I
recently had an
argument with my
girlfriend - she didn't believe me that there could possibly be different
degrees (
cardinalities) of
infinity.
It is often difficult to describe
concepts like this one to people who are not
familiar with reasonably advanced
mathematical concepts.
Here's how I tried to describe it:
"Imagine the numbers 1..2..3..4.., right up to
infinity"
(I now tried to explain the concept of infinity as the least upper bound of the natural numbers, but this was a mistake - she already had a perfect grasp of the concept of infinity.)
"Now, you know that there are numbers that lie between those numbers - like 1.5, 2.75, 3.14159265 or even
pi?
"
"yes"
"Well, how many of them do you think there are between
zero and
1?"
"Mmm... infinity I suppose?
"OK, so how many numbers are there between 0 and
infinity, including those numbers between the
integers?"
"um... infinity?"
"Well... yes, but do you see that this infinity is
bigger than the original infinity we just talked about?"
"I suppose"
OK - so the first
infinity has a
cardinality of
aleph-
nought and the second of
aleph-
one. I still don't think my
girlfriend really believes that they are of
different sizes!
This argument began by the familiar "I love you, I love you more... I love you infinity" discourse. I did, unfortunately follow this by the remark "I love you aleph-one. I was accused of being a gimp...