Infinity and So On
Perhaps your first taste of infinity was standing between two mirrors, or gazing into the night sky, or simply trying to count as high as you could. We'll talk with mathematician Dr. David Eisenbud about infinity - after this on Earth and Sky.
DB: This is Earth and Sky, on infinity. If you've ever gazed out into the night sky, you may have felt dizzy as you imagined how far it all goes. The idea of infinity doesn't extend just to space, but to numbers, too. And even though infinity goes on and on indefinitely, it has a precise mathematical definition. Infinity is defined in terms of sets - collections of things, or "elements." We spoke with Dr. David Eisenbud of Berkeley's Mathematical Sciences Research Institute, about infinite sets:
(Tape 0:07:30-0:07:43) You can tell a set is infinite if you can throw away one element and still have a set of the same size. So infinity plus one - any infinity plus one - is equal to that same infinity.
DB: Say you have the set of all numbers. Now throw away the number one. You still have an infinite set. In fact, it's still the same size infinity. But not all infinities are the same size. Again, Dr. Eisenbud:
(Tape 0:05:35-05:46) Amazingly enough there are lots of different sizes of infinities. There's a smallest infinity, and that's the number of counting numbers, numbers like 1, 2, 3, 4, 5. (Tape 0:05:54-0:06:03) And for example the number of points in a line is a bigger infinity. But there are bigger infinities than that, too.
DB: Dr. Eisenbud added that there's no biggest infinity. There's a mathematical theorem that says that you can always make bigger and bigger infinities. So the possibilities are…endless. Thanks to Dr. David Eisenbud for speaking with us. And with thanks to the National Science Foundation, I'm Deborah Byrd, for Joel Block, for Earth and Sky.
Author: Beverly Wachtel
Thanks to the following individual for aiding in the preparation of this script:
Dr. David Eisenbud
Director, Mathematical Sciences Research Institute
If you enjoyed this program, you may be interested in the following:
Mathematical Sciences Research Institute website:
Infinity: A Platonic Realms Mini-text:
Counting to Infinity: