# Why are certain sets defined?

I was reading a little bit about Galois theory and set theory in my free time, and something that I noticed is how 'constructed' these branches of mathematics seem. For example, arithmetic, algebra and calculus to me seem to be very 'natural'. I know this isn't a very mathematical thing to say, as all of math is human. When studying set theory and Galois theory you encounter a lot of sets which have to satisfy a certain set of rules. For example:

Group is a set of elements, satisfying the following rules:

• For any 2 elements $x$ and $y$ in group $G$ we also have $xy$ in the group $G$.
• There is an identity element.
• Associativity.
• Every element $x$ in $G$ has a unique inverse $y$.

To me such sets don't feel very natural, as I can also make one myself:

Bazooper is a set of elements, satisfying the following rules:

• Every Bazooper $B$ is a group.

• 2nd, 3rd, ..., nth constraint that I made up myself.

That's why I have trouble studying all these different type of sets, because they're not really intuitive to me. What is the thought process behind defining a certain set?

• Think about the properties that $\mathbb{Z}$ has under the operation of addition. It satisfies all the structure to be a group. The way that I understand it, is that people find a structure that people care about (aka integers, polynomial rings, etc.) and they try to generalize it). What properties do the real numbers satisfy as a vector space? What about the integers under multiplication and addition? – LASV Dec 9 '13 at 15:32

Groups, rings, fields, all come from abstracting properties of "known objects". For example groups are an abstraction of $\Bbb Z$ and the clock arithmetic of $\mod n$; so are rings. Fields are generalization of $\Bbb Q$ and $\Bbb R$.