What exactly is the meaning of the term 'being that exists' which is associated with the argument from contingency. Can I equate this term with an abstract object (SEP) such as 'element' in the mathematical definition of a set so that I can make a set of the elements of 'being that exists'? Does this break some sort of philosophical principle?
The argument of the first mover was developed in Aristotle and subsequently identified in Islamic and Christian philosophy with an act of God and hence a reasoned argument for the existence of God from rational grounds.
This argument has nothing to do with the ontological status of sets or of the elements of sets. You can take a mathematical realist position here, as the mathematician Connes does, and say that such things exist in an world intelligible to the mind. Or you can take a more agnostic view, as David Hilbert did, and merely take this to mean that the axioms are self-consistent.