I have been reading Topics in Algebra and got really interested when the author says counting is sometimes the most difficult part in mathematics. How does one account for all the possible scenarios in complex situations?

I find one mathematical concept which helps a lot in this regard is that of an equivalence relation. The most striking (yet basic) result of these equivalence relations is that of the disjointedness of the corresponding equivalence classes.

As I read more of the book, I am becoming convinced that I dont have to read it anymore! I dont see why constructing a theory based on a predefined structure of the objects is any good (group theory is built on the notion of a group). Such a theory may have its uses for applications (e.g: Fourier theory) but my tastes are currently towards understanding the limitations of logic itself.

I am done wading through all those endless theorems. I would love to understand that Godel’s incompleteness theorem paper now. Ofcourse, that is an adventure in itself.