! I cannot stress how much I truly enjoyed it." Although the second issue was less action-packed Joyce still thought it was a "great story" and despite only being two installments in he says "I’m willing to bet that this story becomes a classic once it is completed".


title: Irredeemable

text: On May 5, 2016, "Deadline Hollywood" announced that 20th Century Fox was producing a feature film adaptation of the comic book written by Tommy Wirkola and directed by Adam McKay.


title: Irredeemable

text: On March 17, 2022, it was announced that Netflix will be adapting the comic book into a feature film, combining both "Irredeemable" and "Incorruptible", Jeymes Samuel will direct, Kemp Powers writing, and Jay-Z producing with Wirkola & McKay no longer attached.


title: Irredeemable

text: In December 2009, a spin-off titled "Incorruptible" was released, chronicling the reformation of former supervillain—and one of Plutonian's greatest enemies—Max Damage, into a hero after he witnesses the atrocities committed by Plutonian. The series finished in June 2012 with issue #30.


title: Masquerade (2012 film)

text: Masquerade (; lit. "Gwanghae: The Man Who Became King") is a 2012 South Korean period drama film starring Lee Byung-hun in dual role as the bizarre King Gwanghae and the humble acrobat Ha-sun, who stands in for the monarch when he faces the threat of being poisoned.


title: Masquerade (2012 film)

text: With 12.3 million tickets sold, "Masquerade" is the ninth highest-grossing South Korean film. Also, it swept the 49th Grand Bell Awards, winning in 15 categories, including Best Film, Director, Screenplay and Actor.


title: Masquerade (2012 film)

text: The confusing and conspiratorial 15th ruler of Korea's Joseon Dynasty King Gwang-hae (Lee Byung-hun) orders his Secretary of Defense, Heo Gyun (Ryu Seung-ryong), to find him a double in order to avoid the constant threat of assassination. In constant fear of being poisoned, the king becomes obnoxious and threatens everyone around him, including the kitchen maids. Heo gyun finds Ha-sun, a lowly acrobat and bawdy joker who looks remarkably like the king to replace the king occasionally whenever the king is out of the palace. In few days, just as feared, King Gwang-hae is drugged with Poppy by his favorite consort, conspired by the law minister. Heo Gyun proposes Ha-sun fill the role as the king until King Gwang-hae fully recovers and grooms Ha-sun to look and act like the king. While assuming the role of the king at his first official appearance, Ha-sun begins to ponder the intricacies of the problems debated in his court. Being fundamentally more humanitarian than King Gwang-hae, Ha-sun's affection and appreciation of even the most minor servants slowly changes morale in the palace for the better. Over time he finds his voice and takes control of governing the country with real insight and fair judgments. Even Heo Gyun and the Chief Eunuch are moved by Ha-sun's genuine concern for the people, and realize he is a better ruler than Gwang-hae. Ha-sun even fights for the respect of the Queen's safety and protects her and her brother from death sentences. However, his chief opposition, Park Chung-seo (Kim Myung-gon), notices the sudden shift in the king's behavior and starts to ask questions. The queen (Han Hyo-joo) is also conflicted between the real king and the fake king's secret. The Chief Eunuch and the Secretary of Defense ask Ha-sun to leave the country for good. The king was again brought back to the throne to punish the revolts.


title: Masquerade (2012 film)

text: Historically, Gwanghae, the 15th Joseon king from 1608 to 1623, attempted diplomacy through neutrality as China's Ming Dynasty (1368-1644) and Qing Dynasty (1636-1912) set their sights on the country. He also tried his hand at other reforms and reconstruction to try to make the nation prosperous, including an emphasis on the restoration of documents, but met with opposition and was later deposed and exiled to Jeju Island. Since he was deposed in a coup by the Westerners faction, historians did not give him a temple name like Taejo or Sejong.


title: Masquerade (2012 film)

text: The premise behind the film is an interpretation of the missing 15 days in the Seungjeongwon ilgi or Journal of the Royal Secretariat during Gwanghae's reign—designated by his 1616 journal entry, "One must not record that which he wishes to hide." That premise is entirely fictitious in nature. This is because


title: Masquerade (2012 film)

text: Announced in early 2011 and initially titled "I am the King of Joseon", "The Prince and the Pauper"-inspired historical film was to be directed by Kang Woo-suk and star Jung Jae-young as Gwanghae/Ha-sun and Yoo Jun-sang as Heo Gyun, but Kang left the project over differences of opinion with production firm CJ E&M.


title: Masquerade (2012 film)

text: In November 2011, they were replaced by director Choo Chang-min and actor Lee Byung-hun in his first ever historical film. A month later, Han Hyo-joo was cast as Lee's co-star.


title: Masquerade (2012 film)

text: Called by one review as one of the best South Korean costume dramas in years, the film drew praise for being beautifully written and emotionally involving, as well as for its accomplished acting, sure-handed direction, ambitious scale and commercial appeal. It became the second biggest hit film at the 2012 South Korean box office, attracting 8.2 million admissions in 25 days of release, then 9,091,633 after 31 days. On its 38th day, it became the 7th film in Korean cinema history to surpass the 10 million-milestone attendance. At the end of its theatrical run it was listed as Korea's all-time third highest-grossing film with 12,319,542 tickets sold nationwide (as of April 2015, it is currently sixth).


title: Masquerade (2012 film)

text: The film was adapted into a stage play which ran at Seoul's Dongsoong Art Center from February 23 to April 21, 2013. It was produced by Lee Byung-hun's agency BH Entertainment. Bae Soo-bin and musical theatre actor Kim Do-hyun alternated in the lead role of Gwanghae. As part of the promotion for the play, Lee, Bae and Kim were featured in a photo spread in the inaugural issue of "Grazia Korea", published on February 20, 2013.


title: Masquerade (2012 film)

text: Cable network tvN acquired rights for a television adaptation, starring Yeo Jin-goo. Developed for the station by Studio Dragon and produced by GT:st, it is to air in January 2019.


title: Masquerade (2012 film)

text: South Korean rapper Agust D draws from the film in his 2020 track Daechwita. The lyrics reference King Gwanghae, as well as rising through the ranks from poverty and servitude to become a king, while in the accompanying music video Agust D portrays a scarred Joseon tyrant, confronted by a modern doppleganger.


title: Tarantella

text: ' () is a group of various southern Italian folk dances originating in the regions of Calabria, Campania and Puglia. It is characterized by a fast upbeat tempo, usually in time (sometimes or ), accompanied by tambourines. It is among the most recognized forms of traditional southern Italian music. The specific dance-name varies with every region, for instance "Sonu a ballu" in Calabria, "tammurriata" in Campania, and "pizzica" in Salento. Tarantella is popular in Southern Italy and Argentina. The term may appear as ' in a linguistically masculine construction.


