a set N or The non-logical symbols for the axioms consist of a constant symbol 0 and a unary function symbol S. The first axiom states that the constant 0 is a natural number: Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number, while the axioms in Formulario mathematico include zero. The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments. The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function S. Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction. The induction axiom is sometimes stated in the following form: In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section below. Defining arithmetic operations and relations If we use the second-order induction axiom, it is possible to define addition, multiplication, and total (linear) ordering on N directly using the axioms. However, and addition and multiplication are often added as axioms. The respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique using the Peano axioms. Addition Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as: For example: The structure is a commutative monoid with identity element 0. is also a cancellative magma, and thus embeddable in a group. The smallest group embedding N is the integers. Multiplication Similarly, multiplication is a function mapping two natural numbers to another one. Given addition, it is defined recursively as: It is easy to see that (or "1",