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I'm trying to figure out why can't Natural numbers be negative, following the axioms that can define real numbers and from them we can define natural numbers. My book says that natural numbers can't be negative by saying that zero as no predecessor.

Thank you!

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3 Answers

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To summarise the comments:

Because they aren't, by definition.

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Assuming your definition of natural numbers goes something similar to this.

1) if $x$ is a natural number then $x + 1$ is a natural number.

2) $0$ is a natural number.

3) The set of natural numbers has no other elements except those which rules 1 and 2 require to be.

Then negative numbers are not natural numbers for the same reason $\pi$ is not. They don't have to be. They don't follow from rules 1) or 2) and thus are forbidden by rule 3).

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following the axioms that can define real numbers and from them we can define natural numbers.

In my book it's the reverse. We first define the natural numbers, then the integers, then the rationals and lastly the reals.

My book says that natural numbers can't be negative by saying that zero as no predecessor.

And your book is quite right. If you allow negative numbers you get the integers. However, many times the nonnegative integers are of interest. It's nice to have a specific name for that set. We call them the natural numbers.

So it's not really that the natural numbers have to be the way they are, but rather that we have this thing that we occasionally are interested in and we want to give this thing a name, and we call them the natural numbers.

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