C++, Classes, Const, and strange syntax

I was re-reading c++ primer(4th ed.) today - the section on member functions and const references etc, and I came up with this wierd little program:

using std::cout;
using std::endl;

class ConstCheater
{
public:
    ConstCheater(int avalue) : ccp(this), value(avalue) {}
    ConstCheater& getccp() const {return *ccp;}
    int value;
private:
    ConstCheater* ccp;
};

int main()
{
    const ConstCheater cc(7); //Initialize the value to 7
    cout << cc.value << endl;
    cc.getccp().value = 4;    //Now setting it to 4, even though it's const!
    cout << cc.value << endl;
    cc.value = 4;             //This is illegal
    return 0;
}

My question is - why does c++ allow syntax such as this? Why can I edit normal data members in a class when it's declared const? Isn't the POINT of const to make it so that you can't modify values?