I'm watching a lecture series by Conway about the free will theorem. There, as far as I understand, he shows that there exists mathematical objects with certain properties for which there is no function on the entire object which satisfies certain symmetry properties which are analogous to the properties of measuring spin. So the conclusion to him is that the object "makes the answer up on the fly" and brings this into connection with the randomness of quantum mechanics when measuring the spin of a particle.
Question: Because there is also no definite answer to something like the position of a particle, are there any objects like this with regard to measuring the position of a particle, where in some sense there is no function on the entire object which satisfy certain "coordinate properties"? Because I really can't think of any.