What Are Examples of Alternate Orderings in Mathematics?

[gmax137]

In another thread

I don't have a specific example in mind, but there could be a solution in domains other than the integers and with an ordering other than our normal ordering. It all depends on which meanings you attach to the symbols you use.

This has me curious about "ordering other than our normal ordering." What does this mean? I take it that "normal ordering (of integers)" is ... 0, 1, 2, 3... Do mathematicians consider alternate orderings like ...0, 2, 1, 3... That doesnt seem to make sense to me, that's more like changing the names. Or is it like complex numbers, where it isn't really clear what Z1 > Z2 means.

I think I'm looking for pointers to where "alternate orderings" would be described / discussed.

Thanks

 

[mathman]

Other than normal has to be user defined!

 

[jbriggs444]

In another thread

This has me curious about "ordering other than our normal ordering." What does this mean? I take it that "normal ordering (of integers)" is ... 0, 1, 2, 3... Do mathematicians consider alternate orderings like ...0, 2, 1, 3...

Yes. Mathematicians consider orderings like that. Look at this article about total orders.

Thinking about a different order for the integers does not affect their arithmetic properties. You still have 2 + 2 = 4 even though 4 may not be the number after the number after 2 in the new ordering.

For instance, you could sort all of the even numbers up front and put all of the odd numbers behind.

Or you could put 42 up front and leave all of the other numbers in the standard order behind.

Or you could set up a bijection (a one to one mapping) between the integers and the rational numbers and sort the integers in order by their counterpart rational number according to the chosen bijection.

The resulting ordering might not have a "first integer". That would mean that it is not a "well ordering".

You might weaken the order so that you do not require that all integers be comparable at all. That sort of thing would be a "partial order"