Why does std::set seem to force the use of a const_iterator?

Consider the simple program below, which attempts to iterate through the values of a set using NON-const references to the elements in it:

#include <set>
#include <iostream>

class Int
{
public:
   Int(int value) : value_(value) {}
   int value() const { return value_; }
   bool operator<(const Int& other) const { return value_ < other.value(); }
private:
   int value_;
};

int
main(int argc, char** argv) {
   std::set<Int> ints;
   ints.insert(10);
   for (Int& i : ints) {
      std::cout << i.value() << std::endl;
   }
   return 0;
}

When compiling this, I get an error from gcc:

test.c: In function ‘int main(int, char**)’:
test.c:18:18: error: invalid initialization of reference of type ‘Int&’ from expression of type ‘const Int’  
for (Int& i : ints) {  
              ^  

Yes, I know I'm not actually trying to modify the elements in the for loop. But the point is that I should be able to get a non-const reference to use inside the loop, since the set itself is not const qualified. I get the same error if I create a setter function and use that in the loop.

A set is like a map with no values, only keys. Since those keys are used for a tree that accelerates operations on the set, they cannot change. Thus all elements must be const to keep the constraints of the underlying tree from being broken.