Ethics

[Pyrrho]

That is a ridiculous standard, both because it can never be met, due to the fact that there is probably nothing upon which everyone agrees (ever hear of the Flat Earth Society?), and also because it is generally irrelevant to what is true.  Agreement does not mean that one has reached the truth.

I'm more or less saying that in other fields of inquiry we are given a set of axioms that cannot be denied less we fall into self-refutation. If defining "goodness" or "rightness" was actually in any way comparable to determining facts such as the shape of the earth, we would be living in a much different world. 

You are begging the question with that.  You are assuming that they are not matters of fact, but have given no evidence for that claim.  However common it may be to believe what you believe, that isn't evidence that you are correct.

Where is your proof, for example, that Bentham is wrong?

Also, "mind-independent" does not mean the same thing as "objective."  "People have minds" is an objective statement (whether it is true or false), and it clearly deals with minds, so it is not "mind-independent."

"People have minds" is an objective statement because its truth value doesn't depend on any single person's biases, feelings, or interpretations, and in that sense, is "mind-independent." 

"Health and wealth are the greatest goods," on the other hand, is a claim of which the truth depends entirely on the attitudes and feelings of the person assessing it. 

Again, you are begging the question.  Someone like Bentham would tell you that the truth or falsehood of the claim "health and wealth are the greatest goods" is not dependent upon anyone's assessment.  Bentham would say that the truth or falsehood of it is determined by whether it leads to the greatest happiness for the greatest number or not.  It makes no difference what anyone's personal opinion is about it, according to Bentham.

Now, if you want to use the phrase "ethical objectivism" in a nonstandard way, I cannot stop you from doing so, but if you want to effectively communicate with others, it is good to keep to standard usage of expressions as much as possible.

To clarify, I'm speaking of ethical theories that purport moral statements to contain the same value of "truthiness" as claims of objective fact, as in the examples pertaining to beer that you provided, or the three-dimensional geometry of planets. 

According to Bentham, the truth or falsehood of ethical claims is objective in the same way.  It is a question of whether the thing in question promotes the greatest happiness for the greatest number.  It is not dependent on how anyone in particular feels about the thing, or whether anyone believes it promotes the greatest happiness for the greatest number.  It is objective, according to Bentham.  (And, again, his view is just one example; I am neither endorsing his view nor am I saying he is wrong.)

As for the concept of solvable, you are going to have a very hard time determining whether or not I drank a gallon of beer yesterday.  Your inability to solve the problem (or, if you prefer, your inability to know the truth or falsehood of the statement) has no bearing on whether it is true or false.  Likewise, if Bentham is correct, it is entirely possible that you may never know it.

Right. My point is that nobody presumes this is the case when it comes to ethical disagreements.

There have been many philosophers who have written on ethics and have thought that they have given the right answer.  And many have thought that they have given sufficient reasons why others should agree with them.  Dismissing them offhand is not a proof that they are wrong; it is just begging the question.

 Everyone then becomes an expert at what ought to be. 

Many people believe they are experts on ethics.  But that does not make them experts on it.

It is a very curious thing, but very often, in philosophy, people imagine themselves to be experts on various topics, without bothering to have studied them.

Ethics and religion are two separate areas of thought, which you should know from having read Plato's Euthyphro.

What gave you the impression that god or religion even entered my mind here?

I do not know if it entered your mind or not, but, not only has it certainly has entered the mind of someone in this thread, it is a common thing for religionists to claim that it is only through religion that there can be an objective morality, and many atheists who reject an objective morality seem to accept that claim.  So I may be being pro-active in your case, but we are not the only ones in this thread, so it is worth mentioning for the benefit of others who might be reading this.

For the benefit of others who might be reading this:

To be clear, whether there is an objective morality or not is entirely independent of whether or not there is a god.  The only kind of morality for which god would be important would be one in which morality depends on god, as, for example, in the divine command theory, in which a thing is good if god commands it, and bad if god commands one not to do it.  It may be called the "bully theory of morality" or the "might makes right theory of morality," as one does what god says because god will get you if you don't.  For more on that theory, see:

https://en.wikipedia.org/wiki/Divine_command_theory

But most ethical theories do not depend on there being a god at all, and most are not incompatible with there being a god (e.g., utilitarianism, Kantianism, etc.).