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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: enter image description here

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

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So how is this different from a variable? –  Michael Greinecker Oct 1 '12 at 18:34
Why is this set theory related? –  Asaf Karagila Oct 1 '12 at 18:47
How is it applicable in set theory?? –  Asaf Karagila Oct 1 '12 at 18:51
@AsafKaragila: people come here (and ride buses) who don't know about set theory. Asking set-theory questions and getting explanations as to whether it is really a set theory question is part of the learning process. Be gentle :-) –  robjohn Oct 1 '12 at 20:12
It's good to be nice to newcomers. –  Nick Alger Oct 1 '12 at 20:12

2 Answers 2

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.

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A placeholder whose value we do not care about, so long as it is the member of a particular set, is not exactly a variable. It just says "insert any element here". It does seem as if the pattern $N.x.x$ does simultaneously correspond to $N.4.4$ and $N.5.5$, if we happen not to care what $x$ is, only that it is nonnegative. I.e. although $N.5.5$ and $N.4.4$ are different spellings, what if we consider them equivalent under the given formalism. –  Kaz Oct 1 '12 at 20:43
A placeholder that we don't care about is a variable that we don't care about with constraints that we deem important to specify for some reason anyways. –  Arkamis Oct 1 '12 at 21:48
@EdGorcenski: It seems that I didn't add some important explanations before. Please take a look at my edit 1. It might help you to understand why it is not a variable. –  altern Oct 2 '12 at 17:56
@altern: Your explanation and diagram have finally convinced me that what you mean is precisely a variable. –  Charles Oct 2 '12 at 18:17
@Charles: ok then. is there any plausible definition of 'variable'? let's take it and see whether my case satisfies all the conditions for the definition of variable –  altern Oct 2 '12 at 18:28

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.

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does the concept of cylinder set assume that there might be no number from $N^+$ in a tuple? For example, $(1, \\__\ , 3)$? –  altern Oct 2 '12 at 18:56
The cylinder set would contain every possibility where the first and third numbers are fixed to be 1 and 3 and the second number is free. Ie, cylinder set = $\{(1,0,3),(1,1,3),(1,2,3),(1,3,3),(1,4,3),...\}$. –  Nick Alger Oct 2 '12 at 20:41
that's great. but as you might see in my diagram, $1.x.3$ is a real entity, it is not a pattern. $1.x.x$ is also a real entity, $1.0.x$ too. are those entities might exist in cylinder set? as far as I understand after reading about cylinder sets at the wikipedia, those entities are not able to be a part of cylinder sets. –  altern Oct 3 '12 at 20:05
The cylinder set is a real entity too, just one that has an infinite number of elements. Sets with infinite elements used to bother me too, but it's no big deal when you realize there are a lot of things like this. For example a line in the plane has an uncountable infinite number of points, but it's no problem to think of the whole line as a single "thing". –  Nick Alger Oct 4 '12 at 0:14
Line in the plane is a great example. In my case it would be rather subway line with stations at it. Station on the subway line is more than just one dot, right? It's complex structure that needs to be referenced somehow with a relation to the line where this station is located. So, is there some kind of a hint of how I could use cylinder sets with the possibility of establishing internal relations (especially inheritance) between elements? –  altern Oct 6 '12 at 7:37

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