Groucho Marxism

When children first start learning about numbers, a question they often ask is: what is the biggest number? At which point we adults usually explain that there is no biggest number as whatever number you can think of, you can always add 1 to make an even bigger number. But what if that wasn’t actually true? What if there was a number that you can’t add 1 to make an even bigger number? At first this idea strikes us as absurd. However we must bear in mind that numbers are a human invention and we are therefore free to define them, and operations on them, as we please. There is nothing to prevent us defining our number system in such a way that, instead of numbers going on forever, at some point they simply stop.

The natural numbers – 0,1,2,3, and so on – and the operations on them are defined formally using something called the Peano axioms, named after the 19th century Italian mathematician Giuseppe Peano. The first axiom asserts the existence of at least one natural number; the next four are general statements about equality of natural numbers; the next three are statements about natural numbers expressing the fundamental properties of the ‘successor operation’; and the final axiom is a statement of the principle of ‘mathematical induction’ over the natural numbers. The successor operation is codified using a function S() which maps the set of all natural numbers to itself, with the interpretation that S(n) = n+1.

These axioms were chosen because they best represent our intuitive understanding of how natural numbers work. But there is nothing to prevent us from changing any of these axioms and seeing what happens when we do. So let’s do that. One of the Peano axioms says that for all natural numbers m and n, S(m) = S(n) if and only if m = n; in words, the successor function always takes different output values given different input values. What if we replaced this axiom with its negation? The negation of this axiom says that there exist natural numbers m and n such that m ≠ n and S(m) = S(n); in words, there are (at least) two distinct numbers which yield the same result when the successor function is applied to them.

Intuitively, we may take one of these numbers – say m – to be the largest number, and the other – say n  – to be the largest number minus 1. We can also stipulate that S(m) = S(n) = m – in words, adding 1 to m yields m – and that m is the only natural number with this property. As in standard Peano arithmetic, we may define addition of natural numbers using the successor function by a+0 = a and a+S(b) = S(a+b). Then m+0 = m; m+1 = m+S(0) = S(m+0) = S(m) = m; m+2 = m+S(1) = S(m+1) = S(m) = m; and so on. Therefore m+a = m for any natural number a. Similarly, we have 0+m = 0+S(m) = S(0+m), so 0+m = m; 1+m = 1+S(m) = S(1+m), so 1+m = m; 2+m = 2+S(m) = S(2+m), so 2+m = m; and so on. Therefore a+m = m for any natural number a.

Again, as in standard Peano arithmetic, we may define multiplication recursively using the successor function by a·0 = 0 and a·S(b) = a+a·b. Then m·0 = 0; m·1 = m+m·0 = m+0 = m; m·2 = m+m·1 = m+m = m; and so on. Therefore m·a = m for any natural number a ≠ 0. We have 0·m = 0·S(m) = 0+0·m, which doesn’t tell us anything about 0·m, so we can stipulate that 0·m = m·0 = 0. For any natural number a ≠ 0 we have a·m = a+a·m, so a·m = m. Thus addition and multiplication with respect to the largest number m work as follows: m+a = a+m = m for any natural number a; m·0 = 0·m = 0; and m·a = a·m = m for any natural number a ≠ 0. These are identical to the rules usually used when applying addition and operation to infinity: ∞+a = a+∞ = ∞; ∞·0 = 0·∞ = 0; and ∞·a = a·∞ = ∞ for a ≠ 0.

The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory. The standard construction of the naturals, due to the Hungarian mathematician John von Neumann, starts from a definition of 0 as the empty set, ∅, and defines the successor function S() by: S(a) = a∪{a}. Under this definition, a = {0,1,2,…,a-1} for each natural number a. In standard set theory the so-called axiom of infinity ensures that the set of natural numbers is infinite – i.e. that numbers go on forever. In a previous blog post I suggested that to better root mathematics in the material world we should replace the axiom of infinity with its negation, which in turn would mean that numbers do not go on forever. So perhaps the idea that there is a largest number is not so absurd after all.