Can we do sorting in less then Nlog(N) time?

Suppose we are using a set ( it can be multiset if there are duplicate values), as we know insertion in a set is log(n) time where n is the current length of the set, so if we insert n elements in a set then overall complexity will look like log1+log2+log3…log(n). We can surely say that it is less than log(n)+log(n)…log(n)=nlog(n), can anyone tell where I am doing wrong?