Is there such math concept as potential zero?

Question

It seems that I need to use concept of potential zero in my work and I want to know whether I could reference some other works in order to fully understand what I'm dealing with.

Specifically for my purposes I have defined concept of potential zero $x$ as a placeholder for the non-negative integer. Placeholder might be replaced by one non-negative integer. Placeholder is different from the set of non-negative integers $\mathbb{N}^0$ and used in conjunction with reference to actual non-negative integer. For example, pattern $N.x.x$ (where $N$ is a single non-negative integer) defines set of such numbers/entities as $0.x.x$, $1.x.x$, $2.x.x$, ..., $N.x.x$. Pattern $N.x.x$ can be used to generate other kinds of patterns (for example, $N.M.x$, $N.x.K$ and $N.M.K$, where $M$ and $K$ are single non-negative integers) describing other different kinds of sets. Note that potential zero $x$ is different from an empty set $ \varnothing\ $ because it is a placeholder only for single non-negative integer. And it is not a variable because such numbers as $1.x.x$, $5.x.35$ are used for referencing real existing entities where $x$ is used as an actual number, so it is not unknown, but defined.

I'm aware of such math concepts as and potential and actual infinity. That makes me think that such concepts as potential zero and actual zero have been discovered as well. I would like to know about the person who described this first and where I could read more about potential zero. After all, does such math concept exist at all?

Edit 1: I would like to extend my explanation of what I consider to be potential zero in order to exclude suggestions to consider $x$ to be variable instead. If it was a variable, I would call it a variable and would not ask stupid questions on math.stackexchange.com, right? But it is not a variable.

1. It is not unknown even though I use $x$ symbol (commonly used for denoting 'unknown' values) which , in fact, is used as a part of names used for referencing 'real' entities ($10.x.x$, $1.5.x$, $20.x.21$, etc).
2. It is a placeholder for single non-negative integer, not for the set of possible values or other 'invisible' values.
3. It is expected to be substituted with a sequence instead of one value or set of values. Placeholder is about to be replaced with 0 first, then by 1, 2 and so on. So, $x$ turns out to be the number before 0 with a 'potential' to 'start' new sequence of non-negative integers.

Take a look at the following diagram, it might help you to understand what I need this all for:

Also there is extended version of the diagram representing described set of principles.

Answer 1

You believe that the usage of a variable is a situation in which the value is unknown. This is strictly untrue; for the equation $4x^2+234789238476x-2324872831230=0$, the value of $x$ are in fact defined and have specific values -- you may not yet know them without calculation, but those values exist.

When one derives, writes, or otherwise creates a relationship with a variable, it is usually possible to determine whether well-defined values of that variable exist, if they're unique, and under what conditions and considerations. For instance, one may use the Peano Existence Theorem to decide the existence of solutions of a differential equation. We may not necessarily be able to compute that solution, either due to lack of education, lack of interest, or lack of elementary closed form solution, but we know that such a solution does in fact exist. And if it is unique, then the "variable" assumes a single, specific value at every point in time or space (or whatever your independent axis represents). Lack of knowing is not the same thing as "unknown"---or more aptly, lack of knowing is not the same thing as "unknowable."

Your pattern $N.x.x$ should instead be written as a tuple $(N,x,x) \in \Bbb X^3$, where $\Bbb X$ is a suitably defined set, and where we can define restrictions on the domain of $x$.

Your "placeholder," as you have defined it, is a uniquely-determined variable. You may not know or be able to know its value, but that is true for variables in general. The only difference between your notation $N.x.x$ and the notation $(N,y,y,),\ y \in \Bbb N$ is that in the latter we have not sufficiently restricted the uniqueness of $y$ based on extrinsic criteria that you must most certainly have to place a suitable restriction on your "placeholder" $x$.

All you're saying is "$x$ can represent a natural number, but it cannot represent all of them such that the patten $N.x.x$ cannot simultaneously be used to represent $N.4.4$ and $N.5.5$." In such a case, we should simply write $N.x.x = \{ N.y.y \mid y\ \textrm{satisfies the conditions of } x\}$. If you're truly interested in analyzing the range of values of $N.x.x$, then you could look at some abstract algebra, or some computer-sciency language theory stuff that honestly, I know little about. Nevertheless, the abstract theory of grammar uses constructions like this regularly.

In the end, what you have is exactly a variable with contextual restrictions that do not allow it to assume all values of its domain. This is not a new concept/construction. It is your prerogative to call it "absolute zero" if you'd like, as long as you are consistent with the definition. But you will probably have more success in communicating your idea by attempting first to frame it in the more general and accepted language framework of mathematics, and then make the case that it is sufficiently interesting and important to merit its own terminology.

Answer 2

If I understand your concept correctly, the closest analog would be cylinder sets for the product space $\Pi_{i=1}^3 \mathbb{N}^{+}$.

Also greatly related is the concept of conditioning on a random variable.