Let me clarify: $\mathsf {ZF}$ and other fundational theories often use logic to define the basic concepts like $\implies, \exists$ etc... even if they are a circular definition.
My question is: it possible to present for example $\mathsf {ZF}$ without using logic but only the language of sets?
My attempt is to only allow to write on an infinitely long paper only these three characters on a paper: "$\{$", "," and "$\}$" and write down all the valid composition of the chosen characters but i feels like the axioms are lost in a sea of all possible statement.
Is there a better way of doing this? like maybe using a "rule" to encapsulate better the idea, maybe defining a Turing machine?