I'm reading the first chapter of the PDF "Foundations of Mathematics" written by Mohammad Safdari.
In this chapter the author is developing the logical foundation of set theory. In the page 3 he mentions one of the primitive notions to develop such foundation: the notion of variable. However, even though the author doesn't define the notion of equality between variables, we find the following passage:
Page 32: Axiom 1.8.(iv) "If $z$ is a variable different from $x,y$, then [...]
I think that variables represented by different symbols should ALWAYS be different but could represent the same set. For example, the variables $x$ and $y$ can never be equal since the symbols $x$ and $y$ are clearly different. However we can use those variables to represent the same set.
In my opinion, to implement such ideia it's necessary to get rid of the notion of free and bound variables (see Primitive Notion 1.4 and the Axiom 1.4 which is in the page 27).
My question is: is it possible to define the notion of substitution (see the Primitive Notion 1.5 and the axioms 1.5-1.9 starting on page 27) without the notion of free and bound variables?
Thank you for your attention!
To make more explicit what is in the comments here: The symbols $x$ and $y$ in the passage you quoted are not variables in the formal language (the "primitive notion" you mention that is on page 3). The variables of the language are $x_1, x_2, \ldots$. The symbols $x$ and $y$ do not represent sets within the formal system we are describing. $x$ and $y$ are "meta-variables" (i.e. "variables" in the outside language we are using to describe our formal language) used to stand for the variables in the language. For example, we may have $x = x_5$ and $y = x_8$. When the author says, "$z$ is a variable different from $x$ and $y$", that means if we have $x = x_5$ and $y = x_8$, then $z = x_i$ for some $i \ne 5, 8$. The equality $x = y$ just means that they are both $x_i$ for the same $i$, i.e., literally the same variable. The "=" here is not an "=" within the language (as in representing the same set); it just means they represent the same symbol.