title: Tarantella

text: In the Italian province of Taranto, Apulia, the bite of a locally common type of wolf spider, named "tarantula" after the region, was popularly believed to be highly venomous and to lead to a hysterical condition known as tarantism. This became known as the "tarantella". R. Lowe Thompson proposed that the dance is a survival from a "Dianic or Dionysiac cult", driven underground. John Compton later proposed that the Roman Senate had suppressed these ancient Bacchanalian rites. In 186 BC the tarantella went underground, reappearing under the guise of emergency therapy for bite victims.


title: Tarantella

text: The stately courtship tarantella danced by a couple or couples, short in duration, is graceful and elegant and features characteristic music. On the other hand, the supposedly curative or symptomatic tarantella was danced solo by a victim of a "Lycosa tarantula" spider bite (not to be confused with what is commonly known as a tarantula today); it was agitated in character, lasted for hours or even up to days, and featured characteristic music. However, other forms of the dance were and still are dances of couples usually either mimicking courtship or a sword fight. The confusion appears to derive from the fact that the spiders, the condition, its sufferers ("tarantolati"), and the dances all have names similar to the city of Taranto.


title: Tarantella

text: The dance originated in the Apulia region, and spread throughout the Kingdom of the Two Sicilies. The Neapolitan tarantella is a courtship dance performed by couples whose "rhythms, melodies, gestures, and accompanying songs are quite distinct" featuring faster more cheerful music. Its origins may further lie in "a fifteenth-century fusion between the Spanish Fandango and the Moresque "ballo di sfessartia"." The "magico-religious" tarantella is a solo dance performed supposedly to cure through perspiration the delirium and contortions attributed to the bite of a spider at harvest (summer) time. The dance was later applied as a supposed cure for the behavior of neurotic women ("Carnevaletto delle donne").


title: Tarantella

text: There are several traditional tarantella groups: "Cantori di Carpino", "Officina Zoé", "Uccio Aloisi gruppu", "Canzoniere Grecanico Salentino", "Selva Cupina", "I Tamburellisti di Torrepaduli".


title: Tarantella

text: The tarantella is most frequently played with a mandolin, a guitar, an accordion and tambourines. flute, fiddle, trumpet and clarinet are also used.


title: Tarantella

text: The tarantella is a dance in which the dancer and the drum player constantly try to upstage each other by playing faster or dancing longer than the other, subsequently tiring one person out first.


title: Tarantella

text: Tarantism, as a ritual, is supposed to have roots in the ancient myths. Reportedly, victims who had collapsed or were convulsing would begin to dance with appropriate music and be revived as if a tarantula had bitten them. The music used to treat dancing mania appears to be similar to that used in the case of tarantism though little is known about either. Justus Hecker (1795–1850), describes in his work "Epidemics of the Middle Ages": A convulsion infuriated the human frame [...]. Entire communities of people would join hands, dance, leap, scream, and shake for hours [...]. Music appeared to be the only means of combating the strange epidemic [...] lively, shrill tunes, played on trumpets and fifes, excited the dancers; soft, calm harmonies, graduated from fast to slow, high to low, prove efficacious for the cure. The music used against spider bites featured drums and clarinets, was matched to the pace of the victim, and is only weakly connected to its later depiction in the tarantellas of Chopin, Liszt, Rossini, and Heller.


title: Tarantella

text: While most serious proponents speculated as to the direct physical benefits of the dancing rather than the power of the music, a mid-18th century medical textbook gets the prevailing story backwards, describing that tarantulas will be compelled to dance by violin music. It was thought that the "Lycosa tarantula" wolf spider had lent the name "tarantula" to an unrelated family of spiders, having been the species associated with Taranto, but since "L. tarantula" is not inherently deadly, the highly venomous Mediterranean black widow, "Latrodectus tredecimguttatus", may have been the species originally associated with Taranto's manual grain harvest.


title: Tarantella

text: The Balanchine ballet "Tarantella" is set to the "Grande Tarantelle for Piano and Orchestra", Op. 67 () by Louis Moreau Gottschalk, reconstructed and orchestrated by Hershy Kay. The profusion of steps and the quick changes of direction that this brief but explosive pas de deux requires typify the ways in which Balanchine expanded the traditional vocabulary of classical dance.


title: Endurance

text: Endurance (also related to sufferance, resilience, constitution, fortitude, and hardiness) is the ability of an organism to exert itself and remain active for a long period of time, as well as its ability to resist, withstand, recover from and have immunity to trauma, wounds or fatigue. It is usually used in aerobic or anaerobic exercise. The definition of 'long' varies according to the type of exertion – minutes for high intensity anaerobic exercise, hours or days for low intensity aerobic exercise. Training for endurance can reduce the ability to exert endurance strength unless an individual also undertakes resistance training to counteract this effect.


title: Endurance

text: When a person is able to accomplish or withstand a higher amount of effort than their original capabilities their endurance is increasing which to many personnel indicates progress. In looking to improve one's endurance they may slowly increase the amount of repetitions or time spent, if higher repetitions are taken rapidly muscle strength improves while less endurance is gained. Increasing endurance has been proven to release endorphins resulting in a positive mind. The act of gaining endurance through physical activity has been shown to decrease anxiety, depression, and stress, or any chronic disease in total. Although a greater endurance can assist the cardiovascular system it does not imply that any cardiovascular disease can be guaranteed to improve. "The major metabolic consequences of the adaptations of muscle to endurance exercise are a slower utilization of muscle glycogen and blood glucose, a greater reliance on fat oxidation, and less lactate production during exercise of a given intensity."


title: Endurance

text: In military settings, endurance is considered the ability of a force to sustain high levels of combat potential relative to its opponent over the duration of a campaign.


title: Endurance

text: Different types of endurance performance can be trained in specific ways. Adaptation of exercise plans should follow individual goals.


title: Endurance

text: Calculating the Intensity of exercise the individual capabilities should be considered. Effective training starts within half the individual performance capability. Performance capability is expressed by maximum heart rate. Best results can be achieved within 55 up to 65% of maximum heart rate. Aerobic, anaerobic and further thresholds are not to be mentioned within extensive endurance exercises. In general surveillance of training intensity is achieved through measuring the heart rate.


title: Lauren Lake's Paternity Court

text: Lauren Lake's Paternity Court (originally known as Paternity Court) is a nontraditional court show in which family lawyer and legal analyst Lauren Lake heard and ruled on paternity cases and rendered DNA test results.


title: Lauren Lake's Paternity Court

text: The show was produced by 79th & York Entertainment and Orion Television, and was distributed by MGM Domestic Television Distribution. "Paternity Court" was executive produced by David Armour. The series was MGM's first first-run syndication series to come to the market after a lull period in this arena.


