A function from to is called additive if for every . Clearly any function of the form is additive. Are there any other additive functions?

An easy induction shows that if then for every positive integer . (This is true with , since for every .) It is also easy to prove that (since e.g. ). And from this it follows that , so , for every positive integer . Another easy induction shows that for every real number and every positive integer . Therefore, , from which it follows that for every real number and every positive integer . From these observations it follows that if then for every rational number .

At this point it seems to be hard to deduce anything about other values of . Indeed, there doesn't seem to be much obstacle to defining to be anything we like. If we set to be , then an argument similar to the argument of the previous paragraph shows that for every rational number , but no number of the form is rational except when , so this is not going to conflict with the choices we have already made. We will of course be forced to set to be , but there is no problem in doing that.

Let us be slightly more explicit about why this isn't a problem: it is because if then and (since otherwise we would find that , which is rational). This will enable us to see more clearly what is going on later.

The discussion so far strongly suggests that there should be a function that is additive but not of the form . We haven't yet defined one, since by no means every real number is of the form with and rational. But we have produced a partially defined additive non-linear function, and the method we have used is rather flexible. Indeed, if we pick another number, such as , say, where is not yet defined, then we can extend the definition to all numbers of the form with by setting for some arbitrarily chosen .

More generally, we could construct a sequence of numbers with , with the property that no is a rational linear combination of . And then we could define , for arbitrarily chosen , which would tell us that for every sequence of rational numbers.

The trouble is, even when we have built an infinite sequence in this way, we still haven't defined for all real numbers, since the set of rational linear combinations of those numbers is countable. However, we can still continue to build our function, since we can pick a new real number that is not a rational linear combination of the and choose a value for . And then we can choose that is not a rational linear combination of the and , and so on. But again we find that even if we produce an infinite sequence of we have still defined for only countably many real numbers.

A good way of regarding what we are doing is this: we are considering the real numbers as a vector space over the rationals, and we are trying to build a basis for this vector space, where this means a collection of real numbers such that every real number is a rational linear combination of numbers in in precisely one way. Then if we define the values of however we like for the numbers in and define the values of in the obvious way for rational linear combinations of those numbers, we have a function from to that is linear over the rationals, and hence additive, but not necessarily of the form .

Now , considered as a vector space over , is certainly infinite-dimensional. In fact, it has uncountable dimension. So does it have a basis? (All we mean by "has uncountable dimension" is "cannot be spanned by countably many vectors," so it is not true by definition that it has a basis.) Inspired by finite-dimensional vector spaces it is tempting to say, "Pick a maximal linearly independent set," since such a set is not just linearly independent but also spans the whole space, since if it didn't we could just pick an element that did not belong to its linear span and we could add it to the linearly independent set, contradicting maximality.

So now we seem to be done: we are looking for a basis of over , and all we need to do to find a basis of any vector space is take a maximal linearly independent set.

But why should a maximal linearly independent set exist? Isn't that exactly the difficulty we were facing earlier: we could carry on picking more and more rationally independent real numbers but we never seemed to reach the point where we could no longer continue?

Let us now interrupt this example for a more general discussion of Zorn's lemma and how to use it.