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.
- 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?