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Encyclopedia > Hilbert's paradox of the Grand Hotel

Hilbert's paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (18621943): Robert Boyles self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist. ... The infinity symbol ∞ in several typefaces The word infinity comes from the Latin infinitas or unboundedness. ... David Hilbert (January 23, 1862, Wehlau, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... 1862 was a common year starting on Wednesday (see link for calendar). ... 1943 (MCMXLIII) was a common year starting on Friday (the link is to a full 1943 calendar). ...


In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. Now, imagine a hotel with an infinite number of rooms. One might assume that the same problem will arise when all the rooms are occupied. However, in an infinite hotel, the situations "every room is occupied" and "no more guests can be accommodated" do not turn out to be equivalent. There is a way to solve the problem: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1. Unlike a finite hotel, in an infinite hotel, being "full" in the sense that every room contains a person is not the same as being "full" in the sense that there is no space for another person. Note that a movement of an infinite number of guests would constitute a supertask. Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In philosophy, a supertask is a task occurring within a finite interval of time involving infinitely many steps (subtasks). ...


It would seem to be possible to make place for a countably infinite number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, etc., and all the odd-numbered new rooms will be free for the new guests. However, this is where the paradox lies. Even in the previous statement, if an infinite number of people fill the odd numbered rooms, then what amount is added to the infinity that was already there? Can one double an infinite number? Also, for example, say the infinite number of new guests do come and fill all of the odd-numbered rooms, and then the infinite number of guests in the even-numbered rooms leaves. An infinite number has just been 'subtracted' from an infinite number, yet an infinite number of people remain. This is where Hilbert's Hotel is paradoxical. In mathematics the term countable set is used to describe the size of a set, e. ...


If a coutably infinite number of coaches arrive, each with an countably infinite number of passengers, you can even deal with that: first empty the odd numbered rooms as above, then put the first coach's load in rooms 3n for n = 1, 2, 3, ..., the second coach's load in rooms 5n for n = 1, 2, ... and so on; for coach number i we use the rooms pn where p is the i+1-st prime number. You can also solve the problem by looking at the license plate numbers on the coaches and the seat numbers for the passengers (if the seats are not numbered, number them). Regard the hotel as coach #0. Interleave the digits of the coach numbers and the seat numbers to get the room numbers for the guests. The guest in room number 1729 moves to room 1070209. The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel. In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...


Some find this state of affairs profoundly counterintuitive. The properties of infinite 'collections of things' are quite different from those of ordinary 'collections of things'. In an ordinary hotel, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel the 'number' of odd-numbered rooms is as 'large' as the total 'number' of rooms. In mathematical terms, this would be expressed as follows: the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. In fact, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets this cardinality is called aleph_0 (aleph-null). In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ...


An even stranger story regarding this hotel shows that mathematical induction only works in one direction. No cigars may be brought into the hotel. Yet each of the guests (all rooms had guests at the time) got a cigar while in the hotel. How is this? The guest in Room 1 got a cigar from the guest in Room 2. The guest in Room 2 had previously received two cigars from the guest in Room 3. The guest in Room 3 had previously received three cigars from the guest in Room 4, etc. Each guest kept one cigar and passed the remainder to the guest in the next-lower-numbered room. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

Contents

The cosmological argument

A number of defenders of the cosmological argument for the existence of God, such as William Lane Craig, have attempted to use Hilbert's hotel as an argument for the physical impossibility of the existence of an actual infinity. Their argument is that, although there is nothing mathematically impossible about the existence of the hotel (or any other infinite object), intuitively (they claim) we know that no such hotel could ever actually exist in reality, and that this intuition is a specific case of the broader intuition that no actual infinite could exist. They argue that a temporal sequence receding infinitely into the past would constitute such an actual infinite. This article or section does not cite its references or sources. ... William Lane Craig (born August 23, 1949) is an American philosopher, theologian, New Testament historian, and Christian apologist. ...


However, the paradox of Hilbert's hotel involves not just an actual infinite, but also supertasks; it is unclear whether this claimed intuition is really the physical impossibility of an actual infinite, or merely the physical impossibility of a supertask. A causal chain receding infinitely into the past need not involve any supertasks. See Thomas Aquinas' Summa Theologiae for details about infinite regressions and the existence of God. Saint Thomas Aquinas [Thomas of Aquin, or Aquino] (c. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


In fiction

The novel White Light, by mathematician/science fiction writer Rudy Rucker, includes a hotel based on Hilbert's paradox. White Light is a work of science fiction by Rudy Rucker published in 1980 by Ace Books. ... Science fiction is a form of speculative fiction principally dealing with the impact of imagined science and technology, or both, upon society and persons as individuals. ... Rudolf Rucker, Fall 2005. ...


Stephen Baxter's science fiction novel Transcendent has a brief discussion on the nature of infinity, with an explanation based on the paradox — modified to use starship troopers rather than hotels. Stephen Baxter at the Science-Fiction-Tage NRW in Dortmund, Germany, March 1997 Stephen Baxter (born in Liverpool, 13 November 1957) is a British hard science fiction author. ... Transcendent (ISBN 0345457919) is a science-fiction novel by Stephen Baxter. ...


Geoffrey A. Landis' Nebula Award-winning short story "Ripples in the Dirac Sea" uses the Hilbert hotel as an explanation of why an infinitely-full Dirac sea can nevertheless still accept particles. Geoffrey A. Landis emerged in the late 1980s as one of the foremost scientist-writers in the science fiction genre. ... The Nebula is an award given each year by the Science Fiction and Fantasy Writers of America (SFWA), for the best science fiction/fantasy fiction published in the United States during the two previous years. ... The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles possessing negative energy. ...


In Peter Hoeg's novel Smilla's Sense of Snow, the titular heroine reflects that it is admirable for the hotel's manager and guests to go to all that trouble so that the latecomer can have his own room and some privacy. Peter Høeg, born on May 17, 1957, is one of Denmarks most celebrated contemporary writers of fiction. ... Smillas Sense of Snow (also published as Miss Smillas Feeling for Snow), is a book by Danish author Peter Høeg. ...


The booklet The Cat in Numberland by mathematician/philosopher Ivar Ekeland presents Hilbert’s paradox as a tale for children, in the tradition of Lewis Carroll. It is illustrated by John O’Brien. Lewis Carroll. ...


See also

The inspiration for the name of the principle: pigeons in holes. ...

External links

  • Welcome to the Hotel Infinity! — The paradox told as a humorous narrative, featuring a hotel owner and a building contractor based on the feuding 19th-century mathematicians Georg Cantor and Leopold Kronecker
  • Argument that Hilbert's paradox is not really paradoxical; criticism of Craig's use of it to attack the possibility of an actual infinity

Alternative meaning: Nineteenth Century (periodical) Vagina The 19th century lasted from 1801 through 1900 in the Gregorian calendar. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ...


 
 

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