#include <iostream>
#include <vector>
#include <sstream>

using namespace std;

class Ordinal
{
public:
    Ordinal(): ordinal{} {}

    string to_string() const
    {
	if( ordinal.empty() )
	{
	    return "";
	}
	stringstream strstream{};
	strstream << ordinal[0];
	for( auto it{ ++ordinal.cbegin() }; it != ordinal.cend(); ++it )
	{
	    strstream << ", " << *it;
	}
	return strstream.str();
    }

    Ordinal successor() const
    {
	Ordinal successor{};
	return ++successor;
    }

    Ordinal& operator++()
    {
        ordinal.push_back(ordinal.size());
	return *this;
    }

    Ordinal operator++(int)
    {
        Ordinal tmp { *this };
        ++(*this);
        return tmp;
    }
    
    Ordinal operator+(Ordinal const& other) const
    {
	Ordinal sum{*this};
	for( long unsigned int i{0}; i<other.ordinal.size(); ++i )
	{
	    ++sum;
	}
	return sum;
    }
    
private:
    vector<unsigned int> ordinal;
};

ostream& operator<<(ostream& os, Ordinal const& ordinal)
{
    return os << ordinal.to_string();
}

int main()
{
    Ordinal zero{};
    cout << "0: " << zero << endl;
    Ordinal const one{zero.successor()};
    cout << "1: " << one << endl;

    // three starts as 0
    Ordinal three{};

    // increment it to two
    ++(++three);

    // copy the current three, thus getting two and then increment three
    Ordinal two { three++ };
    
    cout << "2: " << zero + two << endl;
    cout << "2: " << two + zero << endl;
    cout << "3: " << three << endl;
    cout << "4: " << two + two << endl;
}