Difficulty in understanding significance of Grelling's Paradox. Background: I'm a math rookie, yet to enrol in university. I randomly started reading Mendelson's Introduction to Mathematical Logic, when I stumbled upon this paradox in the introductory section:

Grelling's Paradox: An adjective is called autological if the property denoted by the adjective holds for the adjective itself. An adjective is called heterological if the property denoted by the adjective does not apply to the adjective itself. For example, 'polysyllabic' and 'English' are autological, whereas 'monosyllabic' and 'French' are heterological. Consider the adjective 'heterological'. If 'heterological' is heterological, then it is not heterological. If 'heterological' is not heterological, then it is heterological. In either case, heterological is both heterological and not heterological.

I'd like to understand the following:

*

*What is the source of logical fallacy in this paradox? If I formulate a set $A$ of all adjectives and subsets $A_a$ and $A_h$ corresponding to autological and heterological adjectives, respectively, then it could be the case that $\text{(heterological)}\in A-(A_a\cup A_h)$, i.e., it belongs to neither of the two sets(unless $A_a\cap A_h=\emptyset$ and $A_a\cup A_h=A$).

*On a lighter note, I'd like to know about the mathematical significance of this paradox, and how it's dealt with in modern set theories.

Although I understand the answer(s) could be very abstract, please add a simpler analogy along with a necessary technical explanation, if possible.
 A: If $A, A_a,$ and $A_h$ actually "make sense" - more on this below - then we clearly have that $A_a$ and $A_h$ partition $A$: $A_h$ is defined to be $A\setminus A_a$. So your proposal doesn't work.
The fix is that $A_a$ and $A_h$ are in fact more complicated than they appear. We only have a paradox if the adjective "heterological" is in $A$. But it turns out that this doesn't happen: basically, in order to define heretologicity we need to use a truth predicate for $A$ and we don't have one of those in $A$ itself.

Here's one way to see the paradox in action.
Let $\ulcorner\cdot\urcorner$ be your favorite Godel numbering function and let $Form$ be the set of all first-order formulas in the language of arithmetic. For simplicity, let's write "$\mathbb{N}$" for the structure $(\mathbb{N};+,\times,0,1,<)$. Then the set $$X=\{\ulcorner\varphi\urcorner: \mathbb{N}\models\neg\varphi(\underline{\ulcorner\varphi\urcorner})\},$$ the version of $A_h$ for first-order formulas of arithmetic, cannot itself be definable by a first-order formula of arithmetic: if $X$ were defined by some formula $\theta$ of first-order arithmetic, that is if we had $$X=\{n: \mathbb{N}\models\theta(\underline{n})\}$$ for some formula $\theta$ of first-order arithmetic, we would get a contradiction by considering whether $\mathbb{N}\models\theta(\ulcorner\theta\urcorner)$.
More generally, we can generalize the particular setting above to any setting where we have some logic $\mathcal{L}$, some structure $\mathfrak{A}$, and some appropriate "coding" mechanism of $\mathcal{L}$-formulas into $\mathfrak{A}$. Getting the details right takes some thought, but the point is that Grelling's paradox illustrates a fundamental "stepping-up" phenomenon that we can't avoid: the Grelling set for a particular logic/structure/coding system is not definable in that structure by a formula of that logic.
(Note that $X$ can indeed be defined in broader contexts: for example, it's definable in $\mathbb{N}$ by a formula of second-order logic, and it's definable by a first-order formula in the universe of sets, of which $\mathbb{N}$ forms a very small piece.)
A: Logically speaking, Grelling's Paradox is just Russell's Paradox--only the names have been changed.
For better or worse, natural language allows us to define things that cannot logically exist. Logically, just like there can exist no set of all those sets that are not elements of themselves (Russell's Paradox), there can exist no word that describes all those words that do not describe themselves (Grelling's Paradox).
Russell's Paradox:
$\neg \exists r:  [Set(r) \land \forall a: [Set(a) \implies [a\in r
 \iff a\notin a]]]$
We can substitute:

*

*$Set(x) \to Word(x)~~~$ (unary predicate or domain of discourse)


*$x \in y \to Describes(y,x)~~~$ (binary predicate)


*$r \to h~~~$ ($h$ = "heterological")
Grelling's Paradox:
$\neg \exists h:  [Word(h) \land \forall a: [Word(a) \implies
[Describes(h,a)  \iff \neg Describes(a,a)]]]$
