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Vladimir_Nesov comments on What is Eliezer Yudkowsky's meta-ethical theory? - Less Wrong
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Compare with formal systems giving first-order theories of standard model of natural numbers. You can't specify the whole thing, and at some point you run into (independent of what comes before) statements for which it's hard to decide whether they hold for the standard naturals, and so you could add to the theory either those statements or their negation. Does this break the intuition that there is some intended structure corresponding to natural numbers, or more pragmatically that we can still usefully seek better theories that capture it? For me, it doesn't in any obvious way.
It seems to be an argument in favor of arithmetic being objective that almost everyone agree that a certain a set of axioms correctly characterize what natural numbers are (even if incompletely), and from that set of axioms we can derive much (even if not all) of what we want to know about the properties of natural numbers. If arithmetic were in the same situation as morality is today, it would be much harder (i.e., more counterintuitive) to claim that (1) everyone is referring to the same thing by "arithmetic" and "natural numbers" and (2) arithmetic truths are mind-independent.
To put it another way, conditional on objective morality existing, you'd expect the situation to be closer to that of arithmetic. Conditional on it not existing, you'd expect the situation to be closer to what it actually is.