Simply put first order logic is enough to do almost all of mathematics and is certainly enough to cover all the results of an undergraduate degree. More genera

Simply put first order logic is enough to do almost all of mathematics and is certainly enough to cover all the results of an undergraduate degree. More generally speaking the term "first order logic" is ambiguous by itself since we are not specifying our domain of discourse. For example, take the group axioms (e.g. there exists an identity element, every element has an inverse, etc.). In this context our domain of discourse is the elements of a single group. Hence while we can say things such as "there exists only one identity element," which is existential quantification over the elements of the group, we cannot say things such as "there exists a group with only one element" since that is quantification over groups themselves and disallowed by first order logic. However, if we change our domain of discourse and instead e.g. use ZFC (a first order theory) as our starting point and simply augment ZFC with the language of group theory, then both statements are expressible, since both elements of a group and a group itself are sets in ZFC. So insofar as something like ZFC is enough to do almost all of mathematics, first order logic is enough to do almost all of mathematics.