All Discussions Tagged 'Hilbert' - Think Atheist2018-10-16T19:36:53Zhttp://www.thinkatheist.com/forum/topic/listForTag?tag=Hilbert&feed=yes&xn_auth=noHilbert's Hotel, where there's always a room even when every room is takentag:www.thinkatheist.com,2014-01-11:1982180:Topic:14252832014-01-11T19:52:23.253ZUnseenhttp://www.thinkatheist.com/profile/Unseen
<p>How can that be? </p>
<p><em>(T)he Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them. Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on. That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).</em></p>
<p><em>Now suppose infinitely many new guests arrive, sweaty and short-tempered. No problem. The unflappable manager moves the…</em></p>
<p>How can that be? </p>
<p><em>(T)he Hilbert Hotel doesn’t merely have hundreds of rooms — it has an infinite number of them. Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on. That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).</em></p>
<p><em>Now suppose infinitely many new guests arrive, sweaty and short-tempered. No problem. The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on. This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests.</em></p>
<p>Is your poor head swimming yet? Well, <a href="http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/?_r=0" target="_blank">read this article about Hilbert's Hotel</a> and you'll see how the manager creatively solves even tougher problems.</p>
<p>What it adds up to is that infinity is a very strange concept indeed.</p>
<p>Mathematics can't exist without it, and yet both mathematicians and philosophers have been wrestling with this question:</p>
<p>Is infinity real? Or, to put it another way, is there any such thing as a real infinity?</p>