TITLE: Demonstrating That Two Infinities Are Equal
AUTHOR: Eugene Wallingford
DATE: November 25, 2019  6:06 PM
DESC: 
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BODY:
 I remember first learning as a student that some infinities are bigger
 than others.  For some sets of numbers, it was easy to see how.  The
 set of integers is infinite, and the set of real numbers is infinite,
 and it seemed immediately clear that there are fewer integers than
 reals. Demonstrations and proofs of the fact were cool, but I already
 knew what they showed me.

 Other relationships between infinities were not so easy to grok.
 Consider:  There are an infinite numbers of points on a sheet of
 paper.  There are an infinite numbers of points on a wall.  These
 infinities are equal to one another.  But how?  Mathematician Yuri
 Manin demonstrates how:

<BLOCKQUOTE><EM>
 I explained this to my grandson, that there are as many points
 in a sheet of paper as there are on the wall of the room.  "Take
 the sheet of paper, and hold it so that it blocks your view of
 the wall completely.  The paper hides the wall from your sight.
 Now if a beam of light comes out of every point on the wall and
 lands in your eye, it must pass through the sheet of paper.
 Each point on the wall corresponds to a point on the sheet of
 paper, so there must be the same number of each."
</EM></BLOCKQUOTE>

 I remember reading that explanation in school and feeling both
 amazed and enlightened.  What sorcery is this?  So simple, so
 beautiful.  Informal proofs of this sort made me want to learn
 more mathematics.

 Manin told the story quoted above in an interview a decade or so
 ago with Mikhail Gelfand,
<A HREF="https://www.ams.org/notices/200910/rtx091001268p.pdf">
   We Do Not Choose Mathematics as Our Profession, It Chooses Us</A>.
 It was a good read throughout and reminded me again how I came to
 enjoy math.
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