title: Lauren Lake's Paternity Court

text: "Paternity Court" premiered on Monday, September 23, 2013. The court show's first day of taping was on June 13, 2013. By the 2nd season, Lake's name was added to the title of the series, "Lauren Lake's Paternity Court". Production also moved from Los Angeles to the Georgia Public Broadcasting studios in Atlanta that season.


title: Lauren Lake's Paternity Court

text: In 2019 following its 6th season, the show received a Daytime Emmy Award for Outstanding Legal/Courtroom Program. After its 7th season in early 2020, however, "Lauren Lake's Paternity Court" ended production, confirmed as cancelled by 2021. MGM discarded of all of its courtroom programming following financial struggles from the COVID-19 pandemic.


title: Lauren Lake's Paternity Court

text: Reports of the series first emerged in December 2012. As early as December 2012, 9 months before its debut, the court show had already been sold in 75% of the country. Stations acquired the show on an all-barter basis with 3½ minutes of local and 3½ minutes of national advertising time in every episode. By August 2013, the show was sold in 92% of the country.


title: Lauren Lake's Paternity Court

text: An article in "Broadcasting & Cable" talking about the conception of the show listed the paternity test-focused episodes of the daytime talk show "Maury" as a direct inspiration for "Paternity Court", as Bryan said in the article that the high ratings for "Maury" among women 25–54 and the popularity of the court show genre made fusing the two concepts possible. Weigel Broadcasting president Neal Sabin, whose station group was among the first to take the program, thought it was a natural fit for the court show-heavy lineup on his stations, saying it was 'a little bit Maury and a little bit court-y'.


title: Lauren Lake's Paternity Court

text: Producers of the series have argued at the same time that "Paternity Court" and "Maury" do contrast, as "Paternity Court" does not focus on the narrative of "Maury" in building tabloid drama solely from the "who's your daddy?" question posed by paternity tests and the issues of multiple partners possibly being so with only bare follow-ups by that show's staff, but instead uses the tests on their show to build long-term relationships in a healthy manner once those results are revealed. As reported in late 2012, court programming is the second highest-rated genre on daytime television. Bryan has stated the goal of "Paternity Court" is to reinvigorate the court show genre.


title: Lauren Lake's Paternity Court

text: "Paternity Court" is a half-hour hybrid of a talk show using the court show plaintiff/defendant format. Lake talks to the show's litigants and decides cases based upon the results of DNA tests. While the show's title is "Paternity Court", it also looks into other situations using DNA verification, such as probate disputes over wills, which are litigated under a binding arbitration arrangement.


title: Lauren Lake's Paternity Court

text: In late January 2013, creator David Armour revealed several details of the MGM conflict-resolution strip with Lauren Lake:


title: Lauren Lake's Paternity Court

text: Unlike most present-day court shows which typically have two cases in each episode, "Paternity Court" only focuses on one case per episode, though a second case can make an uncommon appearance. Lake takes time before and after the results to speak with her litigants.


title: Lauren Lake's Paternity Court

text: Original plans for "Paternity Court" were to film the series from a real-life courtroom, though this later changed to a traditional studio setting. The first season of the show was taped from Sunset Bronson Studios in Los Angeles, next to the "Judge Judy" set, leasing over the space previously used for Big Ticket Television's "Judge Joe Brown" until its 2013 cancellation. Beginning with season 2, production moved to Georgia Public Broadcasting Studios in Atlanta. The site was previously utilized by "Swift Justice with Nancy Grace" in the 2010–11 season. As with many Atlanta-based productions, the show also received tax credits from the Georgia Film Office for producing the show in Georgia, and the GFO had an end credit for their support.


title: Lauren Lake's Paternity Court

text: "Lauren Lake's Paternity Court" ended production after the 2019–20 television season (the show's 7th season). Due to the financial decline of the show's distributor, MGM, "Paternity Court" ended production along with its sister shows "Couples Court with the Cutlers" and "Personal Injury Court". Resulting from the 2020 COVID-19 pandemic, the MGM studio also went up for sale. The show, however, airs in reruns but ratings for the show are not calculated due to nonexistence of a partnership between MGM and Nielsen ratings. The end of the aforementioned court shows are also rooted in MGM's station clientele consisting of lower-rated networks like The CW and MyNetworkTV.


title: Lauren Lake's Paternity Court

text: After its conclusion, Lake launched a podcast series about her show in July 2020, featuring highlights of its "most popular, shocking, and provocative cases" in conjunction with Audio Up.


title: Lauren Lake's Paternity Court

text: It was announced on June 22, 2021, that Lake will return to television in a revival of the courtroom series, "We the People", previously helmed by Gloria Allred. Lake will preside over her own edition of the series "We the People with Judge Lauren Lake." Similarly to its first installment with Allred, the court show will be produced by Byron Allen's Entertainment Studios, airing on JusticeCentralTV. The show is set to join a series of sister television court shows of that network including "America's Court with Judge Ross", "Justice for All with Judge Cristina Perez", "Justice with Judge Mablean", "Supreme Court with Judge Karen" and "The Verdict with Judge Hatchett".


title: Lauren Lake's Paternity Court

text: "Paternity Court" has two sister shows: "Couples Court with the Cutlers", which was hosted by Keith and Dana Cutler and follows a similar approach of using evidence and testing to prove or disprove infidelity; and "Personal Injury Court", which is hosted by Gino Brogdon and saw personal injury cases. All three court shows ended production in the wake of the COVID-19 pandemic, later confirmed as cancelled.


title: Soen

text: Soen is a Swedish progressive metal supergroup. Their debut album "Cognitive" was released on 15 February 2012.


title: Soen

text: Initially formed in 2004, it was in May 2010 that the formation of Soen was announced. The original lineup consisted of former Opeth drummer Martin Lopez, ex-Death, Testament and Sadus bassist Steve Di Giorgio, Willowtree vocalist Joel Ekelöf and guitarist Joakim Platbarzdis.


title: Soen

text: The group's first song, "Fraccions", was released on the band's official website in October 2010. Their first video, for the song "Savia", was posted on their Facebook page in February 2012. Their second video is a moody acoustic version of the song "Ideate". The band's third video is for the song "Delenda".


title: Soen

text: Drummer Martin Lopez has described Soen's music as "melodic, heavy, intricate and very different than everything else." Their debut, "Cognitive", was released in February 2012 receiving positive critical reception. The members' technical proficiency was praised the most, though many reviewers noted extreme stylistic similarities to the music of Tool.


