Quick Math Test

Peano axioms

The system of the natural numbers, \(\Bbb{N}\), can be characterized with 3 basic axioms.

First, it is a system (or structure) of the type: \(⟨ 𝑁, 𝑆, π‘’βŸ©\), where \(N\) is the domain, \(S\) a function from \(Nβ†’ N\), and \(e\) a special element, such that:

  1. \(e βˆ‰ π‘…π‘Žπ‘›(𝑆)\);
  2. \(S\) is one-to-one (also called β€œinjective”); and
  3. any subset \(A\) of \(N\) that contains \(e\) and is closed under \(S\) is the whole of \(N\)

Axiom 1: the special element \(e\) is not the output of the function \(S\) when evaluated at any element of \(N\). In simpler terms, when we input any element of the domain \(N\) to our function \(S\) we never get \(e\) as the output value. It is clear that, interpreted in the usual way, \(S\) turns out to be the (immediate) successor function, and \(e\) is the number \(0\). So this axiom, interpreted, says that \(0\) is not the successor of any other natural number.

This differentiates the natural numbers from, say, the system of the integers, where \(0\) is the successor to \(-1\).

Axiom 2: the function \(S\) being injective means that no two distinct elements of \(N\) can have the same image under \(S\). Simply put, no two elements (numbers) can have the same successor. This rules out loops, like in modular arithmetic, where two numbers can have the same successor. (Take the 12-hour clock: the \(0\) hour has hour \(1\) as successor, but so does hour \(12\).)

Axiom 3: for a set to be closed under a function means that whenever we input an element of that set into our function we get another element from the same set. This rules out getting extraneous elements.

Therefore, if \(A\) is a set contained within \(N\) and both contains \(e\) and is closed under \(S\), then in fact \(A = N\).

Another way of putting this is to say that nothing but \(N\) itself can contain and be closed under \(S\). If (contrary to fact) there were a smaller set \(A\) fulfilling those conditions, then \(A= N\). For which it also follows that \(N\) is the smallest set that fulfills those conditions.