I think that our dispute first lies in whether an “unbounded infinite” is valid as a concept — Bob Ross
Yes, I think this is really the issue. Lets see if we can put this in terms of math.
You already mentioned that the infinite X is bounded if we use actual numbers. The only way to really capture an unbounded infinite is not to use numbers at all, but the relation itself, where is is not limited by any number or dimension. I have no problem with this. What I will attempt to demonstrate is that there is only one unbounded infinite, and the X "without numbers" is it.
Your original bounded infinite could be represented as
X = Y with limit 5. Here we have X is fine as long as it doesn't equal 5. But if X is bounded as soon as numbers are used, then as soon as a number is used in the equation, it is also bounded. So X = Y with a limit of 5 is a bounded infinite by the limit.
But lets go further. X = Y is really a limit of "Whatever Y is, X is. We can say we won't assign actual numbers to X, but there is a number, a bound within the formula itself that acts exactly on a limit. That limit is that Y will always be X, and Y cannot be anything but X.
The above may be confusing, so let me add another detail. 2X = Y. Now we explicitly have a number in which Y will always be double X. Even if we don't use actual numbers in X or Y, this double explicitness is a limit, or a bound. Referencing the previous X = Y, lets change it to 1X = 1Y, which is equivalent.
Ok, if X, unnumbered is an unbounded infinity, while all the rest are bounded, can we have multiple unbounded infinities. Can I just say Y without using actual numbers and have that different from saying X without actual numbers? Besides the symbol itself, they are both identical. X is unbounded, and Y is unbounded. They are not bounded in relation to one another. If they are not bounded in relation to one another, they are not different from one another. Neither has any limits, so they are both the same.
Lets now translate that to words, context, and meaning. As soon as you put a limit in words, context, or meaning, you are no longer talking about an unbounded infinite. You are talking about a bounded infinite.
Now, this still doesn't convey the whole idea fully. We now have to change it to words, meaning, and context. To represent X, we need unstated words, unstated meaning, and unstated context. The moment we state anything, any "number", we are now within a bounded infinite limited by the expression of that word, meaning, and/or context.
Can we have a sine qua nons for an unbounded infinite. Yes, but there is only one. That would be "not X". If not X were true, then X would not follow. Anything more specific may be a sqn for a bounded infinite, but it cannot be a sqn for an unbounded infinite.
The same applies to the principle of regulation. Within X words, Y meaning, and Z contexts we are still bound by words, meaning, and context. Let simplify this further. W = { X, Y, and Z } all without "numbers" or explicit individual representations. W is still bound by X, Y, and Z. The only way for W to be unbounded is just "W".
So I do not think it can be shown the Principle of Regulation is a sqn. There are specific words, such as principle, regulation, of, that are understood within a particular bounded infinite meaning, and in particular bounded infinite contexts. Can thinking things within this limit form and use conclude the logic of the principle of regulation is necessary. Absolutely. But can this be concluded from "W" alone? No, I don't believe it can.
To clarify on
p1. A unbounded infinite is a concept — Bob Ross
No, I'm not stating this. I'm stating an unbounded infinite is not a concept. The moment we create a concept within it, we are now within a bounded infinite. As such, there is only one unbounded infinite. Anytime any explicit infinite is proposed, it is by nature bounded.
That being said, this does not mean you should give up on the principle of regulation as a basis for a theory. I think it is a fine starting point, and I know I, and probably many in this discussion would love to see where your mind takes this. I would hate it to be stopped by something as trivial as a debate over infinity.
You shouldn't need sqn's to prove the principle of regulation to logically thinking minds. And even if you do, perhaps its something you could come back and show later? Is the concept of a SQN within an unbounded infinite absolutely needed to continue your line of thought from the PoR proposal? If you just started the sentence with, "If we have the ability to derive, the principle of regulation logically arrives," would that hamper what you want to do? I feel you have so much more to say, and possibly introduce greater thoughts that I would hate to see stopped over focusing on what may be a technical, and perhaps unnecessary detail to show us what you have planned.
In my experience in philosophy, it is easy to get stuck on approaches that seem necessary to us when first formulating the idea, but as we evolve the idea, were perhaps not as necessary or important as we thought to those who are reading our papers. Consider your readers so far. Very few have argued against the PoR, but almost everyone has a problem with your views of infinity. Now we may all be wrong, and you may be correct. But is it necessary at this time to focus on the infinite as such, or can this be shelved or stated another way that allows your readers to focus on the first premise they can readily accept?
I realized, to keep it brief, that even if I concluded that there was no foundation to derivation, or no derivation, it is all by means of the principle of regulation (or whatever one wants to call it). — Bob Ross
This right here is where I think you should go into detail. Prove not only to yourself, but that none of us can conclude anything differently. If you do this, I don't think anyone is going to need the infinite. How in the absence of derivation must we all necessarily have the principle of regulation? If I am not a being able to derivate, could I conclude I could not derivate?'
I look forward to your work Bob.