Ordered numbers - limited or not

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highmath
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How many ways we can order numbers?
Is there infinity ways to order numbers?
Is there a proof of it?
 
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Well, two numbers, for example, can be ordered in two ways. If we allow partial orders, that is, allow elements $x$ and $y$ such that none of $x<y$, $y<x$ and $x=y$ holds, then there are three ways to order two numbers.
 

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