Hilbert's paradox of the Grand Hotel

Hilbert's paradox of the Grand Hotel is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and that this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture and was popularized through George Gamow's 1947 book One Two Three... Infinity.

The paradox

Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms, where the pigeonhole principle would apply.

Finitely many new guests

Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from his current room n to room n+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests.

Grand Hotel

A grand hotel is a large and luxurious hotel, especially one built in a traditional architectural style. More specifically, Grand Hotel may refer to:

Hotels

Europe

Grand Hotel (film)

Grand Hotel is a 1932 American Pre-Code drama film directed by Edmund Goulding. The screenplay by William A. Drake is based on the 1930 play of the same title by Drake, who had adapted it from the 1929 novel Menschen im Hotel by Vicki Baum. As of 2014, the film is the only one to have won the Academy Award for Best Picture without it or its participants being nominated in any other category.

The film was remade as Week-End at the Waldorf in 1945, and also served as the basis for the 1989 stage musical of the same title. During the 1970s, a remake, to be set at Las Vegas' MGM Grand Hotel, was considered.

Grand Hotel has proven influential in the years since its original release. The line "I want to be alone", famously delivered by Greta Garbo, placed number 30 in AFI's 100 Years...100 Movie Quotes. Also, the phrase "Grand Hotel theme" has come to be used for any dramatic movie following the activities of various people in a large busy place, with some characters' lives overlapping in odd ways and some of them remaining unaware of one another's existence. Such "grand hotel" films have been set at airports, aboard ocean liners, in large department stores, etc., as well as in hotels. Neil Simon used the format in both play and film versions of Plaza Suite, California Suite, and London Suite. In 2007, the film was selected for preservation in the United States National Film Registry by the Library of Congress for being "culturally, historically, or aesthetically significant."

Grand Hotel (album)

Grand Hotel is the sixth studio album by Procol Harum. Released in 1973, it signalled a change of direction for the band. Guitarist Dave Ball who had joined the band for their live album the previous year left shortly after the photo shoot for the proposed album's cover to be replaced by Mick Grabham. Grabham's head was superimposed on the front and back cover of the album on Ball's body. Although the band had gone through significant personnel changes in previous years, the band would enter its most stable phase with this line up.

The album reached No. 21 on the Billboard album charts.

Although on the surface "Grand Hotel" appears to be a concept album according to lyricist Keith Reid the "concept" doesn't extend beyond the title tune.

The single "A Souvenir of London" was banned by the BBC for its reference to venereal disease in the lyrics of the song. Reid claimed that the song was really inspired (regardless of how it turned out and was interpreted) by a visit to a souvenir shop near George Martin's Air Studios. "Almost every album has had at least one comic song...and this one was a bit tongue in cheek" Reid stated as part of an interview for the 2009 CD reissue.

Grand Hotel (Mackinac Island)

The Grand Hotel is a historic hotel and coastal resort on Mackinac Island, Michigan, a small island located at the eastern end of the Straits of Mackinac within Lake Huron between the state's Upper and Lower Peninsulas. Constructed in the late 19th century, the facility advertises itself as having the world's largest porch. The Grand Hotel is well known for a number of notable visitors, including five U.S. presidents, Russian presidents Vladimir Putin and Dmitri Medvedev, inventor Thomas Edison, and author Mark Twain.

Grand Hotel is a member of Historic Hotels of America, the official program of the National Trust for Historic Preservation.

History

In 1886, the Michigan Central Railroad, Grand Rapids and Indiana Railroad, and Detroit and Cleveland Steamship Navigation Company formed the Mackinac Island Hotel Company. The group purchased the land on which the hotel was built and construction began, based upon the design by Detroit architects Mason and Rice. When it opened the following year, the hotel was advertised to Chicago, Erie, Montreal and Detroit residents as a summer retreat for vacationers who arrived by lake steamer and by rail from across the continent. At its opening, nightly rates at the hotel ranged from US$3 to US$5 a night.