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Peano's Axioms
authored by Premmi and Beguène Previous Topic: An Axiomatic Study of Numbers Introduction Thinking of numbers intuitively brings to mind the simplest and most fundamental set of numbers, namely the set of natural numbers. These numbers are used to count objects like cars, books, pens, etc. If we associate natural numbers such as etc. with counting, […]
In order to appreciate the intellectual feat of Peano, it’s worth noting that utilizing only his axioms, we are able to prove all the established properties of the natural numbers. Before we discuss Peano’s Axioms in detail, it is a useful exercise to explore an alternative way to describe natural numbers, distinct from the usual intuitive notion of counting. This formulation is equivalent to the axiom of induction and serves the same purpose of removing unwanted elements from the superset \mathbb{N}_0, ensuring that it contains only natural numbers.
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