section_path: Axiomatic foundations
section_name: Axiomatic foundations
paragraph_text: Axiomatic foundations
  
  Axiomatic foundations of arithmetic try to provide a small set of laws, called axioms, from which all fundamental properties of and operations on numbers can be derived. They constitute logically consistent and systematic frameworks that can be used to formulate mathematical proofs in a rigorous manner. Two well-known approaches are the Dedekind–Peano axioms and set-theoretic constructions.
  
  The Dedekind–Peano axioms provide an axiomatization of the arithmetic of natural numbers. Their basic principles were first formulated by Richard Dedekind and later refined by Giuseppe Peano. They rely only on a small number of primitive mathematical concepts, such as 0, natural number, and successor. The Peano axioms determine how these concepts are related to each other. All other arithmetic concepts can then be defined in terms of these primitive concepts.
  
  Numbers greater than 0 are expressed by repeated application of the successor function s {\displaystyle s}. For example, 1 {\displaystyle 1} is s (0) {\displaystyle s(0)} and 3 {\displaystyle 3} is s (s (s (0))) {\displaystyle s(s(s(0)))}. Arithmetic operations can be defined as mechanisms that affect how the successor function is applied. For instance, to add 2 {\displaystyle 2} to any number is the same as applying the successor function two times to this number.
  
  Various axiomatizations of arithmetic rely on set theory. They cover natural numbers but can also be extended to integers, rational numbers, and real numbers. Each natural number is represented by a unique set. 0 is usually defined as the empty set ∅ {\displaystyle \varnothing }. Each subsequent number can be defined as the union of the previous number with the set containing the previous number. For example, 1 = 0 ∪ { 0 } = { 0 } {\displaystyle 1=0\cup \{0\}=\{0\}}, 2 = 1 ∪ { 1 } = { 0, 1 } {\displaystyle 2=1\cup \{1\}=\{0,1\}}, and 3 = 2 ∪ { 2 } = { 0, 1, 2 } {\displaystyle 3=2\cup \{2\}=\{0,1,2\}}. Integers can be defined as ordered pairs of natural numbers where the second number is subtracted from the first one. For instance, the pair (9, 0) represents the number 9 while the pair (0, 9) represents the number -9. Rational numbers are defined as pairs of integers where the first number represents the numerator and the second number represents the denominator. For example, the pair (3, 7) represents the rational number 3 7 {\displaystyle {\tfrac {3}{7}}}. One way to construct the real numbers relies on the concept of Dedekind cuts. According to this approach, each real number is represented by a partition of all rational numbers into two sets, one for all numbers below the represented real number and the other for the rest. Arithmetic operations are defined as functions that perform various set-theoretic transformations on the sets representing the input numbers to arrive at the set representing the result.