User blog:Edwin Shade/Surreals and the Nature of Number

In a book I took out of the library today, the notion of surreal numbers was mentioned. I did not understand it fully, but from what I gathered, the surreal numbers are a number system which is composed of two sets of numbers, which in turn define a specific point on a hyper-extended version of the number line. How on earth can such "numbers" like $$\omega^{\pi}$$ and $$\sqrt{\epsilon_0}$$ be considered "numbers" though, when there's no way to produce them using the usual laws of ordinal arithmetic, nor is there any way to treat them as anything buy ? Now I understand $$\{\{|\}|\{|\}\}$$ is supposedly some number greater than all negative numbers yet less than positive numbers yet which is not 0, but that seems like a blatant contradiction. These surreal numbers appear to be just labels for arbitrary collections of sets rather than actual, existing numbers, namely the ones that even if they can not be constructed within our reality, are at least imaginable in the realm of thought, such as $${\omega}3$$ or $$\pi^{\pi}$$. I certainly can see no way to conceptualize the ordering $$\omega^{\pi}$$, nor can I make sense of a "number" that is as contradictory as *, (the aforementioned  $$\{\{|\}|\{|\}\}$$).

In short, what exactly should be considered a number, (the definition must conform to our intuition concerning quantity and order yet at the same time be mathematically sound) ?