title: Soen

text: Martin Lopez commented on the Tool similarity: "We are inspired by Tool but I consider them not only a band but a genre. Besides that, I really don't think there's anything to argue about, we make good music whoever it would remind you of. Some people tend to dislike rather than like and are looking for faults instead of seeing music for what it is, good or bad".


title: Soen

text: "Cognitive" was released on Spinefarm records on 15 February 2012. The album was mixed by David Bottrill (Tool, Coheed and Cambria, Silverchair, Smashing Pumpkins, Muse, etc.) and mastered by João Carvalho. There is also a bonus track on the Japanese edition called "Writhen". Soen's second album, "Tellurian" was released on 4 November 2014. The album was produced by Platbarzdis, mixed again by David Bottrill and mastered at Bob Ludwig's Gateway setup in the U.S. by Adam Ayan. "Lykaia" is the third album released by Soen on 3 February 2017. The whole album was recorded using analog equipment.


title: Bart Millard

text: Bart Marshall Millard (born December 1, 1972) is an American singer and songwriter who is best known as the leader of the band MercyMe. He has also released two solo albums: "Hymned, No. 1", in 2005 and "Hymned Again" in 2008. He received a solo Grammy nomination in the category of Best Southern, Country, or Bluegrass Gospel Album for the latter album.


title: Bart Millard

text: In high school, Millard wanted to become a football player, a dream which ended when he injured both ankles at a high school football game. As a result, Millard took choir as an elective. Millard's father, Arthur Wesley Millard Jr., died in 1991, during Bart's first year of college, and his youth pastor invited him to work with the church's youth group worship band. Millard accepted and worked with the video and audio systems for the group. James (Jim) Bryson played piano for that band and later went on to play with Bart Millard and the worship band on a trip to Switzerland. This trip inspired Millard to pursue a full-time musical career. Millard and two of his friends, James Phillip Bryson and Michael John Scheuchzer, moved to Oklahoma City, and formed MercyMe. Since then, the band has recorded six independent, ten studio, two Christmas, and three compilation albums.


title: Bart Millard

text: Millard made a promise to his grandmother to record an album of hymns before she died, which he did with "Hymned No. 1", and he subsequently shared the story of how he was inspired by his grandmother's faith. Millard made the second of his two hymn albums because he realized that the church he was attending did not sing hymns and he wanted his children to have hymns as a part of their lives.


title: Bart Millard

text: Millard said his dad heavily influenced him in his musical direction with respect to the hymn albums, drawing particularly on Willie Nelson and Louie Prima. Millard stated that these influences would not be appropriate for MercyMe, as MercyMe is similar to Coldplay, while these hymns are more in the vein of Frank Sinatra. Millard said that he embarked on his solo effort in order to give expression to musical styles that would not have been compatible with MercyMe. Millard said that they sang a Hank Williams song called "I Saw the Light" in his church growing up.


title: Bart Millard

text: The title of "Hymned Again" is a "tongue-in-cheek" reference to the first album. The first album was an effort to make songs that he did not particularly care about cooler, and the second album was created to achieve a Kansas City swing/shuffle mood in the vein of Louie Prima, Harry Connick Jr. and Jamie Cullum, which is what was achieved. According to Greer, this album was very reminiscent of "She" by Connick, which was done five or six times according to Millard. Millard said he accomplished this by listening to a stack of albums by these musicians. Millard said the songs on the album "Hymned Again" are in the tradition of the Great Revival era musically, but that this was not done on purpose. Millard said the one original song on the album, entitled "Jesus Cares for Me", was written by Thad Cockrell, a song that, according to the singer, "could’ve been written 50, 60 years ago." Millard nervously asked Vince Gill to participate on the album. On the possibility of a future "Hymned" effort, Millard said, "Man, I hope so."


title: Bart Millard

text: Millard is featured as a vocalist on "I See Love," a 2004 single by Third Day and Steven Curtis Chapman. He is also featured as a backing vocalist on Phil Wickham's 2009 single "Safe."


title: Bart Millard

text: Millard was called the Best Male Vocalist by "Christianity Today" in 2005, for his work on his solo album "Hymned No. 1" and for "The Christmas Sessions" album with his band MercyMe. Millard's "Hymned No. 1" was called the No. 9 best album of the year by "Christianity Today" in 2005.


title: Bart Millard

text: Millard's song "I Can Only Imagine" was inspired by his father's death, and was made into a film also titled "I Can Only Imagine". The film was released on March 16, 2018.


title: Bart Millard

text: Millard is from Greenville, Texas. He and his wife Shannon have five children: Sam, Gracie, Charlie, Sophie and Miles.


title: Surreal number

text: In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field.


title: Surreal number

text: Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book "Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness". In his book, which takes the form of a dialogue, Knuth coined the term "surreal numbers" for what Conway had called simply "numbers". Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book "On Numbers and Games".


title: Surreal number

text: A separate route to defining the surreals began in 1907, when Hans Hahn introduced Hahn series as a generalization of formal power series, and Hausdorff introduced certain ordered sets called η-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962 Norman Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α, and in 1987 he showed that taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.


title: Surreal number

text: If the surreals are considered as 'just' a proper class sized real closed field, Alling's 1962 paper handles the case of strongly inaccessible cardinals which can naturally be considered as proper classes by cutting off the cumulative hierarchy of the universe one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense. There is an important additional field structure on the surreals that isn't visible through this lens however, namely the notion of a 'birthday' and the corresponding natural description of the surreals as the result of a cut-filling process along their birthdays given by Conway. This additional structure has become fundamental to a modern understanding of the surreal numbers, and Conway is thus given credit for discovering the surreals as we know them today—Alling himself gives Conway full credit in a 1985 paper preceding his book on the subject.


title: Surreal number

text: In the Conway construction, the surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers "a" and "b", or . (Both may hold, in which case "a" and "b" are equivalent and denote the same number.) Each number is formed from an ordered pair of subsets of numbers already constructed: given subsets "L" and "R" of numbers such that all the members of "L" are strictly less than all the members of "R", then the pair { "L" | "R" } represents a number intermediate in value between all the members of "L" and all the members of "R".


title: Surreal number

text: Different subsets may end up defining the same number: { "L" | "R" } and { "L′" | "R′" } may define the same number even if "L" ≠ "L′" and "R" ≠ "R′". (A similar phenomenon occurs when rational numbers are defined as quotients of integers: and are different representations of the same rational number.) So strictly speaking, the surreal numbers are equivalence classes of representations of form { "L" | "R" } that designate the same number.


