Hey everyone! This is Nicole. I’m here today to talk to you about our childhood problems, specifically those having to do with
infinity. I don’t know how many of you can relate to this, but I remember when I was a kid, I would have these petty arguments with some friends and,
especially, my older brother. They would always revolve around how you could one up each other. I was thinking that the arguments had to do with jinx but I
can’t actually remember. The argument would go something like this:
You: Jinx times 100.
Other kid: Well times 1,000.
You: Well times infinity! *I have won everything now*
Other kid: Infinity + 1!
You: *gets mad at Kid 2* Well, Infinity times infinity.
Other kid: *gets mad at Kid 1* You can’t do that! That’s not a real number!
Anyway, now that we are more mature adults, we can think about this in terms of reality. Would infinity + 1 really be more
than infinity? Would it stop at infinity? What about infinity times infinity? That sure seems like it should be bigger than infinity. Well, according to Greog
Cantor and his transfinite arithmetic, infinity, especially countable infinity, which generally little kids are talking about countable numbers, you don’t go
up to a little kid and say, “Are you referring to countable numbers or the continuum?” And they say, “Oh, I’m referring to a continuous line not integers.” I
don’t think most little kids even know about that branch of mathematics. If you think about infinitely many countable and plus one more than that, well you
never really ending with infinity so plus one more doesn’t alter the never-ending string, it still never ends, so it’s still infinity and since that property
holds with addition you could say, “That’s the same as infinity plus two and infinity plus infinity.” Which using algebra, that’s just 2*infinity. Let’s refer
to infinity as omega or the set of whole numbers. Cantor figured out that infinity plus infinity is just infinity or omega plus omega is just omega. What about
omega squared, omega times omega? So we’ve got 2*omega and then we just keep adding so we get omega*3, omega*4, until we get to omega times omega. Well, that’s
just repeated addition, isn’t it? Infinitely many times. You are still getting an infinity out of that (Aczel, 2000) 1 .
John H. Conway (2007) 2 at Princeton University in 1969 had a new
thought, ‘What about surreal numbers?’ (a term he coined) You have omega, or infinitely many whole numbers, but what is the largest finite number before you
hit omega? That should be omega-1, but what really is omega-1? If we look at our transfinite arithmetic, would omega-1 suddenly become finite or would it
remain infinite?
John H. Conway continued to explore the realm of surreal numbers that he discovered. You can go past omega-1 to omega-2 or
omega-1,000,000. Between the finite and the infinite, somewhere, we have all of these other numbers. We have omega/2, ¾omega and what about infinitesimally
small numbers like 1/omega, which we know approaches zero? However, between 0 and 1/omega we have 1/(2*omega) and then 1/(omega^omega). It never ends. How can
we comprehend all of these infinitesimally small numbers or the difference between the bridge of the finite to the infinite? Where does it make that jump and
that distinction? He talks about between all of the finite numbers and omega the simplest number between them is the squared root of omega.
I have a quote by Conway (2007) 2 here that I really like. He says,
“There are more surreal numbers than you could have ever dreamt of.” It is something thoroughly unexpected, which is why he claims it has taken so long to
discover surreal numbers. It is so natural to think, ‘What comes after infinity? Well, just plus one.’ But not many people think about what comes right before
infinity. Conway says, “You have to break some rules when you go from the finite to the infinite, but you can keep some others.” So that’s something for
everyone to think about. Where is the difference between the rules of the finite and the infinite and how do they interact?
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