function that cannot be expressed using finite characters There are functions that cannot be expressed using finite characters. For example while function $x^3$ can be written using finite characters there exists a sequence of cartesian pair, describing function, that cannot be written ysing finite characters. Is there any way in set theory or logic to express hat a function cannot be expressed using finite characters? Or the other way around?
I know that a function with infinite domain is
expressed with infinite cartesian pairs in set
theory. However, a function like $x^3$ can be
written as $x^3$ - written as finite characters. So
is there any way to express that a function, while
usually expressed with infinite Cartesian pairs,
can also be expressed in a concise form like
$x^3$?
 A: There are at least three different concepts you could be shooting for here.
One is whether the function has a closed form, that is, whether it can be defined in the form $f(x)=\cdots$, with some definite mathematical expression involving $x$ to the right of the equals sign. This is mostly an informal concept because we usually don't specify exactly which operations and functions we allow in a "mathematical expression". Basic arithmetic, certainly. Sines and cosines, probably. Logarithms? Bessel functions? General definite integrals (usually not)? How about arbitrary real constants -- most of these don't themselves have a finite description, but it is common to consider every polynomial to be a close form nevertheless.
If you specify exactly which operations you allow, "closed form" can be made into a formal concept. But it can still be hard to reason about whether a function given in some other way has a closed form or not.
Second, we can ask whether the function has an algorithm, that is, a (finite) description of a "mechanical" procedure that starts with $x$ and somehow, after finitely much time, results in definite knowledge of which number $f(x)$ is. In yet other words, this asks whether there is a possible computer program that computes the function.
This concept can be formalized quite well. Synonyms for "has an algoritm" include "is computable" or "is recursive". It is one of the foundations of computer science, and also of technical importance in mathematical logic. Also of course, it can be of some practical interest whether a particular function we're looking at is computable or not. Some subtleties creep in when the function in question is a real function of a real variable (rather than say, a function that takes rationals to rationals) because most real numbers cannot be specified as input to the algorithm in finite time. But there are various ways around this that still produce a useful concept.
Finally, we can look at whether the function is definable -- that is, can we describe (in finite space) some property (of any kind) at all that this particular function has but no other functions have? Such a property should be expressed as a logical formula in the language of formal logic.
This sounds like it should be a good formal concept, but there are surprisingly deep problems with it. It turns out that if we really want to allow any kind of (finitely) describable mathematical property to separate our function from other ones, then "definable" cannot itself count as a property, and there is no set of all definable functions.
It is safe enough to argue that a function is definable, by showing a concrete property that provably applies to this function and no other function. But paradox lurks around every corner as soon as we try to make any more general arguments about the concept. Is best left alone, or attacked only with a solid grounding in proof and model theory and axiomatic set theory.
