What exactly is the significance of an axiom that is an open formula? As I understand it, a formula in a first order theory is open if it has at least one free variable, and an open formula is valid if and only if its closure is valid.
What, for example, does the formula x=y assert as an axiom? Axioms are taken as valid by definition (i.e. we agree to accept them as valid without proof). So x=y being valid means "for all x, and for all y, x=y". That is, any two distinct variables in the language of the theory designate one and the same element. So a theory with such an axiom would only have models with one-element domains. Is my understanding correct?
The reason I ask this question is that in his "Mathematical Logic" text, when presenting solutions to the characterization problem for first-order theories in chapter 4, Schoenfield treats open theories separately (a theory is open if all of its non-logical axioms are open formulas), and i'm trying to understand what difference it makes. For that matter, why would you even assert open formulas as axioms when you can just assert their closure (you do need axioms to be either valid or not valid and you can only talk about validity of a formula if it's closed, otherwise it has many meanings).