The word "ought" means a particular thing, refers to a particular function, and once you realize that, ought-statements have truth-values. There's just nothing which says that other minds necessarily care about them. It is also possible that different humans care about different things, but there's enough overlap that it makes sense (I believe, Greene does not) to use words like "ought" in daily communication.

What would the universe look like if there were such a thing as an "objective standard"? If you can't tell me what the universe looks like in this case, then the statement "there is an objective morality" is not false - it's not that there's a closet which is supposed to contain an objective morality, and we looked inside it, and the closet is empty - but rather the statement fails to have a truth-condition. Sort of like opening a suitcase that actually does contain a million dollars, and you say "But I want an objective million dollars", and you can't say what the universe would look like if the million dollars were objective or not.

I should write a post at some point about how we should learn to be content with happiness instead of "true happiness", truth instead of "ultimate truth", purpose instead of "transcendental purpose", and morality instead of "objective morality". It's not that we can't obtain these other things and so must be satisfied with what we have, but rather that tacking on an impressive adjective results in an impressive phrase that fails to mean anything. It is not that there is no ultimate truth, but rather, that there is no closet which might contain or fail to contain "ultimate truth", it's just the word "truth" with the sonorous-sounding adjective "ultimate" tacked on in front. Truth is all there is or coherently could be.