Can someone please give me an example of a set that lies in a $\sigma$-algebra generated by some set other then the $\sigma$-algebra itself, such that this (the first) set can't be obtained by performing countably many set theoretical operations ? (Obviously with "example of a set" I mean "prove that such a set exist", since this set probably isn't obtainable in a constructive way.)
I'm asking this since I've heard countless times, that generally the elements of a $\sigma$-algebra are not really constructively reachable.