#concept
first peano axiom ?
- 0 is a natural number
peano axioms 2-5 ? 2. For every natural number x, x = x. That is, equality is reflexive. 3. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric. 4. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive. 5. For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality.
peano axioms 6-8 ? 6. For every natural number n, S(n) is a natural number. That is, the natural numbers are closed under S. 7. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection. 8. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
peano axiom 9 ? 9. If K is a set such that:
- 0 is in K, and
- for every natural number n, n being in K implies that S(n) is in K,
then K contains every natural number.
References
Notes
The first axiom states that the constant 0 is a natural number:
- 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.
- For every natural number x, x = x. That is, equality is reflexive.
- For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
- For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
- For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality.
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.