Not a lot of time this morning, but if you are interested in seeing how order is used in set theory, John Armstrong is doing a nice series of the creation of numbers at The Unapologetic Mathematician. Today he talks about Archimedean fields and how the real numbers are in the ‘sweet spot’ between being too big (any larger and they wouldn’t be Archimedean) and too small (any smaller and they would not be Dedekind Complete).

Both the Archimedean property and Dedekind Completeness are statements about ordered sets:

The Archmedean Property basically says that no matter how small the number I choose, I can always find a natural number to multiply it by to make the absolute value of the product **greater than** any other (fixed) number.

The property of Dedekind Completeness states that any subset with an upper bound has a **supremum (least upper bound).** Notice that the rational numbers are a lattice, but not Dedekind complete. It is important to be careful here, as there is a related concept called ‘completeness’ or ‘order completeness’

Notice also the role that **order** plays in both of the above definitions! For those of us thumbing along through the pages of Dr. Erdman’s Book, you can find the definitions in Chapter 5: Dedekind Complete 5.2.6; and Chapter 6: Archimedean property of the Reals 6.1.12.

That’s all for today!

Ex animo-

Felicis

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