Does this Set Represent a Natural Number?
JavaScript (Node.js), 53 48 44 bytes



f=a=&gt;(a=eval(a)).every(e=&gt;a[e.length]&amp;&amp;f(e))


Try it online! Test cases mostly shamelessly stolen from @Arnauld's answer. Explanation: If a set represents a natural number, then the natural number it represents must be equal to the size of the set, and (given that the elements are guaranteed distinct) the elements must be the representations of the natural numbers less than it, and these must therefore have shorter lengths. This is trivially true of the empty set of course. Edit: Saved 5 bytes thanks to @Arnauld. Saved 4 bytes thanks to @Cowsquack.