I could try to refine the counterargument in the near future, but regarding the premise of sqrt(2), it is just an example and happens to be my favorite number. I picked it because there is a famous proof by contradiction concerning its existence. The scope of my paper / presentation is the existence of irrational numbers. I am also using Cantor's diagonal proof concerning the uncountability of the real numbers as well as the Riemann sum describing a means of computing the digits of e. I want to use some proof during the paper / presentation, but the e example is perhaps my most important one at this point, even though I don't have a proof concerning it. (This project is still ongoing.)
Zero perhaps is necessary for closure of arithmetical or algebraic systems--to satisfy certain equations, much like sqrt(2). But you're right, that would perhaps be a more contentious and unwieldy example.
Really, it is a philosophical topic, but one that is about mathematics. I am trying to use proof and logic more to try to get into the mathematics at the same time.
I am curious what you mean by the bolded comment above. Are you referring to the existence of sqrt{2} or the fact that it is irrational.
As you probably know there are several ways to develop the real numbers from the rationals. They all have their pluses and minuses. The easiest one to get to sqrt{2} is to use Dedekind cuts. In this case, sqrt{2} corresponds to the cut...
A={r in Q | r<0 or r^2<2} B={r in Q| r>0, r^2>2}.
More to the point, Dedekind's method is a bear when it comes to the algebraic structure of the real line but it almost immediately provides you with the Least Upper Bound Property which gives you all of the core completeness properties in the most economical way.
Once you have completeness, the existence of sqrt{2} follows in many ways. As does the existence of e which you can simply define as the limit of the sequence,
S_n = 1^0/0!+ 1^1/1! + 1^2/2! + ... + 1^n/n!.
This sequence is clearly increasing, and it is easy to show that it is bounded above. Hence by the LUB its limit exists. You can also use this to show that e is irrational, though the proof that it is transcendental is much more difficult.
Zero has a very colorful history.
By the way, if you don't mind me asking where are you going to school? I was curious that you mentioned a "Writing in Mathematics" course. This is unusual but quite novel.