To keep my Platonist hat on... Well if you accept that there is an objective truth to the natural numbers, the set theory axioms you choose will have ramificati

To keep my Platonist hat on... Well if you accept that there is an objective truth to the natural numbers, the set theory axioms you choose will have ramifications for the natural numbers. (Elsewhere in these threads a commentator brings up that Not(Con(ZFC)) will tell you a Turing machine terminates when in our universe it doesn't) Put another way there will always be a purely arithmetical statement of the natural numbers that is independent of your set theory axioms. What is the truth value of that statement? If you think you have a clear and absolute conception of the natural numbers it seems like you should believe that an arithmetical statement has an objective truth value, and if that's true, then that seems like your objective benchmark to decide on how to accept a new axiom (does it prove or refute this true arithmetical statement?). There's your absoluteness.