Question About InfinitiesDate: 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" |
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