Second-order logic and foundations of mathematics

We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts. it relies entirely on informal reasoning. On the other hand. if it is given a weak servant cs, it loses its power in expressing concepts categorically. First-order set theory and seconJ-order logic are not radically different: the latter is a major fragment of the former.