There is also algebraic number theory, but algebra and number theory are related as soon as you start learning about them. However, saying that that "...is concerned with abstract sets and quantifiable properties..." is not a good answer because there are certain topics in number theory that are related to numbers but they come from a totally different area, for example, modular forms from complex analysis being applied to number theory. I would agree with you that there is no connection but I do not know.
