The point of all questions regarding theology is a deeper understanding of theology. That's the reason I started a thread about God killing Dathan/Abiram's children.
Alright then, so you are seeking a deeper understanding of theology.
Very well, then, let's go deep.
Your question was "What would you say if someone asked you to prove that God is good?"
I would start by saying that there are four key words in the question we have to define if we are to have any hope at answering it. Otherwise we may be talking past it.
The first word is "prove".
The second word is "God".
The third word is "is".
And the fourth word is "good".
To start with the first word, "prove" means three things: "test" (as in "proving grounds"), and it means to present arguments that support a role (to present ones "proofs"), and in the strictest mathematical sense, it means to present mathematical statements in logical order, each following necessarily from the last, that ultimately arrive at an inescapable conclusion
assuming the postulates. Even the most rigorous geometric proof requires that one accept the postulates as a given. If one rejects the postulates of geometry, than nothing can be proven.
When speaking of theology in particular, however, most non-religious people have in a mind a fourth meaning of "proof". What they really mean is "Demonstrate to me, using only facts that I accept as facts, and logical arguments that I accept, each following necessarily from the last, whereby any doubt whatever can be removed from a particular statement concerning a divine being or beings. This means that necessarily you must first prove beyond a shadow of a doubt, using only evidence admissible in
my court, and answering every objection that I raise, that God exists at all. And then, having proven that God exists, you must then prove to my full satisfaction that God exhibits the trait that I demand you prove, to wit: goodness."
This is a tall order. It may or may not be impossible, depending on the willingness of the demander to accept certain representations as facts.
Which is why the definition of the word "is" is primordial, for if one cannot prove in the first instance that God
is at all, then obviously one cannot prove that such God is
good, or bad, or indifferent, or purple.
So, before we can define what good is, let alone what God is, we have to first define what
IS is. (The legal defense of a US President against an accusation of perjury once reposed on the statement "It all depends on what
is is.")
When speaking of God, it's not surprising that one of the dialogues that the Hebrew God gave, to Moses, reposed upon the verb "to be", and that in the text this God stated his name in reference to this verb: "I am", or "I will be what I will be" are some of the variant definitions of this name, as is "I am what I am" (God is Popeye?) and "I will be what I choose to be".
If one takes the word "to be" in English and tries to define it, one ends up in a quick circle: "to be" is "to exist", to exist is to be. In Hebrew, the concept of "to live" is tied into "to be"/"to exist", such that "to be", "to exist" and "to live" are all the same word. We, of course, separate "live" out as a biological function, but ancient Hebrew, where we have to look to see ancient writings supposedly revealed by this God that Westerners mostly talk about, does not make the distinctions we make.
So then we have to decide if we're going to stick with what the Hebrew means, or if we're going to insist on our meaning and distinctions.
Of course the hostile questioner is not going to permit the vagueness of "to live" to be included in "to be", and is going to stick with the modern understanding that "living" is a biological subset of "being". Perhaps the rocks and stars live in Hebrew, but they don't in English, and the person asking the question is not going to adjust his thinking about the meaning of words in order to try to see from a viewpoint that he would find cranky in the first place.
To be is to exist, and that which exists, is. "Is" means that which exists. It is not limited to what we perceive or know exists, but what
really exists independent of observation. Using this concrete "hard" concept of existence, then, the answer to the question "If a tree falls in the forest and nobody is there to hear it", is "Of course - physics dictates the creation of vibrations from the fall, which move air molecules and create what would be perceived as sound, and it is that vibration that is sound, irrespective of the fact that there is a sensing organ there to hear it or not."
Our questioner has a firm sense of reality, and reality is generally that which can be perceived, measured and detected. Of course, he is not willing to bind himself to that position, because he is willing to entertain the concept of multidimensional string theory as a real feature of the universe, and an explanation therefor, even though there is no means at present to detect any such dimensions. The fact that they can be mathematically diagrammed and the logic of the mathematics upholds their possibility, and the logical way that they are thought to interact seems to give a rational, non-imaginary way by which the existence of the universe can be explained. Therefore, pure logic, derived mathematically but otherwise not empirically testable
is, after all, acceptable to our questioner. He will not throw out the possible reality of string theory just because it is (at present) completely speculative, because the speculation follows logically and mathematically from that which is perceived.
I mention this because it demonstrates the two ways that one could prove the existence of God to the questioner, and after all, God's mere
existence must be satisfactorily proven first before there can be any meaningful discussion of his attributes (such as goodness). Otherwise we would be spending time discussing the color of the gill slits of nargals and the piebald patterning of bandywomps. The "two ways" are:
(1) Prove God through logic and empirical observation, scientifically, or
(2) God proves himself by making a direct theophany to the questioner and revealing himself as God. (Of course, in such instance, the revelation might be overwhelming, but might not actually be
God, but some other, much more powerful being masquerading as God in order to obtain followership from the questioner. The questioner could legitimately wonder why such a powerful being would even
want the followership of such a lesser developed creature as himself.)
(2) is up to God to do if he chooses. Sometimes he does. Until that moment, we are left with (1) as our only option.
It would be swell, because it would be easier, if we could just confine ourselves to the concept of the Abrahamic God, to discuss whether He exists or not, but that assumption - that if there is a God at all, the Abrahamic God would "obviously" be Him - assumes far too much that our interlocutor will not accept - or at any rate
should not accept if he is really seeking a rigorous proof of God's existence (or lack thereof).
To prove God
is a thing at all, one needs to begin with the prospect of what God would have to be like so that, if you found something with those properties, you would not only be
able to say "Yes, I have found God", but would be compelled to say it.
"Goodness", a subjective quality that would depend on the position of the observer (what is "good" to the lion is often suboptimal for the antelope), would not be a fundamental property necessary to establish the existence of God - it is possible to conceive of (and human history is full of the tales of) a god or gods that were quite "evil" (another subjective term).
So, stripped of value judgments about good and evil, what would something have to be for you to consider that thing "God", and be willing to call it that.
I would say that "God", as you mean the term, would have to have four essential properties:
(1) Omnpotence
(2) Omnipresence
(3) Timelessness ("Eternality" ought to be a word, but it isn't, so "timelessness" will have to suffice)
(4) Omniscience.
If I were to show you a thing that had those four properties, you would agree that I had shown you God as a matter of definition, yes?
That's a question. If I show you something that is omnipotent, omnipresent, eternal and omniscient, would you agree that I have shown you God?
If so, we could then look at that thing to determine whether or not it met whatever our criteria was of good.
So, please let me know the answer to that question: If I demonstrate something that is omnipotent, omnipresent, eternal and omniscient, have I presented you God as a sort of geometric proof: here is a thing that has all of the necessary attributes of God, therefore, this is God, by definition?
Put differently, IF God is those four things, THEN that which is demonstrated to have those four things is God.
Is that an acceptable strategy for a definition of God, and a set of criteria necessary to complete a proof for you?
I've tried to phrase this as a yes or no question, because, after all, you asked for the proof of God's goodness. To prove that, I first have to prove God exists in the first place, and that can be reduced to a set of definitions, I believe.
"God is that which is omnipotent, omnipresent, eternal and omniscient." Agreed?