MHB Ordered numbers - limited or not

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There are multiple ways to order numbers, with two distinct arrangements possible for two numbers in a total order. When considering partial orders, which allow for elements where no definitive relationship exists, the number of arrangements increases to three. The discussion raises the question of whether there are infinite ways to order numbers, suggesting a deeper exploration of ordering principles. Proofs and examples are referenced to illustrate these concepts. Overall, the nature of ordering numbers can vary significantly based on the type of order applied.
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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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