Set theoretic dilemma

How about you start by checking whether [math]S[/math] and [math]N[/math] are both well-defined according to ZFC?

[math]S[/math] can't exist by the Axiom of Regularity

Let [math]S[/math] be a finite collection of symbols representing numerical sets such that [math]\mathbb{N}\in S[/math].
Does [math]S[/math] contain [math]\mathbb{N}[/math] as set of natural numbers?
What is the distinction between set and its symbol?

Define "symbol"

>>9108914
The set of well formed formulas in first order logic exists and is countably infinite.