How to prove the set of natural numbers is closed under addition. [closed]

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Seems so obvious that a natural number is closed under addition. It's just a result of how we count adding apples and apples always gives you whole numbers as adding apples is equivalent to counting the number of apples in two groups of apples . How would you mathematically prove that the set is closed under addition? It seems so obvious that it's probably taken as an axiom.

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2 Answers

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To be able to prove something about addition, you would first have to define it. And the sum of two natural numbers, however you define it, is a natural number.

So it all boils down to how you get to define/construct the natural numbers.

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If you define the natural numbers as the intersection of all inductive sets and thus an inductive set, we can use induction. n + 1 is a natural number for all natural n, and if n +m is a natural number then n+(m+1) is also natural number. So n +m is a natural number for all n and m.

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