Question About Infinities

Date: 24 Mar 1995 11:33:45 -0500
From: Mike Nowell
Subject: Infinities

        From one of our 4th graders:

        If there are two points in space, A and B, and set 1 
equals all the possible paths which lead away from point 
A (this set is assumed to be infinite) and set 2 equals all 
the possible paths which lead away from point A which 
do not pass through point B (this is also assumed to be 
infinite), if I subtract set 2 from set 1, does this prove 
that there are no paths which lead away from point A 
which pass through point B?

Date: 24 Mar 1995 14:17:31 -0500
From: Stephen Weimar
Subject: Re: Infinities

Great question!  It gave us an occasion to take a little 
side excursion into orders of infinity, even though that 
really doesn't come into play here.

The problem gets clearer if we clarify some language.  
To figure out what the difference is between two sets 
is not the same as finding the difference between the 
number of elements in the set.

It is common to define the difference between sets A 
and B as all the elements which are in one but not the 
other.  Note that I wrote difference rather than subtract, 
because sets are not numbers and you don't treat them
exactly the same way.  The number of elements in a set 
is not the set; it is one of the set's characteristics.

So if we want to know the difference between set A and 
set B in your problem, it becomes fairly easy to state:  
all the paths from point A which pass through B.

If we want to know what happens when we subtract the 
number of elements in B from the number in A, then we 
have a separate problem which doesn't tell us anything 
about the elements themselves, just about the number 
of them.

You want to know what happens when we subtract 
infinity from itself.

And it seems as though just about anything could happen 
when you do that.  My friend likes the example of Hilbert's 
hotel.  Imagine a hotel with an infinite number of rooms 
and a person in each room.  Now suppose a thousand people 
leave (imagine the rooms are numbered 1,2,3,... and the 
people in the first thousand rooms leave).  Would there be 
any fewer people?  Would we have to leave some rooms 
empty?  No, we just reassign everyone left to a new room 
(in my example we could do it by having everyone go to a 
room 1000 less than the number they were in).  So here we 
have two infinities.  The nature of infinity is that if you 
take some finite amount away from it, there's still an 
infinite amount left.  And we're tempted to say in this case 
that the difference between these two infinities is 1000.  
But we could do the same thing with 300 people leaving and 
then we'd say the difference between the infinities is 300.  
Maybe you can begin to see that we can substract infinity 
from itself and make the answer anything we want.

Well, mathematicians tend not to like what happens to math 
if the result of an operation is "whatever you want it to be."  
If you can't define what the result is, we say that it is 
undefined, rather than let the answer be anything at all.

So when you ask us what happens when you subtract infinity 
from itself we say that operation is not defined in mathematics.

Dividing by zero is another example of something undefined 
in mathematics.

-- steve "chief of staff"