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Can anyone provide a rigorous definition of ORDER in set theory?

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Set's by themselves do not have a concept of order at all.   {1,2} is considered to be the same set as {2,1}.     Think of {1,2} as being a box containing "1" and "2".

The notation (1,2) denotes the "ordered pair" (1,2) and the ordered pair (1,2) is considered different than the ordered pair (2,1).   

The concept of an ordered pair is not part of the axioms of set theory.    However we can define what the notation (1,2) means in terms of only set theory in a way that is consistent with our intuitive concept of what an ordered pair is.    

There are multiple ways to define (1,2) set theoretically.   One way would be to define (1,2) as the set {{1},{1,2}}.    That works because the sets {{1},{1,2}} and {{2},{1,2}} are different.
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Order as in order relations, generalisations of "less than or equal to" and its friends? Or the notion that the elements of (7, 2, 5, 1) have a specific order whereas {7, 2, 5, 1} do not?

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