title: Surreal number

text: In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: { | }. This representation, where "L" and "R" are both empty, is called 0. Subsequent stages yield forms like


title: Surreal number

text: The integers are thus contained within the surreal numbers. (The above identities are definitions, in the sense that the right-hand side is a name for the left-hand side. That the names are actually appropriate will be evident when the arithmetic operations on surreal numbers are defined, as in the section below). Similarly, representations such as


title: Surreal number

text: arise, so that the dyadic rationals (rational numbers whose denominators are powers of 2) are contained within the surreal numbers.


title: Surreal number

text: After an infinite number of stages, infinite subsets become available, so that any real number "a" can be represented by { "L" | "R" },


title: Surreal number

text: "R" is the set of all dyadic rationals greater than "a" (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.


title: Surreal number

text: where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth.


title: Surreal number

text: Surreal numbers are constructed inductively as equivalence classes of pairs of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set. The construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.


title: Surreal number

text: A "form" is a pair of sets of surreal numbers, called its "left set" and its "right set". A form with left set "L" and right set "R" is written { "L" | "R" }. When "L" and "R" are given as lists of elements, the braces around them are omitted.


title: Surreal number

text: Either or both of the left and right set of a form may be the empty set. The form { { } | { } } with both left and right set empty is also written { | }.


title: Surreal number

text: The numeric forms are placed in equivalence classes; each such equivalence class is a "surreal number". The elements of the left and right set of a form are drawn from the universe of the surreal numbers (not of "forms", but of their "equivalence classes").


title: Surreal number

text: An ordering relationship must be antisymmetric, i.e., it must have the property that "x" = "y" (i. e., "x" ≤ "y" and "y" ≤ "x" are both true) only when "x" and "y" are the same object. This is not the case for surreal number "forms", but is true by construction for surreal "numbers" (equivalence classes).


title: Surreal number

text: The equivalence class containing { | } is labeled 0; in other words, { | } is a form of the surreal number 0.


title: Surreal number

text: Surreal numbers can be compared to each other (or to numeric forms) by choosing a numeric form from its equivalence class to represent each surreal number.


title: Surreal number

text: This group of definitions is recursive, and requires some form of mathematical induction to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable via "finite induction" are the dyadic fractions; a wider universe is reachable given some form of transfinite induction.


title: Surreal number

text: The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no "S" with "i" < 0, the expression formula_2 is the empty set; the only subset of the empty set is the empty set, and therefore "S" consists of a single surreal form { | } lying in a single equivalence class 0.


title: Surreal number

text: For every finite ordinal number "n", "S" is well-ordered by the ordering induced by the comparison rule on the surreal numbers.


title: Surreal number

text: The first iteration of the induction rule produces the three numeric forms { | 0 } < { | } < { 0 | } (the form { 0 | 0 } is non-numeric because 0≤0). The equivalence class containing { 0 | } is labeled 1 and the equivalence class containing { | 0 } is labeled −1. These three labels have a special significance in the axioms that define a ring; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels.


title: Surreal number

text: For every "i" < "n", since every valid form in "S" is also a valid form in "S", all of the numbers in "S" also appear in "S" (as supersets of their representation in "S"). (The set union expression appears in our construction rule, rather than the simpler form "S", so that the definition also makes sense when "n" is a limit ordinal.) Numbers in "S" that are a superset of some number in "S" are said to have been "inherited" from generation "i". The smallest value of α for which a given surreal number appears in "S" is called its "birthday". For example, the birthday of 0 is 0, and the birthday of −1 is 1.


title: Surreal number

text: Comparison of these equivalence classes is consistent, irrespective of the choice of form. Three observations follow:


title: Surreal number

text: The informal interpretations of { 1 | } and { | −1 } are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of { 0 | 1 } and { −1 | 0 } are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled and −. These labels will also be justified by the rules for surreal addition and multiplication below.


title: Surreal number

text: The equivalence classes at each stage "n" of induction may be characterized by their "n"-"complete forms" (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains "every" number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels:


title: Surreal number

text: The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number { 1, 2 | 5, 8 } is therefore equivalent to { 2 | 5 }; one can establish that these are forms of 3 by using the "birthday property", which is a consequence of the rules above.


title: Surreal number

text: A form "x" = { "L" | "R" } occurring in generation "n" represents a number inherited from an earlier generation "i" < "n" if and only if there is some number in "S" that is greater than all elements of "L" and less than all elements of the "R". (In other words, if "L" and "R" are already separated by a number created at an earlier stage, then "x" does not represent a new number but one already constructed.) If "x" represents a number from any generation earlier than "n", there is a least such generation "i", and exactly one number "c" with this least "i" as its birthday that lies between "L" and "R"; "x" is a form of this "c". In other words, it lies in the equivalence class in "S" that is a superset of the representation of "c" in generation "i".


title: Surreal number

text: The addition, negation (additive inverse), and multiplication of surreal number "forms" "x" = { "X" | "X" } and "y" = { "Y" | "Y" } are defined by three recursive formulas.


title: Surreal number

text: This formula involves the negation of the surreal "numbers" appearing in the left and right sets of "x", which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form. This only makes sense if the result is the same, irrespective of the choice of form of the operand. This can be proved inductively using the fact that the numbers occurring in "X" and "X" are drawn from generations earlier than that in which the form "x" first occurs, and observing the special case:


title: Surreal number

text: This formula involves sums of one of the original operands and a surreal number drawn from the left or right set of the other. It can be proved inductively with the special cases:


title: Surreal number

text: which by the birthday property is a form of 1. This justifies the label used in the previous section.


title: Surreal number

text: Multiplication can be defined recursively as well, beginning from the special cases involving 0, the multiplicative identity 1, and its additive inverse -1:


title: Surreal number

text: The formula contains arithmetic expressions involving the operands and their left and right sets, such as the expression formula_9 that appears in the left set of the product of "x" and "y". This is understood as the set of numbers generated by picking all possible combinations of members of formula_10 and formula_11, and substituting them into the expression.


title: Surreal number

text: for positive "y". Only positive "y" are permitted in the formula, with any nonpositive terms being ignored (and "y" are always positive). This formula involves not only recursion in terms of being able to divide by numbers from the left and right sets of "y", but also recursion in that the members of the left and right sets of itself. 0 is always a member of the left set of , and that can be used to find more terms in a recursive fashion. For example, if "y" = 3 = { 2 | }, then we know a left term of will be 0. This in turn means = is a right term. This means


title: Surreal number

text: It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:


title: Surreal number

text: With these rules one can now verify that the numbers found in the first few generations were properly labeled. The construction rule is repeated to obtain more generations of surreals:


title: Surreal number

text: For each natural number (finite ordinal) "n", all numbers generated in "S" are dyadic fractions, i.e., can be written as an irreducible fraction , where "a" and "b" are integers and 0 ≤ "b" < "n".


