June 17, 2014

Taliesin
Taliesin

Not exactly infinite.  There is something both grand and sad about Frank Lloyd Wright.  He was brilliant with light (windows, views, sight-lines, recessed lighting!)  But something always feels a bit off-kilter to me, a bit heavy, a bit not-quite.

And thinking about infinity can do that to one’s equilibrium.  Warning: very long post.  A complex idea is coming.  Go slowly.  Ponder it all.  Life is amazing.  Ideas are amazing.  There aren’t an infinite number of tangible objects (previous post) but I think there are an infinite number of ideas.  And perhaps language-symbolic sentences to express those ideas.  Maybe next post on linguistics.  But back to mathematics.

We all understand, at some level, that the counting numbers (integers) can go on and on and on…forever.  You can always keep adding one more…forever.  (If you had forever, another mind-boggling notion.)  We call that “infinity.”  There is an infinite number of counting numbers.  Let that notion settle a bit.  (Not being able to personally count forever doesn’t mean the numbers don’t go on forever.)

But there’s more than integers.  There are the “real” numbers.  The ones we usually think of with the decimal points (in base 10, we’ll stick mostly with that just for now) and the zillion digits after them.  Like 1/3:  0.33333333333 etc.  The threes there go on forever, as it were.   Other real numbers are:

0 (zero),  19,   1.6,   pi,   461. 8192947, and 238,996,428,778.0034827.  You can make up a zillion real numbers.  And more…

Here’s the amazing thing.  The SIZE of the set of integers (the group of them) is infinite.  They go on forever.  The integers are part of the reals (I named zero and 19 there, which are integers), but the set of integers is SMALLER than the set of reals.  The set of real numbers are a BIGGER infinity!  I’ll show you how/why.

Here is a proof using a technique by a 19th century mathematician named Georg Cantor:

Say you had “forever” and you made a list of real numbers, here in base 2 (which uses only 1s and 0s.)  If you haven’t used base 2 (computers do), here’s how you count  from 1 to 7 in base 2:

1,  10,  11,  100,  101,  110,  111

The usual columns that you think of as 1s, 10s, and 100s are now 1s, 2s, and 4s, etc.  (If this is confusing, ask a computer or math nerd to explain further.)

So, let’s suppose you have made what you think is an exhaustive list of the real numbers in base 2.   But you haven’t!  Here is the beginning of your list, just the first 11 numbers you happened to come up with.  (You need, as it were, a very big forever for this!)

Cantor's Diagonal
Cantor’s Diagonal

The upper chart is the beginning of an infinitely long chart that purports to list all the real numbers.  First real number, any old one you came up with,  is called s1; the second one  is called s2 etc.  There are 11 real numbers listed here in base 2.  You claim to have listed them all, in a ginormous chart that goes on forever.  You can keep listing them…it doesn’t matter…because:

In the separate box at the bottom is a BRAND NEW number  (s= 10111010011… in blue type) that doesn’t appear ANYWHERE in the list!  It was formed by choosing the opposite digit of each of the red digits in the big chart. The first entry is changed in the first position, the second entry in the second position, and so on.  This new number demonstrates that the list you thought was complete — ISN’T!

Think about it.  It’s beautiful.  It’s why I thought I wanted to be a mathematician.  Are art and writing easier?  More poetic?  This is the poetry of ideas!

Just incidentally, this sort of argument, called Cantor’s diagonal, is used in other proofs.  You can check it out in Wikipedia:

http://en.wikipedia.org/wiki/Cantor’s_diagonal_argument

Whew!  I’m tired.

 

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