TITLE: Mathematical properties not expressible in set-theoretic terms
QUESTION [7 upvotes]: It was once assumed that all of mathematics could be expressed in set-theoretic terms.
I understand that this is no longer the case.
Can anyone provide examples of mathematical properties which resist set-theoretic representations.
I would especially be interested in properties of numbers - not arithmetic properties, obviously.

REPLY [3 votes]: Perhaps what you are referring to is the use of set theory as foundations for mathematics and the existence of other foundations. Other foundations do exist, but that does not mean set theory is insufficient. It's more that simply other foundations exist. Some foundations can be better suited for some purposes and not for others. It depends a lot on what you wish to accomplish. 
A recent major advancement in non-set-theoretic foundations of mathematics is homotopy type theory, see here.