I hear that the axiom of choice (AC) derives from 
The generalized continuum hypothesis(GCH).
And also hear that both AC and GCH are independent of 
Zermelo–Fraenkel set theory(ZF).


So, I'm just curious why don't expert mathematicians use ZF+GCH 
instead of ZF+AC(ZFC).



Difference between ZFC and ZF+GCH
In some sense $\sf GCH$ is a limiting axiom. While it solves a lot of things, it also means that certain things we are interested in become false or trivialized. And that's no fun.


For example, forcing axioms like $\sf MA$ become trivial assuming even just $\sf CH$, and stronger forcing axioms like $\sf PFA,MM$ and others become false (since they imply $2^{\aleph\_0}=\aleph\_2$).


So while $\sf GCH$ gives us more power in terms of implications, it also gives us more constraints. On the other hand the axiom of choice gives us a lot of power, since it helps tame infinite objects, but it leaves enough space to other axioms to develop.


And since you're asking, we can take this question one step further. $\sf GCH$ can be derived from the axiom $V=L$ (Godel's axiom of constructibility), so why aren't we just assuming it and that's that? Well, essentially the same answer. While it has a lot of merits, it also imposes a lot of constraints, more than we might want to assume.




---


Here's another (perhaps a better) argument in favor of $\sf AC$ and against $V=L$ and $\sf GCH$.


When working over $\sf ZF$ and you use forcing, then once you force $\sf AC$ to be true, it will remain true in every other generic extension, whereas $V=L$ is immediately false once you add any new sets to the model, and will never again be true if you only allow yourself generic extensions; and $\sf GCH$ is like a switch that we can turn on and off using [possibly class-] generic extensions.


So this is a very good argument in favor of the axiom of choice. It is stable under forcing, which is a lovely technique. On the other hand, limiting yourself to $\sf GCH$ makes forcing more difficult (because you will always have to ensure that the forcing does not violate that), and insisting on $V=L$ will make forcing outright impossible.


