## Ask a Scientist: Is infinity something real?

Otago Daily Times, 6 September 2002

Jamie Hunt, of Cashmere Primary School, asks:-

Is infinity something real?

David Wiltshire, a cosmologist at the University of Canterbury responds:

If we definitely knew that the Universe went on forever, or that it would last forever, we could say that infinity is real. We might be able to answer those questions some day but we cannot do so yet.

There is another way of looking at the question though, by asking what do we mean by "real"? Computers contain both hardware - the stuff inside the boxes - and software - the programs that run on them. Most people would say that software is real even though it is not stuff you can touch, because computers would not work without programs. The laws of physics that describe the way the Universe works are real in a very similar sense to computer programs being real. The mathematical idea of infinity is very important for the laws of physics, so in that sense infinity is real.

Actually there are many different sorts of infinity. Think of counting by ones - 1,2,3,4,5,... - and then think of counting by twos - 2,4,6,8,10,... Both ways of counting get to infinity but if you are counting in ones and a friend is counting in twos, your friend's numbers will get infinitely large twice as quickly. Mathematicians deal with this by calling the first infinity "omega", and the second infinity two times omega.

Not only is it possible to multiply omega by an ordinary number, but one can even multiply infinities together, and divide infinities. For example, imagine cutting a pie into infinitely many small pieces, and then cutting each infinitely small piece into infinitely many pieces again, and again and again... Of course, that is not something you can do with a real pie but it is something that you can do mathematically. By imagining such things mathematicians have discovered numbers which include all possible sorts of infinities, both large and small, and which are very different to ordinary numbers. These numbers are called "surreal numbers" because they are very strange, even for mathematicians. Physicists who think about the laws of the Universe have not yet had to use surreal numbers, but we do know of parts of physics where they might be useful. If that turns out to be true then not only is infinity real, but different sorts of infinity are real too.

Dr David Wiltshire
Department of Physics and Astronomy
University of Canterbury
Homepage: http://www2.phys.canterbury.ac.nz/~dlw24/