Is there a way to implement an Indexer on a class member and not the class itself?

This is strictly for code readability because I would like the idea of having an Indexer property of a variable within a class.

An Example would be

Code:
    public class MyClass
    {
        private List<PlayerClass> player = new List<PlayerClass>();

        public PlayerClass Player[int index]
        {
            get { return player[index]; }
            set { player.Add(value); }
        }    
    }
Then lets say I instantiate a MyClass object

MyClass blah = new MyClass();

I would be able to access the player list like so...

blah.Player[1]

without making the player list public. I would like to keep this member private so that I can use the properties to validate the input (even though that isn't shown in this example). I know I can do this with a method and keep my list private, I was just hoping i could use the indexer notation []. Any ideas?