Thread #16920098
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For there to be order there has to be, at the very least, a binary relation on a set A, such that, (1) for all a in A, a is less than or equal to a, and (2) for all a, b in A, a is less than or equal to b, and b is less than or equal to a.
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>>16920098
What you describe here is order as a sequential organization of elements. There are other senses of the word which aren't related to a notion of an "ordering" like this, and what you describe is a particular class of ordering known as a strong transitive ordering. There are other classes of orderings.
>there has to be, at the very least
Incorrect. You simply need to be able to express a semantically equivalent principle. Some formalisms don't even have real collections, though they can encode them.
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>>16921019
You're right. There are many kinds of order and I was very narrowly only listing a very basic one.
For example, a discreet preorder would only have the equal sign as a binary relation on a set, which would make it impossible to rank anything in such a set. Every single element in that set would be "equal" to every other element in the set. That is not how we typically think of order, since it would have no heirarchy, but it is a kind of ordering.
TL;DR, you are correct, I kneel.