title: Surreal number

text: The set of all surreal numbers that are generated in some "S" for finite "n" may be denoted as "S" = formula_19. One may form the three classes


title: Surreal number

text: of which "S" is the union. No individual "S" is closed under addition and multiplication (except "S"), but "S" is; it is the subring of the rationals consisting of all dyadic fractions.


title: Surreal number

text: There are infinite ordinal numbers β for which the set of surreal numbers with birthday less than β is closed under the different arithmetic operations. For any ordinal α, the set of surreal numbers with birthday less than β = ω (using powers of ω) is closed under addition and forms a group; for birthday less than ω it is closed under multiplication and forms a ring; and for birthday less than an (ordinal) epsilon number ε it is closed under multiplicative inverse and forms a field. The latter sets are also closed under the exponential function as defined by Kruskal and Gonshor.


title: Surreal number

text: However, it is always possible to construct a surreal number that is greater than any member of a set of surreals (by including the set on the left side of the constructor) and thus the collection of surreal numbers is a proper class. With their ordering and algebraic operations they constitute an ordered field, with the caveat that they do not form a set. In fact it is the biggest ordered field, in that every ordered field is a subfield of the surreal numbers. The class of all surreal numbers is denoted by the symbol formula_21.


title: Surreal number

text: Define "S" as the set of all surreal numbers generated by the construction rule from subsets of "S". (This is the same inductive step as before, since the ordinal number ω is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can only be performed in a set theory that allows such a union.) A unique infinitely large positive number occurs in "S":


title: Surreal number

text: "S" also contains objects that can be identified as the rational numbers. For example, the ω-complete form of the fraction is given by:


title: Surreal number

text: The product of this form of with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.


title: Surreal number

text: Not only do all the rest of the rational numbers appear in "S"; the remaining finite real numbers do too. For example,


title: Surreal number

text: The only infinities in "S" are ω and −ω; but there are other non-real numbers in "S" among the reals. Consider the smallest positive number in "S":


title: Surreal number

text: This number is larger than zero but less than all positive dyadic fractions. It is therefore an infinitesimal number, often labeled ε. The ω-complete form of ε (respectively -ε) is the same as the ω-complete form of 0, except that 0 is included in the left (respectively right) set. The only "pure" infinitesimals in "S" are ε and its additive inverse -ε; adding them to any dyadic fraction "y" produces the numbers "y" ± ε, which also lie in "S".


title: Surreal number

text: One can determine the relationship between ω and ε by multiplying particular forms of them to obtain:


title: Surreal number

text: This expression is only well-defined in a set theory which permits transfinite induction up to "S". In such a system, one can demonstrate that all the elements of the left set of ω"S"·"S"ε are positive infinitesimals and all the elements of the right set are positive infinities, and therefore ω"S"·"S"ε is the oldest positive finite number, 1. Consequently, = ω. Some authors systematically use ω in place of the symbol ε.


title: Surreal number

text: "S" is not an algebraic field, because it is not closed under arithmetic operations; consider ω+1, whose form


title: Surreal number

text: does not lie in any number in "S". The maximal subset of "S" that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities ±ω, the infinitesimals ±ε, and the infinitesimal neighbors "y" ± ε of each nonzero dyadic fraction "y".


title: Surreal number

text: This construction of the real numbers differs from the Dedekind cuts of standard analysis in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in "S" with its forms in previous generations. (The ω-complete forms of real elements of "S" are in one-to-one correspondence with the reals obtained by Dedekind cuts, under the proviso that Dedekind reals corresponding to rational numbers are represented by the form in which the cut point is omitted from both left and right sets.) The rationals are not an identifiable stage in the surreal construction; they are merely the subset "Q" of "S" containing all elements "x" such that "x" "b" = "a" for some "a" and some nonzero "b", both drawn from "S". By demonstrating that "Q" is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element of "Q" is reachable from "S" by a finite series (no longer than two, actually) of arithmetic operations "including multiplicative inversion", one can show that "Q" is strictly smaller than the subset of "S" identified with the reals.


title: Surreal number

text: The set "S" has the same cardinality as the real numbers "R". This can be demonstrated by exhibiting surjective mappings from "S" to the closed unit interval "I" of "R" and vice versa. Mapping "S" onto "I" is routine; map numbers less than or equal to ε (including −ω) to 0, numbers greater than or equal to 1 − ε (including ω) to 1, and numbers between ε and 1 − ε to their equivalent in "I" (mapping the infinitesimal neighbors "y"±ε of each dyadic fraction "y", along with "y" itself, to "y"). To map "I" onto "S", map the (open) central third (, ) of "I" onto { | } = 0; the central third (, ) of the upper third to { 0 | } = 1; and so forth. This maps a nonempty open interval of "I" onto each element of "S", monotonically. The residue of "I" consists of the Cantor set "2", each point of which is uniquely identified by a partition of the central-third intervals into left and right sets, corresponding precisely to a form in "S". This places the Cantor set in one-to-one correspondence with the set of surreal numbers with birthday ω.


title: Surreal number

text: Continuing to perform transfinite induction beyond "S" produces more ordinal numbers α, each represented as the largest surreal number having birthday α. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is ω+1 = { ω | }. There is another positive infinite number in generation ω+1:


title: Surreal number

text: The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled ω − 1 on the basis that it coincides with the sum of and . Similarly, there are two new infinitesimal numbers in generation ω + 1:


title: Surreal number

text: At a later stage of transfinite induction, there is a number larger than ω + "k" for all natural numbers "k":


title: Surreal number

text: This number may be labeled ω + ω both because its birthday is ω + ω (the first ordinal number not reachable from ω by the successor operation) and because it coincides with the surreal sum of ω and ω; it may also be labeled 2ω because it coincides with the product of and . It is the second limit ordinal; reaching it from ω via the construction step requires a transfinite induction on


title: Surreal number

text: This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.


title: Surreal number

text: Note that the "conventional" addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals 1 + ω equals ω, but the surreal sum is commutative and produces 1 + ω = ω + 1 > ω. The addition and multiplication of the surreal numbers associated with ordinals coincides with the natural sum and natural product of ordinals.


title: Surreal number

text: Just as 2ω is bigger than ω + "n" for any natural number "n", there is a surreal number that is infinite but smaller than ω − "n" for any natural number "n". That is, is defined by


title: Surreal number

text: where on the right hand side the notation "x" − "Y" is used to mean . It can be identified as the product of ω and the form { 0 | 1 } of . The birthday of is the limit ordinal ω2.


