Is the definition of the following set controversial: $\{x\mid x\in \mathbb N \space \space x \text{ can be described in less than 30 words}\}$ Is the definition of the following set:
$$\{x\mid x\in \mathbb N \space \space x \text{ can be described in less than 30 words}\}$$
problematic? I for one don't see a problem with it, the set is finite and a formula could be written in set theory to describe every such number. Since $x\in \mathbb N$ a "set of all sets" monstrosity can't be built. There is somewhat ambiguity as to what counts as mathematical language, but I see no other problem.
Is this correct or can a contradiction somehow be derived?
 A: The problem is this: "The smallest natural number not in that set" is described in 30 words or less, and is therefore in the set.
A: This is simply not a valid definition. Because "description" is not a mathematical term, and if it is then you haven't specified all the needed information and moreover it will not be a valid definition for defining a set (in the sense that would give you a paradox).
Because ultimately, you want to prove that the set is (1) non-empty; and (2) its least element can be described in under $30$ words, so is not there. Combined these produce a contradiction.
Let's examine the cases of how we can interpret this.
Case I: Description as a natural language.
If by "describe" you mean using English words, then clearly this is not a mathematical definition. We can invent more English words, and in theory we can argue that for every natural number $n$, it is described by the word "Na" repeated $n$ times.
You could argue that English is only the words which appear in the recent version of some dictionary. Fine. $n$ is described by the number of "post-" prefixes you attach to the word "number". So $0$ is "number" and $1$ is post-number and $2$ is post-post-number, and so on. Now every natural number can be described in one word. So your set is empty, and the paradox is avoided. But clearly we have something else in mind, which is some vague and nondescript meaning of "describing a number in such and such many words".
Case II: Description as a formal language.
Recall that in a structure $M$ we say that an element can be defined if there is a formula $\varphi(x)$ such that the only way that $M$ satisfies it, is by assigning $x$ a specific value.
For example in the structure $(\Bbb N,\leq)$ the number $0$ is definable since it is the smallest number, formally $\varphi(x)$ can be expressed as $\forall y(x\leq y)$.
So now you could ask what is the least number which not definable in $\Bbb N$ with a definition shorter than, say, $100$ symbols (I've pushed the $30$ up, since often we need some more space to express things formally).
But now there are two problems:


*

*You did not specify the language and structure. Presumably, this is the language of arithmetic. But you have to be at least explicit about it. In principle the language could have a constant for every natural number, in which case the language is infinite, but we can define every number with just three symbols $x=c$.

*But even worse than that, you are quantifying over definitions. So this collection is not first-order definable, and therefore you cannot use this as a definition to show that the least number in this set is in fact definable with a formula with fewer than $100$ symbols.
So you really need to appeal to the set theoretic universe to even prove that this set is a set (or something stronger than just first-order logic over $\Bbb N$). But then the result is that this is not something that you have stated in first-order logic, so you cannot use this to create a paradox.
Fine, you say, you can work with second-order logic. Then this definition will not be second-order, it will be third-order. And so on, the cycle continues. 
In summary.
The definition of the set is ambiguous, it is either susceptible to free interpretation as a natural language statement (which is certainly not a mathematical statement), or it is not going to define a non-empty set and thus avoids the paradox entirely. Or, if you prefer to think about this as a formal statement, you need to specify the language and structure, but then you quantify over definitions which is no longer usable when you want to appeal to the definition of the set itself in finding a contradiction.
