***it seems to me that it is incumbent upon you to at least give an example of something which cannot be referred to (certainly, if it can be referred to, the reference itself can be used as a label and converting that reference into a number is a trivial issue).***
This is the nature of the problem! If I tell you what cannot be numbered (or labelled), then I have already identified this 'thing'. However, if I have identified it, how do I know this 'thing' is something that exists versus only thought of as existing? For example, if I identified 'red' as a number '1', but later you found out that 'red' includes what you call 'orange', then you might say that I should number 'red' as 1 and 'orange' as 2. However, by me identifying 'red' as 1 (i.e., as I originally did), I am assuming that there is 'red' as something that exists. What you later found by seeing me include 'orange' as 1, is that I was mistaken - at least partially in my identity of 'red' (i.e., you say I need 1 for 'red' and 2 for 'orange'). However, all of this assumes that what we are labelling can even be labelled at all! What if, instead, this process of putting 1 for red, 2 for orange, is an infinite process whereby we never correctly identify 'red'? No matter how many times we place a number on the real 'red', later we always realize that we incorrectly included or excluded certain elements of the real 'red'.
What this suggests is that (P-6) is an assumption. We can assume (P-6) for the 'things' that we think exists, but we cannot assume (P-6) for the 'things' that exist.
***It is very important to my presentation that no limitations be imposed concerning how that numbering is to be accomplished; absolutely all possibilities must be held open! The only thing which is important is that labeling with a number is possible.***
These 2 sentences contradict each other. Considering all possibilities includes the possibility that 'things' cannot be correctly numbered.
***If you stop to think about it, you should comprehend that, even if it takes you a thousand (or even a thousand billion) pages of text to explain exactly what kind of circumstance will yield a "thing" which may exist and cannot be labeled, I could simply transform that text into a binary representation which can easily be interpreted as a number and the thing being discussed has been labeled with a number. I have utterly no interest is what this particular "thing" is, my interest is only in the fact that it can be labeled.***
You could if it is possible to do so. The argument is that it might not be possible. How do you know it is possible? Just because we lack imagination of how it is possible, is not enough to justify (P-6). Our imagination is what might be limited here.
***Let us suppose that there exists some "thing" which cannot be referred to. Explain to me exactly how the "causal tie-in" is accomplished (which you yourself have required in your definition of "exists") without any means of reference to that "thing". Is not the "causal tie-in" itself a reference?***
There might exist an Approximation Rule (AR) to the universe. The rule might look like this:
AR: All 'things' that exist exhibit approximate characterizations. These approximate characterizations are what is grasped by human minds in order to approximately conceptualize the real 'things'. For example, molecules, atoms, nuclei, quarks, etc, are all approximate characterizations of 'things', but they themselves are not 'things'. The 'things' cannot be numbered, labelled, identified, or conceived of by the human mind (or any mind, for that matter) because this is contrary to the nature of 'things'.
If such a rule exists, then (P-6) is false.
***It appears to me that your response betrays a fear of committing to the idea that "things" that exist or are thought to exist can be referred to! That is a rather extreme position to take up and, in my mind, would only be taken up as a last ditch defense to an unreasonable position.***
Dick, read your words: appears, fear, unreasonable. These words are based on the world of things that we think exists. If this wrong, then it doesn't matter how things appear, or what our fears are, or what is unreasonable. The fact of the matter doesn't change due to the way we prefer things to be.
***The whole idea of "meaning" implies understanding of some sort.***
No. Meaning in this sense only implies that some isomorphic relationship obtains. For example, if a relationship exists between a son and his mother, then it is meaningful for the son to say 'mom' to this person. If a mother doesn't exist or is non-referable, then it is not meaningful for a son to say 'mom' to a non-referencable person. We don't have to understand who mother is, we do need the relationship to exist (i.e., son to mother must be referencable, otherwise the terms are meaningless).
Warm regards, Harv