<story><title>The Hilbert Hotel</title><url>http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/</url><text></text></story><parent_chain></parent_chain><comment><author>RiderOfGiraffes</author><text>There are not only different sizes of infinity, as this article is trying to show, but also different types of infinity?&lt;p&gt;In fact, I&apos;m coming to believe that the usual methods of teaching infinity are wrong (not the concepts, just the presentation).&lt;p&gt;People have this intuition that there can&apos;t be anything bigger than infinity.  We reinforce that by showing that the rationals are countable, that Q^2 is countable, in fact Q^n is countable.  Even the algebraics are countable.  We create the intuition that anything infinite is countable.&lt;p&gt;Then we present 2^N and everything goes bizarre.  Some people like that, but others decide that it&apos;s all meaningless.  I think this method of presentation does the majority of people a disservice.&lt;p&gt;But not only is there the set-theoretic infinity, there is also the geometric infinity.  That actually better matches people&apos;s intuition.  Separating the ideas of the set theoretic and the geometric versions of infinity has, I&apos;m finding, huge benefits when trying to help people understand what&apos;s going on.&lt;p&gt;Of course, there&apos;s also cardinal versus ordinal infinities as well.  That&apos;s fun.</text></comment>
<story><title>The Hilbert Hotel</title><url>http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-hotel/</url><text></text></story><parent_chain></parent_chain><comment><author>ynniv</author><text>Hard math is the hardest thing that I can think of.  One can learn most other things with enough effort (ie, repetition), but hard math takes immense thought.&lt;p&gt;On the other hand, it is also rarely necessary.</text></comment>