Peanoaxiomák

Peanoaxiomák, also known as the Peano axioms, are a set of fundamental axioms for the natural numbers published by the Italian mathematician Giuseppe Peano in 1889. They provide a formal foundation for arithmetic and are crucial in mathematical logic and set theory. The axioms define the properties of natural numbers and the successor function.

There are typically five Peano axioms, although variations exist. The first axiom states that 0 is a natural number. The second axiom asserts that every natural number has a successor, which is also a natural number. The third axiom states that 0 is not the successor of any natural number. The fourth axiom posits that distinct natural numbers have distinct successors; if the successor of x is equal to the successor of y, then x must be equal to y. The fifth axiom, the principle of mathematical induction, is perhaps the most powerful. It states that if a property holds for 0, and if whenever the property holds for a natural number n, it also holds for its successor, then the property holds for all natural numbers. These axioms are sufficient to construct the entire system of natural numbers and their arithmetic operations.