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What Are Natural Numbers?

Now I know, as we all learn, natural numbers are positive integers. But how do we define positive?

The simple answer would be, greater than zero. However, that brings up the question of being "greater than". What exactly does "greater than" mean? I've heard definitions that \(a\) is greater than \(b\) if \(a-b\) > \(0\). But that definition has "greater than" in it!

Please help!

Note by Clive Chen
1 year, 2 months ago

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Positive numbers are numbers such that if you take the square root of it, the resulted number is real. Jonathan Hsu · 1 year, 2 months ago

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@Jonathan Hsu So Zero is positive? Clive Chen · 1 year ago

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"Greater than" is a total ordering of the natural numbers.

A total ordering applied to a set \(X\) assigns to any two elements which is "greater" (>) in a way that obeys the following rules:

  • For any \(x,y\), either \(x\ge y\) or \(y\ge x\). (That's the total part)
  • If \(x \ge y\) and \(y\ge x\), then \(x=y\). (So "equal" in the order is the same as equal in the set)
  • If \(x \ge y\) and \(y\ge z\), then \(x\ge z\). ("Greater than" is transitive)

A definition of the natural numbers might include the total order we learned as children \(\ldots,-2,-1,0,1,2,\ldots\) (Although of course there are additional structures on the integers).

So \(b-a>0\) is really just something we've all agreed upon, from this point of view. Maggie Miller · 1 year, 2 months ago

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