title: Surreal number

text: To classify the "orders" of infinite and infinitesimal surreal numbers, also known as archimedean classes, Conway associated to each surreal number "x" the surreal number


title: Surreal number

text: where "r" and "s" range over the positive real numbers. If "x" < "y" then ω is "infinitely greater" than ω, in that it is greater than "r" ω for all real numbers "r". Powers of ω also satisfy the conditions


title: Surreal number

text: Each power of ω also has the redeeming feature of being the "simplest" surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal number "x" there will always exist some positive real number "r" and some surreal number "y" so that "x" − "r"ω is "infinitely smaller" than "x". The exponent "y" is the "base ω logarithm" of "x", defined on the positive surreals; it can be demonstrated that log maps the positive surreals onto the surreals and that


title: Surreal number

text: This gets extended by transfinite induction so that every surreal number has a "normal form" analogous to the Cantor normal form for ordinal numbers. This is the Conway normal form: Every surreal number "x" may be uniquely written as


title: Surreal number

text: where every "r" is a nonzero real number and the "y"s form a strictly decreasing sequence of surreal numbers. This "sum", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number. (Zero corresponds of course to the case of an empty sequence, and is the only surreal number with no leading exponent.)


title: Surreal number

text: Looked at in this manner, the surreal numbers resemble a power series field, except that the decreasing sequences of exponents must be bounded in length by an ordinal and are not allowed to be as long as the class of ordinals. This is the basis for the formulation of the surreal numbers as a Hahn series.


title: Surreal number

text: In contrast to the real numbers, a (proper) subset of the surreal numbers does not have a least upper (or lower) bound unless it has a maximal (minimal) element. Conway defines a gap as { "L" | "R" } such that every element of "L" is less than every element of "R", and "L" ∪ "R" = formula_21; this is not a number because at least one of the sides is a proper class. Though similar, gaps are not quite the same as Dedekind cuts, but we can still talk about a completion formula_31 of the surreal numbers with the natural ordering which is a (proper class-sized) linear continuum.


title: Surreal number

text: is greater than all real numbers and less than all positive infinite surreals, and is thus the least upper bound of the reals in formula_31. Similarly the gap formula_35 = { formula_21 | } is larger than all surreal numbers. (This is an esoteric pun: In the general construction of ordinals, α "is" the set of ordinals smaller than α, and we can use this equivalence to write in the surreals; formula_35 denotes the class of ordinal numbers, and because formula_35 is cofinal in formula_21 we have by extension.)


title: Surreal number

text: With a bit of set-theoretic care, formula_21 can be equipped with a topology where the open sets are unions of open intervals (indexed by proper sets) and continuous functions can be defined. An equivalent of Cauchy sequences can be defined as well, although they have to be indexed by the class of ordinals; these will always converge, but the limit may be either a number or a gap that can be expressed as


title: Surreal number

text: with "a" decreasing and having no lower bound in formula_21. (All such gaps can be understood as Cauchy sequences themselves, but there are other types of gap that are not limits, such as ∞ and formula_35).


title: Surreal number

text: Based on unpublished work by Kruskal, a construction (by transfinite induction) that extends the real exponential function exp("x") (with base "e") to the surreals was carried through by Gonshor.


title: Surreal number

text: The powers of ω function is also an exponential function, but does not have the properties desired for an extension of the function on the reals. It will, however, be needed in the development of the base-"e" exponential, and it is this function that is meant whenever the notation ω is used in the following.


title: Surreal number

text: When "y" is a dyadic fraction, the power function , may be composed from multiplication, multiplicative inverse and square root, all of which can be defined inductively. Its values are completely determined by the basic relation , and where defined it necessarily agrees with any other exponentiation that can exist.


title: Surreal number

text: The induction steps for the surreal exponential are based on the series expansion for the real exponential,


title: Surreal number

text: more specifically those partial sums that can be shown by basic algebra to be positive but less than all later ones. For "x" positive these are denoted ["x"] and include all partial sums; for "x" negative but finite, ["x"] denotes the odd steps in the series starting from the first one with a positive real part (which always exists). For "x" negative infinite the odd-numbered partial sums are strictly decreasing and the ["x"] notation denotes the empty set, but it turns out that the corresponding elements are not needed in the induction.


title: Surreal number

text: This is well-defined for all surreal arguments (the value exists and does not depend on the choice of "z" and "z").


title: Surreal number

text: The surreal exponential is essentially given by its behaviour on positive powers of ω, i.e., the function "g(a)", combined with well-known behaviour on finite numbers. Only examples of the former will be given. In addition, holds for a large part of its range, for instance for any finite number with positive real part and any infinite number that is less than some iterated power of ω ( for some number of levels).


title: Surreal number

text: A general exponentiation can be defined as , giving an interpretation to expressions like . Again it is essential to distinguish this definition from the "powers of ω" function, especially if ω may occur as the base.


title: Surreal number

text: A surcomplex number is a number of the form , where "a" and "b" are surreal numbers and is the square root of . The surcomplex numbers form an algebraically closed field (except for being a proper class), isomorphic to the algebraic closure of the field generated by extending the rational numbers by a proper class of algebraically independent transcendental elements. Up to field isomorphism, this fact characterizes the field of surcomplex numbers within any fixed set theory.


title: Surreal number

text: The definition of surreal numbers contained one restriction: each element of L must be strictly less than each element of R. If this restriction is dropped we can generate a more general class known as games. All games are constructed according to this rule:


title: Surreal number

text: Every surreal number is a game, but not all games are surreal numbers, e.g. the game { 0 | 0 } is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a field, but the class of games does not. The surreals have a total order: given any two surreals, they are either equal, or one is greater than the other. The games have only a partial order: there exist pairs of games that are neither equal, greater than, nor less than each other. Each surreal number is either positive, negative, or zero. Each game is either positive, negative, "zero", or "fuzzy" (incomparable with zero, such as {1|−1}).


title: Surreal number

text: A move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a fuzzy game for the first player to move.


title: Surreal number

text: If "x", "y", and "z" are surreals, and "x"="y", then "x" "z"="y" "z". However, if "x", "y", and "z" are games, and "x"="y", then it is not always true that "x" "z"="y" "z". Note that "=" here means equality, not identity.


title: Surreal number

text: The surreal numbers were originally motivated by studies of the game Go, and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized "Game" for the mathematical object {L|R}, and the lowercase "game" for recreational games like Chess or Go.


title: Surreal number

text: For most games, the initial board position gives no great advantage to either player. As the game progresses and one player starts to win, board positions will occur in which that player has a clear advantage. For analyzing games, it is useful to associate a Game with every board position. The value of a given position will be the Game {L|R}, where L is the set of values of all the positions that can be reached in a single move by Left. Similarly, R is the set of values of all the positions that can be reached in a single move by Right.


