Mathematics
Constructing the Integers from Peano Axioms
Quick fact
In the standard construction, the integer 0 is defined as the equivalence class of the pair (0,0), and the integer -1 is the equivalence class of (0,1), using a clever rule that treats pairs as differences.
Why this is interesting
You've counted 1, 2, 3... but where does 0 come from? And how do you get -3 when you only have the counting numbers? Surprise: integers are not given—they are built.
Read the full explanation
Understanding Constructing the Integers from Peano Axioms
Start with the natural numbers: 0,1,2,3,... as given by the Peano axioms. These axioms say there is a first number 0, and every number n has a successor S(n) (like n+1). To create negative numbers, we need a way to represent a 'deficit'. Idea: Represent every integer as a pair (a,b) of natural numbers, intended to mean 'a minus b'. For example, (5,2) means 5-2=3, and (2,5) means 2-5=-3. So the pair captures the result of subtraction even when the result is not a natural number. But this representation is not unique: (7,4) also means 3. So we must identify pairs that represent the same integer. We say (a,b) and (c,d) are equivalent if a + d = b + c. For instance, (5,2) and (7,4) are equivalent because 5+4 = 2+7 (both 9). Each integer is then an equivalence class of such pairs. For example, the integer '3' is the set of all pairs (a,b) with a-b = 3, like (3,0), (4,1), (5,2), and so on. This entire construction uses only natural numbers and addition—no subtraction is defined yet!
A deeper explanation
The construction works because we define an equivalence relation on the set of ordered pairs of naturals. The relation is (a,b) ~ (c,d) if a+d = b+c. This relation is reflexive, symmetric, and transitive, making it an equivalence relation. The set of equivalence classes forms the integers. Operations like addition and multiplication are defined on these classes. For example, the class of (a,b) plus the class of (c,d) is the class of (a+c, b+d). Multiplication is trickier: (a,b) times (c,d) is (ac+bd, ad+bc). These definitions respect the equivalence, meaning they give the same result no matter which representative you choose. Why does this matter? It shows that negative numbers are not mystical but can be constructed from something simpler. This is a cornerstone of mathematics: building new number systems from existing ones using set theory. It also ensures that all arithmetic properties we expect of integers—like associativity and distributivity—hold by logical proof, not by assumption. This approach is foundational: once we have integers, we can construct rationals as pairs of integers, then reals via Cauchy sequences or Dedekind cuts, building the entire number line from the ground up.