> But Principia Mathematica was (to a significant degree) a dead-end in the history of mathematics. Most practicing mathematicians have rejected PM's type theory in favour of simpler axiomatic systems such as ZF(C). Even many professional type theorists will quibble with some of the details of Whitehead and Russell's type theory, and argue there are superior alternatives. And you are effectively assuming a formalist philosophy of mathematics, which is highly controversial, many reject, and few would consider "proven".

Yeah, exactly. I intentionally set that trap. You're actually arguing for my point. I've spent comments writing on the axioms of geometry, and you didn't think I was familiar with the axioms of ZFC? I was thinking of bringing up CH the entire time. The fact that you can have alternate axioms was my entire point all along. Most people are just way more familiar with the 5 laws of geometry than the 9 axioms of ZFC.

The fact that PM was an alternate set of axioms of mathematics, that eventually wilted when Godel and ZF came along, underscores my point that defining a set axioms is hard. And that there is no clear defined set of axioms for philosophy.

I don't have to accept your argument against objectivity in ethics, because I can still say that the system IS objective- it just depends on what axioms you pick! ZF has different proofs than ZFC. Does the existence of both ZF and ZFC make mathematics non objective? Obviously not! The same way, the existence of both deontology and consequentialism doesn't necessarily make either one less objective than the other.

Anyways, the Genghis Khan example clearly operates as a proof by counterexample of your example of objectivity, so I don't even think quibbling on mathematical formalism is necessary.