When you use the word 'number', you are introducing a specific kind of symbol that is known within the context of mathematics. You aren't just talking about a label. You can't treat a label such as 'red' like you can treat a label such as '5'. We can reduce 5. I can add to 5. I can count to 5, etc. You can't count to red. You can't reduce red to blue by counting down.
***At the present, the only aspect of "mathematics" which is of interest is the fact that the field has provided an infinite set of symbols which can be used as labels. I have no concern at all with Peano's axioms. As far as I am concerned, we are still well within that category which I refer to as a vague rudder trying to steer one's thoughts.***
That's fine when you want to label things for the purpose of identifying them in some systematic manner, but (6) is stronger than this since it reference (5) which talks about Ultimate Reality. By using the term 'number' you have subtly introduced Peano's axioms (PA) as being true about Ultimate Reality. This is my objection.
So, as a solution, why don't we avoid numbers altogether. I don't quite feel comfortable with this solution, but since you are saying that we are not labeling a 'thing' but only the concept of a 'thing' (e.g., the largest even number), therefore I suppose I can accept (6) as:
(6) It seems possible to label all these 'things' in some general matter yet to be determined.
I'm sure that my (6) is acceptable to you since you have admitted that you do not need the PA function of numbers, but only the labeling function. We simply don't have to worry about what label we use, only that the 'things' in (5) are only label-able in some manner.
***You are claiming the statement itself is unreasonable without giving any reason at all except that you refuse to accept it as reasonable. I think you are trying to read something into it which is not there because you are scared to death that I might slip something by you.***
I don't want you to utilize Peano's axioms with regard to Ultimate Reality. It's as simple as that. For that reason, let's talk about labels and forget the use of numbers. If you need numbers then there's a reason why you need numbers, and that reason - as it is clear in your paper - makes use of PA. You can't use PA since you don't know if those axioms are true with regard to Ultimate Reality.
Warm regards, Harv