title: Surreal number

text: The zero Game (called 0) is the Game where L and R are both empty, so the player to move next (L or R) immediately loses. The sum of two Games G = { L1 | R1 } and H = { L2 | R2 } is defined as the Game G + H = { L1 + H, G + L2 | R1 + H, G + R2 } where the player to move chooses which of the Games to play in at each stage, and the loser is still the player who ends up with no legal move. One can imagine two chess boards between two players, with players making moves alternatively, but with complete freedom as to which board to play on. If G is the Game {L | R}, -G is the Game {-R | -L}, i.e. with the role of the two players reversed. It is easy to show G - G = 0 for all Games G (where G - H is defined as G + (-H)).


title: Surreal number

text: This simple way to associate Games with games yields a very interesting result. Suppose two perfect players play a game starting with a given position whose associated Game is "x". We can classify all Games into four classes as follows:


title: Surreal number

text: The notation G || H means that G and H are incomparable. G || H is equivalent to G−H || 0, i.e. that G > H, G < H and G = H are all false. Incomparable games are sometimes said to be "confused" with each other, because one or the other may be preferred by a player depending on what is added to it. A game confused with zero is said to be fuzzy, as opposed to positive, negative, or zero. An example of a fuzzy game is star (*).


title: Surreal number

text: Sometimes when a game nears the end, it will decompose into several smaller games that do not interact, except in that each player's turn allows moving in only one of them. For example, in Go, the board will slowly fill up with pieces until there are just a few small islands of empty space where a player can move. Each island is like a separate game of Go, played on a very small board. It would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game. This doesn't appear to be easy to do. For example, there might be two subgames where whoever moves first wins, but when they are combined into one big game, it is no longer the first player who wins. Fortunately, there is a way to do this analysis. The following theorem can be applied:


title: Surreal number

text: A game composed of smaller games is called the disjunctive sum of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the disjunctive sum of the addends.


title: Surreal number

text: Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their disjunctive sum. From this he invented the concept of a Game and the addition operator for it. From there he moved on to developing a definition of negation and comparison. Then he noticed that a certain class of Games had interesting properties; this class became the surreal numbers. Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.


title: Surreal number

text: In what is now called the "sign-expansion" or "sign-sequence" of a surreal number, a surreal number is a function whose domain is an ordinal and whose codomain is { −1, +1 }. This is equivalent to Conway's L-R sequences.


title: Surreal number

text: Define the binary predicate "simpler than" on numbers by "x" is simpler than "y" if "x" is a proper subset of "y", "i.e." if dom("x") < dom("y") and "x"(α) = "y"(α) for all α < dom("x").


title: Surreal number

text: For surreal numbers define the binary relation < to be lexicographic order (with the convention that "undefined values" are greater than −1 and less than 1). So "x" < "y" if one of the following holds:


title: Surreal number

text: so that "x" = "y" if and only if δ("x","y") = dom("x") = dom("y"). Then, for numbers "x" and "y", "x" < "y" if and only if one of the following holds:


title: Surreal number

text: For numbers "x" and "y", "x" ≤ "y" if and only if "x" < "y" ∨ "x" = "y", and "x" > "y" if and only if "y" < "x". Also "x" ≥ "y" if and only if "y" ≤ "x".


title: Surreal number

text: The relation < is transitive, and for all numbers "x" and "y", exactly one of "x" < "y", "x" = "y", "x" > "y", holds (law of trichotomy). This means that < is a linear order (except that < is a proper class).


title: Surreal number

text: For sets of numbers, "L" and "R" such that ∀"x" ∈ "L" ∀"y" ∈ "R" ("x" < "y"), there exists a unique number "z" such that


title: Surreal number

text: Furthermore, "z" is constructible from "L" and "R" by transfinite induction. "z" is the simplest number between "L" and "R". Let the unique number "z" be denoted by σ("L","R").


title: Surreal number

text: One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals.


title: Surreal number

text: However, similar definitions can be made that eliminate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule ∀"g" ∈ dom "f" (∀"h" ∈ dom "g" ("h" ∈ dom "f" )) and whose range is { −, + }. "Simpler than" is very simply defined now—"x" is simpler than "y" if "x" ∈ dom "y". The total ordering is defined by considering "x" and "y" as sets of ordered pairs (as a function is normally defined): Either "x" = "y", or else the surreal number "z" = "x" ∩ "y" is in the domain of "x" or the domain of "y" (or both, but in this case the signs must disagree). We then have "x" < "y" if "x"("z") = − or "y"("z") = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom "f" in order of simplicity (i.e., inclusion), and then write down the signs that "f" assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is { + }.


title: Surreal number

text: The sum "x" + "y" of two numbers, "x" and "y", is defined by induction on dom("x") and dom("y") by "x" + "y" = σ("L","R"), where


title: Surreal number

text: The additive identity is given by the number 0 = { }, "i.e." the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number "x" is the number − "x", given by dom(− "x") = dom("x"), and, for α < dom("x"), (− "x")(α) = − 1 if "x"(α) = + 1, and (− "x")(α) = + 1 if "x"(α) = − 1.


title: Surreal number

text: It follows that a number "x" is positive if and only if 0 < dom("x") and "x"(0) = + 1, and "x" is negative if and only if 0 < dom("x") and "x"(0) = − 1.


title: Surreal number

text: The product "xy" of two numbers, "x" and "y", is defined by induction on dom("x") and dom("y") by "xy" = σ("L","R"), where


title: Surreal number

text: The multiplicative identity is given by the number 1 = { (0,+ 1) }, "i.e." the number 1 has domain equal to the ordinal 1, and 1(0) = + 1.


title: Surreal number

text: The map from Conway's realization to sign expansions is given by "f"({ "L" | "R" }) = σ("M","S"), where "M" = { "f"("x") : "x" ∈ "L" } and "S" = { "f"("x") : "x" ∈ "R" }.


title: Surreal number

text: The inverse map from the alternative realization to Conway's realization is given by "g"("x") = { "L" | "R" }, where "L" = { "g"("y") : "y" ∈ "L"("x") } and "R" = { "g"("y") : "y" ∈ "R"("x") }.


title: Surreal number

text: In another approach to the surreals, given by Alling, explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the axiomatic approach to the reals, these axioms guarantee uniqueness up to isomorphism.


title: Surreal number

text: Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.


title: Surreal number

text: Given these axioms, Alling derives Conway's original definition of ≤ and develops surreal arithmetic.


title: Surreal number

text: A construction of the surreal numbers as a maximal binary pseudo-tree with simplicity (ancestor) and ordering relations is due to Philip Ehrlich, The difference from the usual definition of a tree is that the set of ancestors of a vertex is well-ordered, but may not have a maximal element (immediate predecessor); in other words the order type of that set is a general ordinal number, not just a natural number. This construction fulfills Alling's axioms as well and can easily be mapped to the sign-sequence representation.