Weak second order Logic

Question

I was reading the other day (Chapter 3 Introduction), that

sequential calculus is also called weak second order monadic logic with one successor or WS1S.

I understand the difference between first-order and second-order logic and also the one successor parts. What is a "weak" MSO? SO, what is it weaker than?

Answer 1

In case somebody stumbles upon this question, it would be a crime to leave it unanswered. In the Semantics of WSkS (Section 3.3.2 Semantics of the book) it says

the logic SkS is defined ... except that set variables may be interpreted as infinite sets.

By contrast, in case of WSkS, they are finite sets.