I'm currently taking a course on Calculus and Analytic Geometry with a quite rich insight on set theory and propositional, predicate logic. I've noticed that some values have not been numerically defined, hence the wording "undefined". Now, what exactly does undefined mean? Because I've also noticed that most mathematicians tend to use "undefined" when interpreting a value that is approaching both infinity and negative infinity. e.g: the value of the tangent ratio at 90 degrees is interpreted as "undefined", while on the graph we can see that it seems to be approaching infinity from the top and negative infinity from the bottom. Or anything (except zero) divided by zero is also "undefined" since it could be both infinity and negative infinity. Is there some technical definition that I am not aware of? I thought, intuitively perhaps, that it is some sort of special set, the undefined set. Much like the empty set is used to illustrate the fact that no value satisfies a given equation, the undefined set illustrates the fact that the equation is satisfied by exactly two values, i.e \infty and -\infty. So by definition, this undefined set could be \{\infty, -\infty\}. I don't know if either infinity or negative infinity is recognized as a number, but perhaps it could be treated as simply a matter of notation. Hence \infty simply means a perpetually increasing value, and -\infty a perpetually decreasing value. Is this correct, or is there no other ultimate meaning behind "undefined" other than it simply hasn't been given a proper definition?
