Here is a second instalment of reactions to the Horizon programme *To Infinity and Beyond*. (The first instalment is here.)

Many people (not all of them related to me) seemed to like my performance. Thank you – very gratifying!

The most gratifying reaction was being recognised by a stranger on the platform at Mile End station last Sunday morning. The person who recognised me had got as far as telling me that he had been puzzling about infinity, when his train came, so I never heard what his puzzles were.

However, may other people have emailed me with questions. I will try to deal briefly with some of them.

**Question:** Has the universe been proven to be infinite?

Certainly not, rather the reverse. We can only ever be aware of a finite portion of whatever there is out there; our best current theories say that anything else can never affect us in any way, so effectively doesn’t exist. Also, some physicists are coming to believe that there is a smallest scale as well, so that the universe which we are aware of contains only a finite number of bits of information. Infinity only exists in mathematics, in my view.

**Question:** Is infinity a number? If so, in what sense?

Two difficulties here. Firstly, as the programme didn’t explain very well, there are many different sizes of infinity – see below.

We have to think what we mean by a number. The history of mathematics in the last few centuries has seen a steady expansion in the class of things we regard as numbers. From whole numbers, to arbitrary integers (positive and negative), to fractions, to irrational numbers (like the square root of 2, or pi), to complex numbers (including the square root of –1), with more speculative ventures to quaternions (which I mentioned in another posting) and octonions, dual numbers, and such like; then, in a different development, to Cantor’s infinite cardinal and ordinal numbers, Conway’s numbers (which include both infinite and infinitesimal), non-standard models of the natural or real numbers, and so on. So what is a number? This would take far too long to discuss here!

At the minimum, we should be able to add and multiply numbers. Cantor showed that infinite cardinal numbers can be added and subtracted, but the rules are very simple; the sum or product of two infinite cardinal numbers is just the larger of the two. So, using “infinity” to mean the infinity of the natural numbers, we see that infinity plus infinity, and infinity times infinity, are both equal to infinity.

As I explained in the programme, we cannot subtract (or divide) infinite numbers in any sensible way.

What about the larger infinities? I imagine that an intelligent viewer of the programme would be very puzzled by being told successively that

- there is no largest natural number, since whichever number you have, you can get a larger number by adding one to it (unless you follow Doron Zeilberger’s line);
- when you add one to infinity you don’t change it;
- nevertheless, whatever infinity you have, there is a larger one!

So how do you get this larger infinity? The clue was in Hugh Woodin’s presentation of Cantor’s diagonal argument (though this was done rather quickly and its significance not explained). I won’t go into technicalities. Woodin explained that the infinity of decimals is larger than the infinity of natural numbers. This generalises (as Cantor first realised) to say that, given any infinity, there is a construction (somewhat like decimals) which gives a larger infinity.

**Question:** Isn’t there a mathematical theorem that proves that infinity exists?

The answer to this, which you may regard as cheating, is that we now no longer view theorems (as Euclid did) as universal and unarguable truths about reality, but as statements which follow logically from explicitly stated assumptions called axioms. So, if a mathematical theorem shows that infinity exists, this existence must be somehow implicit in our assumptions.

My example was based on Peano’s axioms for the natural numbers. Peano begins with some very simple axioms about the *successor function* (moving from one number to the next), and goes on to more complicated axioms for addition and multiplication and mathematical induction. But the simple ones will do here. The

first two axioms state:

- Zero is not the successor of any natural number.
- Any other natural number is the successor of a unique natural number.

From these two axioms it follows that the set of natural numbers is infinite. (But note that Zeilberger would not accept the first axiom.) Note also that the axioms do not mention infinity explicitly.

**Question:** If the universe is infinite and matter cannot be created, how is there enough matter for the infinitely many planets?

I mention this here because I was asked this question by Becky, and I must apologise to her because my reply isn’t getting to her for some reason.

First of all, as I explained, I don’t think the universe is infinite in any meaningful sense. Second, some cosmologists have said that matter can be created (this view was championed by Fred Hoyle but has lost out to the Big Bang theory).

But if the universe is infinite, and matter can’t be created, then infinitely much matter must have been there from the start. This is only impossible if you imagine the Big Bang starting with a single point. Perhaps at the moment it happened, it filled the whole infinite universe!

**Question:** If the natural numbers have a boundary (zero), how can they be infinite?

This confuses two sorts of infinity: unboundedness (such as an infinite universe would exhibit) and mathematical infinity, i.e. non-finiteness. There are infinitely many natural numbers, in my view (and that of almost all mathematicians). Even an infinite journey begins with a single step!

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