Does structuralism hold that the statement of the Continuum Hypothesis has a truth value? If no, how does it differentiate between my hash of BB mod 2 statement and CH?
I think there may be some variation in the answer across the different strands of structuralism.
Modal structuralism would say that there are two different possible mathematical structures. One where the CH is true and one where the negation of CH is true. Any statements dependent on the CH will have to be qualified with the mathematical structure being studied. If there is some statement X that is true if CH but false if not CH, then that is fine. It is no different than saying the angles of a triangle add to 180 in Euclidean space but not in hyperbolic space.
Your hash of BB mod 2 statement is different because it is true in all the mathematical structures we study (or not expressible in certain restricted structures). Also, it is not independent of those structures. So we simply say it is true. But if there were some logically possible mathematical structure where it had the opposite truth value, then one would need to qualify which mathematical structure one was referring to when talking about its truth.
Does structuralism hold that the statement of the Continuum Hypothesis has a truth value? If no, how does it differentiate between my hash of BB mod 2 statement and CH?
I think there may be some variation in the answer across the different strands of structuralism.
Modal structuralism would say that there are two different possible mathematical structures. One where the CH is true and one where the negation of CH is true. Any statements dependent on the CH will have to be qualified with the mathematical structure being studied. If there is some statement X that is true if CH but false if not CH, then that is fine. It is no different than saying the angles of a triangle add to 180 in Euclidean space but not in hyperbolic space.
Your hash of BB mod 2 statement is different because it is true in all the mathematical structures we study (or not expressible in certain restricted structures). Also, it is not independent of those structures. So we simply say it is true. But if there were some logically possible mathematical structure where it had the opposite truth value, then one would need to qualify which mathematical structure one was referring to when talking about its truth.