Better yet, just consider all deduction as "a fun language game," rather than having any intrinsic truth or falsity. If you assume a whole bunch of stuff and th

Better yet, just consider all deduction as "a fun language game," rather than having any intrinsic truth or falsity. If you assume a whole bunch of stuff and the axiom of choice, you can get these interesting things over here. If you assume that same bunch of stuff but not the axiom of choice, you get this other interesting stuff over there. Which assumptions to use at any given time just depend on what you're trying to do. If I'm trying to count my sheep, I just need enough axioms to get me addition of natural numbers. If I'm trying to model some mechanical trajectory, maybe I ought to grab some reals. Got circles? Adding imaginary numbers to the reals makes that easy, even though imaginary numbers are totally "fake." Using sentences that might not make sense? Maybe avoiding the excluded middle is a good idea.