Armless thought processes

I asked my father today, ‘How do you know that your arm is your arm and not the arm of another, that it is part of your ‘self’? He replied, ‘Because I can intend for it to move and as result of this intention it moves.’
This seemed sensical enough so I gave it some further thought.
So, if our arm is ours, is a fraction of our self, because we can control it, decide its movement after a plan is formed, then what of the cup on the table? If we see this cup, we want for it to be somewhere that it is not, we attach it as an adjunct to our very own arm, and based on an intention move it from one place to another, after which it is then detached, has this cup been part of our ‘self’? We have intended for it to move, it has moved in accordance with the will.
If the answer is yes, this cup has been part of us. This would seem to make the entire environment which we have altered, by whim and fancy, part of us. We would therefore stretch out almost infinitely from a central point into our environment, which is not separate at all from our selves.
If the answer is no, the cup is external, it has been moved by me, it is not inherently me. The this would seem to make our ‘self’ infinitely small, as our arm, our skin, our skeleton, our viscera are environmental, are moved by intention, but are not ‘me’. But where am I? This would make me an infinitely small spot in the centre of the things I move by intention.
If we are to reject the negative, accept that the cup has been an additional finger, then what of ‘others’? If I were to have a spiteful thought, to shove, push, knock into some body, they have moved, moved because I wanted them to, not because they have their own plan of motility. In that contact have they been me? Where has their ‘self’ been placed for that moment? Would it be extending into the things which they have made their own plans for? If they made action to steady themselves on an unstable chair, which was then to crash to the floor, would this be where they lay?
And so I say to my father, ‘Well what if by some awful misfortune, some accident, you have your arm paralyzed, you are no longer able to move this appendage, and it will hang limply by the rest of your ‘self’. Is this arm still you, more than simply yours?’ My father responds, ‘Yes it is still my arm, for I have moved it, it has been my intention, even if it is not so now. It is no-one else’s arm; it is still part of me.’
And so I consider. If our possession is lingering, on these objects of our, ‘It is my arm because I have had the intention and I have made physical change’, then does this cup remain a part after the moves have been made?
If this is to be the case then are these particles of self overlapping? If I have left this cup, and my good friend then enters the room. If he/she/they decide that this is a ridiculous place to allow a cup to be, they then attach themselves to the cup. The cup is incorporated; it is moved by their intention back from whence it came. Has this part of self, of ‘me’, been overridden, wiped clean? Or are these selves now overlapping, accumulating in the ceramic of this very cup?
Remember the really nasty one in Terminator? I’m just not sure at all as to whether I am very very big, or very very very small.