Great idea, but as terrific as the participants of that site are (I especially like the Frege reference - whom you should read), I think we would find it just as difficult to communicate on that forum due to the way it is set-up.
Regarding your post, the majority of our differences seem to circulate around the word 'possibility'. Possibility has never been defined by you, despite my request to do so. As a result, you are using more than one meaning for the term. In one instance you use possibility to mean "it might be the case, so include that item just in the situation that such a case is valid". In other instances you use possibility to mean "it is the case for some entity to do so, even if we cannot do it" (or, "it is the case that logic allows this procedure to perform this function such that the case is valid"). Also, I noticed a confusion on your part as to what is possible.
Let me show you how the above takes effect:
***We don't know "enough"? Harv, we don't "know" anything! You don't seem to understand that right here you are making an assumption that plurality of objects does not make sense. My position (that the things which make up "Ultimate Reality" can be labeled with the same symbols which label numbers) makes no such assumption. It is possible you are correct and, in that case 'Ultimate Reality' consists of something which need only be referred to with one label; however, it is also possible you are wrong and that multiple labels are reasonable. My picture takes in both possibilities whereas yours does not.***
It is possible for whom and under what circumstances? We aren't certain we know anything about Ultimate Reality. So, from whose reference are you saying it is possible? Define your term.
***On the other hand, what you want essentially assumes that it is impossible to do so; forever and ever by any entity. If we are going to include all possibilities, you need to present a proof that Ultimate Reality cannot consist of multiple elements. It should be clear to you that my position is to include all possibilities.***
Is it possible (i.e., it might be the case) that some entity can know Ultimate Reality beyond the concept itself? I don't know. I would consider that as something that might be the case, but I don't consider it a valid assumption that it is possible (i.e., some entity can do it) that Ultimate Reality can be known beyond the concept itself. Notice how the latter interferes with (7):
(7) It is possible to refer to that portion of true "Ultimate Reality" on which any individuals beliefs are based via a set of unknown numbers.
We cannot consider (7) possible in the sense that we know that this can be done by some entity. We do know that it is possible that some entity might be able to do (7), but we have no idea if it is the case or not. You shouldn't twist the meaning of the term 'possibility' from 'might be the case' to 'some entity can do this'.
***H: (6). It is possible to label 'Ultimate Reality' with exactly the same symbols commonly used to label numbers. This requires only one number. D: If you understood what I am proposing, you would understand that suggestion is clearly not valid! If you are going to insert the constraint that "this requires only one number", you better be prepared to prove one number is sufficient to cover all possibilities. I don't think you have any reason to restrict "all possibilities" to the single case where 'Ultimate Reality' consists of no sub categories of any kind. That strikes me as a rather insupportable assumption and, if you are rational, you should agree with that position.***
Maybe you don't understand what I mean by (6). I mean that instead of using the English phrase 'Ulimate Reality' to refer to whatever exists, we can use a number instead. For example, instead of calling my shoe by the word 'shoe', I could label it '6', e.g., "Let me put my '6' on and then I'll go outside". I can label 'Ultimate Reality' by the number 9, and then say "I have no idea what comprises what '9' represents". This is the most we can say is possible about 'Ultimate Reality'. We cannot say it is possible to label the components of '9' since we have no idea what '9' is other than that 'exists'. We don't even know if '9' is divided up into logical categories.
***The issue is not what "we" can talk about but rather "what can" be talked about.***
What 'can be' is used as a form of the word 'possible'. So, you must define what you mean by 'can be'. Are you referencing some mental entity that can to do such and such? If so, you must say that this is what you are referencing and then prove to me that this is within the capability of a mental entity. If you mean that logic allows it so that it can be such and such, then show me why logic allows it.
***If an analysis is to be applicable to any possible situation, that analysis cannot be constrained to the concept that 'Ultimate Reality' cannot be expressed in terms of 'portions'. Not unless you can prove that is a fact of 'Ultimate Reality'. Essentially you are assuming that your constrained concept of 'Ultimate Reality' is "the valid concept".***
You want to turn this around like I'm putting an unnecessary constraint to what you are trying to do, but what you are trying to do is making invalid assumptions. Unlike mathematics, you cannot define the concept and then see if such and such is possible based on the definition. Rather, you cannot define Ultimate Reality and you cannot assume a possibility without knowing that something is or isn't possible.
***H: Any further breakdown of 'Ultimate Reality' is an assumption based on our experience of the world we experience, and we have no way to know if this experience extends to the concept of 'Ultimate Reality'. D: At this point it is very clear that you do not understand what I am setting up. I am not performing a breakdown at all. I am setting up an attack which includes the possibility that such a breakdown might be possible; that is quite a different issue. My analysis applies equally well to the case where such a breakdown is not possible as it does to one where the breakdown is possible. It is you who is making an assumption. You are assuming that such a breakdown simply is not possible by anybody for all time. My question is, how did you come to know that?***
It is simple. I know the basis of logic, and therefore I understand the basis of proof and knowledge. Knowledge is based on proof, and proof is based on logical deduction. Now, I ask myself, does Dick know that Ultimate Reality is a logical structure? Answer: No, Dick does not know that. Conclusion: Dick has no idea if he can logically deduce something about Ultimate Reality. Since Dick's (7) is based on this assumption, we cannot accept (7).
Warm regards, Harv