Infinity

Joe:   Today, I hear that Philosophy Talk is tackling infinity.   That's a mind-blowing topic that I've been obsessed with lately.  I been getting intimations of the infinite everywhere.   Think of the all-knowing mind of the infinite god, or the beginningless past and the ceaseless future or space without boundary or matter that can be divided into smaller and smaller bits without obvious limit!  Infinity surrounds us and engulfs us! 

Blow:  Well,  I hate to tell you Joe but except for the non-existent mind of god, the things you just mentioned are finite rather than infinite.  The past only stretches so far back as the big bang.  The future extends only so far as the heat death of the universe.  Space is finite.   And matter is not infinitely divisible. 

Joe:  Don't spoil my intimations, dude.   You're such a philistine.  Are you really denying that anything is infinite?  Do you really mean to suggest that there is no infinity in nature?   What would life be like without the infinite?

Blow: No I don’t mean to say that there is no infinity.  But if you really want to experience the infinite, don't muck around in nature.  Look to the realm of pure number.  That’s where infinity really lies. 

Joe:  I’m not sure I agree with you Blow in denying that nature contains the infinite.   But to settle this, why don’t we start out by defining infinity. 

Blow:  That’s a piece of cake. The infinite is that which is not finite. 

Joe:   But wait a second – you've only told me what infinity is not.  That doesn’t tell me anything positive and definite about it.  Suppose I ask you to define the color green.  And you said green is the color that is not red, and not blue, and not orange and so on for every other color you can think of.   That would tell me a lot about what green is not.  But it doesn’t tell me much about what green is.  I don’t want to know what infinity is not.  I want to know what it is.   I want a concrete definition to help me recognize infinity when I come across it. 

BLOW:     I hate to tell you Joe,  but you’re not very likely to come across infinity.  Every number you’ve ever counted to has been finite.  Every extent of space you’ve ever crossed has been finite.  And every span of time you’ve ever experienced has been finite too.

 JOE:    Okay, what’s your point?

BLOW:    Well, take your color analogy and apply it to the case of number. If we followed your lead, we would define infinity like this.  Infinity is the number that is larger than any finite number you can think of.  Infinity is not 1, not 2, not 87, not even a million.  What is it?  It’s precisely the number that is not any of the finite numbers and is larger than every one of them.

JOE:  But that definition is even worse than my negative definition of green, isn’t it?   There are only so many colors.  So you could pick out all but one and say green is the missing one.  And when you came across a color that wasn’t any of the ones you already know.  You could say, “Ah, there it is! That’s the one that wasn’t on the original list.  So that must be green.”  But you can’t do that with infinity and the numbers, because you never run out of numbers.  You never get to the next-to-last number.

BLOW:  In spite of yourself, Joe, you’re starting to get it.  You’re starting to understand the concept of infinity.

JOE:    What do you mean?  I’m more confused than ever. 

BLOW:  Well, ask yourself how many numbers there are.

JOE:    Lots.  There are lots and lots of numbers.

BLOW:      There’s more than a lot of numbers.  There is an infinite number of numbers.  And what shows this is the very fact that you cannot exhaust them by running through them “one-by-one.”   Just try. I’ll give you as much time as you want to do it. 

JOE:   That reminds me of a little story.  A friend of mine had a son who was four at the time.  And he asked his mommy,  “Mommy how old are you?”   And I say, “Tommy,  think of the biggest number you can think of.  That’s how old your mommy is.”  And Tommy got a wide-eyed look in his eyes and said,  “Mommy,  are you 17?”    And I said,  “Tommy, is 17 really the biggest number you can think of? What if you added one to 17.  Would you get a number bigger than 17?”   And Tommy,  said,  “Oh! So Mommy, are you 100?”  And I said,  “Is that really the biggest number you can think of, Tommy?  What if you added one to 100?  What would you get?”   By now Tommy seemed to realize that we could keep playing this game.  So he said, really rapidly,  ”Mommy,  you must be a zillion trillion billion million, billion trillion.”

BLOW:  Poor Tommy.  That was kind of a mean trick to play on a 4 year old kid.  But you almost got him to grasp the idea of infinity by getting him to see that he could not exhaust the numbers.

JOE:  So  are you equating infinity with sheer  “inexhaustibility?”  Is that your positive and definite definition of the infinite – something you can't run out of? 

BLOW:        No,  I wouldn’t equate infinity with inexhaustibility.  That’s just one of the properties of the infinite.  But infinity has lots of other cool properties.

JOE:   Like what? 

BLOW:  For example, if you take half of infinity, you get infinity again.  If you double infinity you get – guess what?

JOE:  Infinity again?

BLOW  Right.  So there’s something else positive and definite to say about infinity.   Infinity is that amount which you can’t decrease through division and can’t increase through multiplication. 

JOE: If you can’t increase infinity through multiplication,  that means that infinity is  that which there is nothing greater than, right?

BLOW:   You can’t quite say that, because infinities actually come in different sizes – some larger, some smaller!  And as far as we know, there is no “largest” infinity.  Think of the rational numbers and the real numbers.   There are an infinite number of rationals and an infinite number of reals.   But you know what,  there are a lot more real numbers than rational numbers. 

JOE:  Wait a minute!  Wait a minute!  Something's not right.  You just said you can’t increase infinity by multiplying it and can’t decrease it by dividing it.  So how can there be different sizes of infinity?

 BLOW:  I’m afraid I’m going to have to keep you hanging there,  Joe, cause I gotta go run and catch Philosophy Talk.  Why don’t you come along?  Maybe you’ll learn something about the infinite.